Algebraic higher symmetry and categorical symmetry -- a holographic and entanglement view of symmetry

In this paper, we introduce the notion of an algebraic higher symmetry, which is beyond higher group. We show that an algebraic higher symmetry in a bosonic system in $n$-dimensional space is characterized and classified by a local fusion $n$-category. We also show that a bosonic system with an algebraic higher symmetry can be viewed as a boundary of a bosonic topological order in one-higher dimension. This implies that the system actually has a bigger symmetry, called the categorical symmetry, which is defined by the categorical description of the one-higher dimensional bulk. This provides a holographic and entanglement view of symmetries. A categorical symmetry is characterized by a non-degenerate braided fusion $n$-category (describing excitations in an anomaly-free bosonic topological order in one-higher dimension). For a system with a categorical symmetry, its gapped state must spontaneously break part (not all) of the symmetry, and the state with the full symmetry must be gapless. The holographic point of view leads to (1) the gauging of the algebraic higher symmetry; (2) the classification of anomalies for an algebraic higher symmetry; (3) the duality relations between systems with different (potentially anomalous) algebraic higher symmetries, or between systems with different sets of low energy excitations; (4) the classification of gapped liquid phases for bosonic systems with a categorical symmetry, as gapped boundaries of a topological order in one higher dimension that characterizes the categorical symmetry. This classification includes symmetry protected trivial (SPT) orders and symmetry enriched topological orders (SET) with an algebraic higher symmetry.

The notion of a symmetry plays a very important role in physics. In quantum system, symmetry is a set of linear constraints on the allowed Hamiltonians. By a "symmetry", we usually mean a global symmetry, where we have an unitary operator U acting on the whole space (i.e. a symmetry transformation) which give rise to the following linear constraint on the Hamiltonians U H = HU . If one digs deeper, however, one finds that there are in fact several different kinds of global symmetries. In quantum field theories, we have anomaly-free global symmetries (gaugeable global symmetries) and anomalous global symmetries (not-gaugeable global symmetries or 't Hooft anomalies 1 ). In lattice systems, we have on-site symmetries (where the symmetry transformation has a composition on lattice sites U = ⊗ i U i ) and non-on-site symmetries. 2,3 These different kinds of global symmetries are closely related. Consider a low energy effective field theory of a lattice model. The on-site symmetries in the lattice model becomes the anomaly-free global symmetries in the effective field theory, since the lattice on-sitesymmetry is always gaugeable. The non-on-site symmetries in the lattice model become the anomalous global symmetries in the effective field theory. 3 For the symmetries related to spacial transformation, such as the lattice translation symmetry and point group symmetry, sometimes they become anomalous symmetry in the effective field theory, and sometimes they are anomaly-free. In this paper, we consider only internal symmetries instead of symmetries related to spacial transformations.
There are also gauge symmetries in field theories and lattice theories. But they are not symmetries in quantum systems, and should not be called symmetry at all.
Recently, in Ref. 4, the notion of a global symmetry is generalized to a k-form symmetry, which acts on all closed subspaces of codimension k and becomes the identity operator if the closed subspaces are contractible. It was stressed that many results and intuitions for global symmetries (the 0-form symmetries) can be extended to higher-form symmetries. In fact, a closely related higher symmetry had been studied earlier (but under various different names), where exactly solvable lattice Hamiltonians commuting with all closed string and/or membrane operators were constructed to realize topological orders [5][6][7][8][9][10][11] . We call the a lattice symmetry generated by a k-codimensional operator as a k-symmetry. Similar to a k-form symmetry, a k-symmetry acts on closed subspaces of codimension k, but it does not become the identity operator when the closed subspaces are contractible. A higher symmetry is a symmetry in a lattice model. A higher symmetry reduces to a higher form symmetry in gapped ground state subspaces (i.e. in low energy effective topological quantum field theory).
The emergence of higher symmetries was also studied before (again with a different name) 12 , where it was found that, unlike usual global symmetry (i.e. 0symmetry), the emergent higher symmetries cannot be destroyed by any local perturbations. Such a topological robustness was used to show that the emergent gapless U (1) gauge bosons are robust against any local perturbations -a topological version of Goldstone theorem 12 . See Ref. [13][14][15][16][17] for recent discussions of lattice higher symmetries, their emergence, anomalies and a classification of associated higher symmetry protected phases on lattice.
In this work, we study a new kind of symmetries that is beyond higher groups. We refer to the new symmetries as algebraic higher symmetries, and refer to higher groups as group-like higher symmetries. Algebraic higher symmetries include group-like higher symmetries as special cases.
Group-like higher symmetries and algebraic higher symmetries can both be generated by p-dimensional operators B i (W p ), where W p is a p-dimensional closed submanifold and B i (W p ) only acts on the degrees of freedom near W p . For a group-like higher symmetry, the k-dimensional operators satisfy a group-like algebra while for an algebraic higher symmetry, they may satisfy a more general multiplication algebra 18 In this case, the symmetry generator B i (W p ) may have neither inverse nor unitary. Such kind of algebraic symmetries was studied in 1+1D conformal field theory via non-invertible defect lines (where invertible defect lines are known to connect to symmetry) [19][20][21][22][23] . In Sec. III, we discuss a lattice model described by a Hamiltonian H, where the above algebraic higher symmetry does show up, i.e.
Then we discuss unbroken algebraic higher symmetry from a point of view of local fusion higher category, as well as the associated categorical symmetry 18 . In Sec. VI B, we obtain a classification of gapped liquid phases for systems with a categorical symmetry. It includes the classification of symmetry protected trivial (SPT) phases and that of symmetry enriched topological (SET) phases for algebraic higher symmetry. In Sec. VII, we describe the emergence of categorical symmetries from topological orders, when the excitations have a large separation of energy. Throughout this work, we use nd to denote the spacial dimension and (n+1)D to denote the spacetime dimension, and the following convention of notations: • nD topological orders: A n , B n , C n (mathsf font); • fusion n-categories: A n , B n , C n (mathcal font); • braided fusion n-categories: A n , B n , C n (euscript font).
Throughout this paper, superscripts always mean the spacetime dimension. Also in this paper, we only consider finite algebraic higher symmetry. We mostly consider bosonic systems, except in Secs. VI C 3, VI C 4, and VI C 6. So when we say, for example, SPT orders, we mean SPT orders in bosonic system. In Secs. VI C 3, VI C 4, and VI C 6, our results apply to both bosonic and fermionic systems, and even anyonic systems, via a more general notion of algebraic higher symmetry.

II. SUMMARY OF MAIN RESULTS
In this section, we summarize the main results and introduce concepts and notations along the way.

A. Category of topological orders
A category of nD topological orders (see Appendix E) is a collection of topological orders (called the objects of the category), plus the domain walls between different topological orders, and domain walls in the same topological order (where they are called codimension-1 excitations). Those domain walls are called the 1-morphisms of the category. The domain walls on the domain walls (the 2-morphisms), etc are also included. The top morphisms are n-morphisms, which are local operators (satisfying certain symmetry constraints) acting on a spacetime point (x, t). The top morphisms can also be viewed as instantons in spacetime.
We denote the n-category of nD potentially anomalous topological orders by TO n . 1-morphisms in TO n given by potentially anomalous 1-codimensional gapped domain walls, so on and so forth. See Ref. 24 for more details. As mentioned above, an object A n ∈ TO n is an nD topological order (see Appendix A). We use mathsf font A n , B n , C n to denote nD topological orders, i.e. A n , B n , C n ∈ TO n . The superscript n represents the spacetime dimension and may be dropped if it is manifest from the context. We denote the trivial nD topological order by 1 n (see Appendix C), and denote the stacking of two nD topological orders A n and B n by A n ⊗ B n . The data 1 n and ⊗ endow the n-category TO n with a structure of a symmetric monoidal n-category.
A topological order is called anomaly-free if it can be realized by lattice models in the same dimension. 3,25 An (n−1)d domain wall between two anomaly-free nd topological orders is called anomaly-free if it can be realized by a (n−1)d lattice wall between two nd lattice-model realization of two adjacent nd topological orders. The collection of all nD anomaly-free topological orders plus their anormaly-free domain walls etc, form the symmetric monoidal subcategory of anomaly-free nD topological orders, denoted by TO n af (see Appendix D and Ref. 24 for more details).
We set the convention of the superscript: ΩC n+1 = Ω(C n+1 ). Excitations of codimension-1 can be fused but not braided. If we exclude the 1-codimensional excitations, we obtain a braided fusion (n−1)-category, which is precisely the looping ΩC n of C n (see Appendix F). The fusion n-category C n does not carry the full information about the nd topological order, since C n only describes the excitations within the nd topological order. There are different topological orders (that differ by stacking invertible topological orders [25][26][27] ) which have identical excitations. To fully describe an nd topological order C n+1 , we need not only the information about the excitations C n , but also the additional information on invertible topological orders. We can also say that nd potentially anomalous topological orders (without any symmetry) are classified, up to invertible topological orders, by fusion n-categories 24,25,28 (see Proposition 35 and 39). It was pointed out in Ref. 25 that a potentially anomalous nD topological order C n+1 uniquely determines an anomaly-free topological order M n+2 in one-higher dimension where C n+1 can be viewed as a boundary of M n+2 . This boundary-bulk relation is the holographic principle of topological order : "anomaly" = "topological order in one-higher dimension" 3 (which is quite different from viewing anomaly as a non-invariance of path integral measure). We denote this relation by (see Proposition 34). Since M n+2 is anomaly-free, its bulk is trivial, i.e.
In other words, Bulk is a "categorified" differential. In fact, we have a stronger version of the holographic principle: the excitations in the topological order C n+1 , described by the fusion n-category ΩC n+1 , can already uniquely determine the bulk anomaly-free topological order M n+2 . We denote the map from ΩC n+1 to M n+2 as bulk(ΩC n+1 ) = M n+2 .

C. Algebraic higher symmetry
For a 0-symmetry given by a group G in spatial ndimension, point-like excitations that carry the usual symmetry charges are defined by the representations of G. We denote the category of these representations by 1Rep(G). These excitations can be fused and braided, and can be condensed to form higher dimensional excitations, called condensation descendants. All these excitations form a (symmetric) fusion n-category, denoted by nRepG.
The 0-symmetry that acts on the whole space can be generalized so that the generalized symmetries can act on any loops, any closed membranes, etc. We call the new symmetries algebraic higher symmetries, which are beyond higher groups. In Sec. III and IV, we discuss some examples of algebraic higher symmetries. But a formal definition in terms of symmetry transformations is not easy to formulate.
An algebraic higher symmetry is anomaly-free if there exists a symmetric gapped Hamiltonian whose ground state is a non-degenerate product state. Or in other words, the symmetric gapped ground state can be nondegenerate for any closed space manifolds. The excitations on top of such a symmetric ground state are called charge objects, that carry "representations" of the algebraic higher symmetry. However, now the excitations (the charge objects) may be point-like, string-like, membrane-like, etc. In particular, for an algebraic ksymmetry that acts on closed subspace of codimension-k, its charge objects has dimension-k.
We propose that an anomaly-free algebraic higher symmetry in nd boson systems is completely characterized by the excitations on top of its symmetric product state (which is used as a definition of algebraic higher symmetry in this paper). Those excitations form a fusion n-category R (called the representation category of the symmetry) that is equipped with a top-faithful monoidal functor R β → nVec, where nVec is the trivial fusion ncategory describing point-like, string-like, etc excitations in a product state without any symmetry (see Appendix F). Such a fusion n-category R is said to be local.
The anomaly-free bosonic algebraic higher symmetries are classified by local fusion n-categories R, i.e. by the data R β → nVec (see Fig. 1a and Sec. V B).
For simplicity, in this paper, we usually drop β and use the representation category R to describe an algebraic higher symmetry. For example, a finite 0-symmetry G in n-dimensional space has a representation category nRepG and can also be referred as a nRepG symmetry.
In this paper, we mainly discuss anomaly-free algebraic higher symmetry. For simplicity, by an algebraic higher symmetry we mean an anomaly-free algebraic higher symmetry unless indicated otherwise.  1. (a) An anomaly-free algebraic higher symmetry in nd bosonic systems is fully characterized by its charge objects, which form a local fusion n-category R. (b) Under the holographic point of view, an algebraic higher symmetry is associated to a categorical symmetry characterized by a braided fusion category given by M = Z1(R). Physically, M describes the excitations in a one-dimension-higher topological order, which has a boundary with excitations described by R. All possible low energy states in a system with an algebraic higher symmetry R have a one-to-one correspondence with all possible boundaries of topological orders with excitations M = Z1(R).
We further generalize the notion of algebraic higher symmetry, by introducing a notion of V-local fusion ncategory (see Def. 4), which has a top-faithful surjective monoidal functor R β → V, where V is a fusion n-category. When V = nVec, R describes algebraic higher symmetry in nd bosonic systems. When V = nsVec, where nsVec is the fusion n-category of super n-vector spaces, R describes algebraic higher symmetry in nd fermionic systems.
The anomaly-free fermionic algebraic higher symmetries are classified by the data R β → nsVec, where R is a fusion n-category.
More general choices of V can describe systems formed by anyons or other higher dimensional topological excitations. So the notion of a generalized algebraic higher symmetry allows us to study the symmetry of bosonic and fermionic systems at equal footing. It is interesting to see that the boson, fermion, and anyon statistics can be encoded in a generalization of algebraic higher symmetry.

D. Dual symmetry
An algebraic higher symmetry is a part of a more general notion: categorical symmetry. Before explaining categorical symmetry, let us explain a simpler notion of dual symmetry. It was pointed out in Ref. 18 that an nd system with 0-symmetry G also has a dual algebraic (n−1)symmetry denoted byG (n−1) .
We may use the holographic view to understand the appearance of the dual symmetry. We note that the symmetric sub-Hilbert space of a G-symmetric system in n-dimensional space can be viewed as a boundary of a one-higher-dimensional G-gauge theory 18 denoted by GT n+2 G . The fusion (the conservation) of the bulk pointlike gauge charges in the G-gauge theory gives rise to the 0-symmetry G. The bulk GT n+2 G also has (n−1)d gauge flux. The fusion (the conservation) of the bulk gauge flux in the G-gauge theory gives rise to the algebraic (n−1)-symmetryG (n−1) (see Sec. III). We stress that both the 0-symmetry G and the dual algebraic (n−1)-symmetryG (n−1) are present at all the boundaries if we view the boundaries as boundary Hamiltonians or a boundary conditions 18 (for details, see next subsection). However, for a gapped boundary, viewed as a quantum ground state, one of the 0-symmetry and algebraic (n−1)symmetry, or some of their combinations must be spontaneously broken 18,32 .
If we condense all gauge flux, we obtain a boundary with the 0-symmetry G and the spontaneously broken algebraic (n−1)-symmetryG (n−1) . The boundary excitations are described by nRep(G). This boundary correspond to the usual G-symmetric product state whose excitations are also described by nRep(G).
If we condensed all gauge charges, we obtain a boundary with the dual algebraic (n−1)-symmetryG (n−1) and the spontaneously broken 0-symmetry G. The boundary excitations are described by a local fusion n-category nVec G . This is the usual spontaneous G-symmetry breaking state. The non-trivial fusion (the conservation) of the symmetry breaking domain walls is also described by nVec G , which gives rise to the dual algebraic (n−1)-symmetryG (n−1) . Thus the dual symmetryG (n−1) can also be represented by its representations category, which is just the category nVec G of G-graded n-vector spaces. For such a boundary, the dual algebraic (n−1)-symmetrỹ G (n−1) is not spontaneously broken.
If the boundary Hamiltonians have both the 0symmetry G and the dual algebraic (n−1)-symmetrỹ G (n−1) , we should see a boundary phase where both the 0-symmetry G and the dual algebraic (n−1)-symmetrỹ G (n−1) are not spontaneously broken. Indeed, such a boundary phase does exist, and it must be gapless. This is because to get a gapped boundary, we must condense enough bulk excitations at the boundary, which break one of the 0-symmetry and algebraic (n−1)-symmetry, or some of their combinations. If we do not condense any bulk excitations, the boundary can only be gapless. 33,34 .
We see that it is better to view a system with Gsymmetry as a boundary of the G-gauge theory in onehigher-dimension. This holographic point of view allows us to see the accompanying dual symmetry (i.e. the algebraic (n−1)-symmetryG (n−1) ) clearly. Using a categorical language, the point-like excitations carrying group representations (the charge objects) in an nd G-symmetric product state generate a local fusion n-category nRepG. The same local fusion n-category nRepG also describes the excitations on a boundary of G-gauge theory GT n+2 G , i.e. GT n+2 G = bulk(nRepG) (see Appendix G). This links the 0-symmetry G to the G-gauge theory GT n+2 G in one higher dimension. The boundary with excitations nRepG can be obtained from GT n+2 G by condensing the gauge flux. GT n+2 G has another boundary whose excitations are described by another fusion n-category nVec G . This boundary is obtained by condensing gauge charges. In this case, the gauge-flux excitations are not condensed, and their non-trivial fusion gives rise to the dual algebraic (n−1)-symmetryG (n−1) . In fact, nVec G is the representation category that describes the charge objects of the dual symmetryG (n−1) . In summary, we have We see that both 0-symmetry G and its dual algebraic (n−1)-symmetryG (n−1) share the same G-gauge theory GT n+2 G in one higher dimension. Thus, we can view GT n+2 G as a combined symmetry, denoted by G ∨G (n−1) . The combined symmetry is referred as categorical symmetry.
We would like to mention that a structure similar to categorical symmetry was found previously in AdS/CFT correspondence, [35][36][37] where a global symmetry G at the high-energy boundary is related to a gauge theory of group G in the low-energy bulk. In this paper, we stress that the categorical symmetry encoded by the bulk is actually a bigger symmetry than the usual symmetry at the boundary. We developed a categorical theory for this holographic point of view, which allows us to gauge the algebraic higher symmetry, classify the anomalies for a given algebraic higher symmetry, identify which algebraic higher symmetries are "equivalent" under duality maps, identify duality relations for low energy effective theories, and classify SET/SPT orders with a given algebraic higher symmetry.

E. Categorical symmetry -a holographic of view of symmetry
The above is just the simplest example of categorical symmetry. We can generalize the above discussion, and show that, when restricted to the symmetric sub-Hilbert space, an nd system with an algebraic higher symmetry R can be viewed as a boundary of an anomaly-free topological order M = bulk(R) (see eqn. (G6)). This allows us to see that our system actually has a bigger categorical symmetry, characterized by a braided fusion n-category M = Ω 2 M describing excitations of codimensionin 2 or higher in M.
The categorical symmetry associated to an nd algebraic higher symmetry R is given by a braided fusion n-category M = Z 1 (R) (see Fig. 1b and Appendix G).
We note that the dimension-0, dimension-1, etc excitations described by M in the bulk topological order M, can be viewed as the dimension-0, dimension-1, etc excitations on the boundary, if they do not condense. The fusion rule of those bulk excitations corresponds to the conservation law, which leads to the categorical symmetry of the boundary (where the boundary is viewed as a system). Although we introduce categorical symmetry via a topological order in one higher dimension, in fact, as pointed out in Ref. 18, a categorical symmetry can be defined by the patch symmetry transformations without going to one higher dimension. So the categorical symmetry is really a property of our nd system.
To be more precise, we know that a symmetry is just a set of linear constraints that select a class of allowed Hamiltonians. This class of Hamiltonians has some common low energy properties determined by the symmetry. Similarly, a categorical symmetry also correspond to a class of Hamiltonians, with some common low energy properties determined by the categorical symmetry. But here, we select the class of Hamiltonians differently. We choose a bulk topological order in one higher dimension with excitations described by a braided fusion n-category M, and choose a bulk lattice Hamiltonian to realize the bulk topological order such that the bulk energy gap approaches to ∞. Then we consider all the boundary Hamiltonians relative to the bulk Hamiltonian (which are referred as all the lattice boundary conditions). This class of boundary Hamiltonians (i.e. the lattice boundary conditions) is the class of the Hamiltonians determined by the categorical symmetry M. We say that those boundary Hamiltonians (i.e. lattice boundary conditions) all have the categorical symmetry M, by definition.
In the last subsection, we have argued through examples that the class of nd lattice Hamiltonians with an algebraic symmetry R can be simulated by the boundary Hamiltonians of a bulk topological order with excitations M = Z 1 (R). The precise meaning of "being simulated" is that, for any lattice Hamiltonian with the algebraic symmetry R, we can find a boundary Hamiltonian of the bulk topological order M = Z 1 (R) such that these two Hamiltonians have identical low energy properties. In this sense, we say that the class of Hamiltonians with an algebraic higher symmetry R also have the categorical symmetry M = Z 1 (R). We say that M = Z 1 (R) is the categorical symmetry associated to the algebraic higher symmetry R.
Form the above discussion, we also see that a categorical symmetry M can exist by itself, and determine the class of boundary Hamiltonians with their common low energy properties. An categorical symmetry M do not have be associated with an algebraic higher symmetry (i.e. M do not have to be the center of a local fusion n-category). Thus we propose that categorical symmetries for nd bosonic systems are classified by a braided fusion n-category M such that Z 1 (ΣM) = (n+1)Vec (i.e. M describes the excitations in an anomaly-free topological order, see Sec. V E).
Here ΣM is the delooping of M explained in Appendix F.
Categorical symmetry has some special properties: 1. For a system with a categorical symmetry M, its gapped ground state must spontaneously break the categorical symmetry partially.
2. It is impossible to spontaneously break the categorical symmetry completely in a gapped state, and possibly, nor in a gapless state.
3. For a system with a non-trivial categorical symmetry, the symmetric ground state must be gapless.
To see an example of categorical symmetry, Ref. 18 shows that a Hamiltonian on n-dimensional lattice with 0-symmetry Z 2 also has a (n−1)-symmetry Z (n−1) 2 . So the system actually has a larger Z 2 ∨ Z (n−1) 2 categorical symmetry. Such a Z 2 ∨ Z (n−1) 2 categorical symmetry is described by a braided fusion n-category Ω 2 GT n+2 Z2 that describes the excitations of codimension-2 and higher, in the Z 2 topological order (or Z 2 gauge theory) in one higher dimension.
The above system can have a gapped phase where Z (n−1) 2 is spontaneously broken and Z 2 is not broken, which is the usual Z 2 symmetric phase. 18 Using the categorical language, we may say that this phase has an (unbroken) algebraic higher symmetry characterized by the local fusion n-category R = nRepZ 2 (which is nothing but the usual Z 2 0-symmetry).
The system can also have a gapped phase where Z 2 is spontaneously broken and Z (n−1) 2 is not broken, which is the usual Z 2 symmetry broken phase. 18 This phase has an (un-broken) algebraic higher symmetry characterized by the local fusion n-category R = nVec Z2 , which describes the conservation of symmetry-breaking domain walls.
The quantum critical point of the Z 2 symmetry breaking transition has the full categorical symmetry Z 2 ∨ Z (n−1) 2 . In particular, in 1-dimensional space (n = 1), the Z 2 ∨ Z (0) 2 categorical symmetry leads to the emergent Z 2 × Z 2 symmetry for right-movers and left-movers 18 .

F. Emergence of algebraic higher symmetry and categorical symmetry
When an nd lattice system has an on-site G 0symmetry, it also have a G ∨G (n−1) categorical symmetry. But how to have a more general higher symmetry or algebraic higher symmetry R, as well as their associated categorical symmetry M = Z 1 (R)? Although in condensed matter physics, it is hard to have a practical lattice model that have exact algebraic higher symmetry R and the associated categorical symmetry M = Z 1 (R) without fine tuning, it is not hard to have a situation that an algebraic higher symmetry R and its categorical symmetry can emerge at low energies.
We will first discuss the emergence of a categorical symmetry M. Once we have an emergent categorical symmetry M (which may or may not be spontaneously broken), then we can determine the emergent algebraic higher symmetry R (which may or may not be spontaneously broken) by solving the equation Z 1 (R) = M.
Let us consider a topological order (with or without symmetry) on an nd lattice, whose excitations are described by a fusion n-category C, which contains charge objects if we have symmetry. Assuming the topological excitations have a large separation of energy scale, such that all the low energy excitations (point-like, string-like, etc) are described a subcategory C low of C. All other excitations not in C low have large energy gaps. Thus at low energies, we only see the topological excitations in C low and we can pretend the system to have a (potentially anomalous) topological order without symmetry, whose excitations are described by C low .
We see that once we know the low energy excitations C low (which may contain possible charge objects from symmetry), the higher energy lattice regularization becomes irrelevant. Thus we can directly consider a field theory with low energy excitations C low . We ask what is the low energy emergent categorical symmetry?
The answer is very simple: the low energy effective categorical symmetry for a field theory with low energy excitations C low is given by a braided fusion n-category M low = Z 1 (C low ). In addition to assigning a categorical symmetry M low to the field theory, we can also assign a bulk-topological-order (bTO) M low = bulk(C low ) (see eqn. (G6)) to the field theory. The bTO M low carries more information than the emergent categorical symmetry, since M low can fully determine M low = Ω 2 M low .
Here by field theory, we mean a theory whose UV regularization is not specified. When we say a field theory have a property, we mean that there exist a UV regularization of the field theory, such that the regularized theory has the property. It is possible that the same field theory with a different regularization may not have the property. The emergent bTO M low and the emergent categorical symmetry M low are very useful. Given a set of low energy excitations C low , we like to ask, when the low energy excitations condense, what kind new phases are possible? What kind of critical points are possible at the phase transitions? Do we have any principle to address those issues? The answer is yes, and the answer is the emergent categorical symmetry and the emergent bTO. This because all the low energy properties of the system are controlled by the emergent bTO M low (or less precisely, by the low energy effective categorical symmetry M low ). The results obtained in this paper on algebraic higher symmetry and categorical symmetry can be applied to such a system with its emergent categorical symmetry M low = Z 1 (C low ). This is one of the most important application of categorical symmetry, and its associated algebraic higher symmetry. In the next section, we con- 2. (a) Two algebraic higher symmetries R and R are dual-equivalent if they have the same categorical symmetry Z1(R) Z1(R ). (a) Two systems with low energy excitations C and C are dual-equivalent if they have the same bTO bulk(C) bulk(C ).
sider two applications along this line. From M low , we also get the low energy emergent algebraic higher symmetries, R low 's, by solving Z 1 (R low ) = M low .

G. Categorical symmetry and duality
A symmetry is useful since it can constrain the properties of a system, such as possible phases and phase transitions, the critical properties at the phase transitions, etc. From the above discussion, we see that the constraint from a symmetry actually comes from the corresponding categorical symmetry. This is because the possible physical properties of a system with an algebraic higher symmetry R are the same as the possible physical properties of a boundary of the topological order in one higher dimension that has excitations described by Z 1 (R) (the categorical symmetry). In particular (see Fig. 1 and 2a), if two symmetries R and R have the same categorical symmetry Z 1 (R) Z 1 (R ), then the two symmetries provide the same constraint on the physical properties, at least within the symmetric sub-Hilbert space. In this case, we say that the two symmetries are dual-equivalent.
As a result, all possible phases in a system with R symmetry have a one-to-one correspondence with all possible phases in a system with R symmetry. In fact, we have a stronger result, all possible states on a system with R symmetry have a one-to-one correspondence with all possible states on a system with R symmetry. Those states include gapped states and gapless states etc. In Ref. 18, some lattice exact duality mappings were discussed for some very simple examples. This duality relation can be an important application of categorical symmetry.
For example, an nd system with G 0-symmetry can be mapped to an nd system with the dualG (n−1) (n−1)symmetry, and vice versa. The G 0-symmetry and thẽ G (n−1) (n−1)-symmetry are dual-equivalent symmetries. Using the categorical notation, we say the nRepG symmetry and the nVec G symmetry are dual-equivalent symmetries.
The above duality result can be generalized even further (see Fig. 2b): Gauging the R-symmetry: stacking two local fusion n-category R over their common bulk Z1(R) gives rise to an nd anomaly-free topological order GT n+1 R , which is the R-gauge theory. Its excitations are described by a fusion ncategory ΩGT n+1 R rev . We like to remark that R's on the two boundaries differ by a parity transformation as indicated by the arrows and superscript rev .
Consider two nd field theories with low energy excitations described by two fusion n-categories C and C respectively. The two systems may have different symmetries described by different charge objects, forming two different subcategories in C and C . The two field theories are dual to each other (i.e. are dual-equivalent), if they have the same bTO bulk(C) = bulk(C ) (see eqn. (G6)), provided that all other excitations remain to have high energies.
Note that the categorical symmetry describes excitations in the bTO. Thus, the bTO contains more information than the categorical symmetry. When two systems have the same bTO, both systems can be simulated by the boundaries of that bTO, and hence the two systems are dual-equivalent. This means that the possible states of the system C (including condensed states, gapless states, etc) have an one-to-one correspondence with the possible states of the system C (see Sec. VII). Those states are just the possible boundary states of the same bTO.
H. Gauging the algebraic higher symmetry and the corresponding R-gauge theory Given an nd product state with an on-site 0-symmetry G (i.e. an anomaly-free 0-symmetry), we can gauge the symmetry to obtain a state with topological order but without symmetry. The resulting topological order is nothing but the G-gauge theory GT n+1 G . The excitations in GT n+1 G are described by a fusion n-category ΩGT n+1 G . In fact ΩGT n+1 G = Z 1 (nRep(G)). Similarly, given an nd product state with an anomalyfree higher symmetry, we can gauge the higher symmetry to obtain a state with topological order but no symmetry. The resulting topological order is described by a higher gauge theory. Now given an nd product state with an anomaly-free algebraic higher symmetry R, can we gauge the higher symmetry to obtain a state with topological order but no symmetry? If we can, then the corresponding topological order is a gauge theory for the algebraic higher Condensing the condensable algebra AR (resp. A R ) produce the R (resp. R) boundary. The functor FR : M → R, induced by moving the bulk excitations to the boundary, maps AR to the trivial excitation on the R boundary. Similar for F R . symmetry R. We denote such a gauge theory by GT n+1 R , the excitations in which form a fusion n-category ΩGT R .
In this paper, we propose a way to gauge algebraic higher symmetry, which gives us a construction of Rgauge theory (by constructing the corresponding topological order). Our approach is based on the holographic view of the R-symmetry, which is very different from the usual gauging based on spacetime dependent symmetry transformations.
Under the holographic point of view, an algebraic higher symmetry R gives rise to a 1-higher-dimensional topological order M such that ΩM = Z 1 (R) (see Fig. 1 Then the topological order obtained by gauging the Rsymmetry in a symmetric product state can be obtained by simply stacking two R boundaries through their common bulk Z 1 (R) (see Fig. 3). This is an algebraic way to gauge an symmetry, which work for 0-symmetries, higher symmetries, and algebraic higher symmetries (see Sec. VI C 5).
The excitations in an nd R-gauge theory GT n+1 R are described by an multi-fusion n-category where Z 0 is the E 0 -center.
Using a similar holographic approach, we can also define the dual symmetry R for an arbitrary algebraic higher symmetry R as follows: R and R are dual to each other, if they have the same bulk Z 1 (R) = Z 1 ( R) = M and if the stacking of R and R through their bulk gives rise to a trivial topological order (i.e. a product state, see Fig. 4, Def. 6, and Prop. 15).

R ⊗
We would like to mention that a gapped boundary of M, such as R, is induced by condensing some excita-tions in M at the boundary. The collection of those condensing excitations form a so called condensable algebra A R . The condensable algebra A R uniquely determines the gapped boundary. A different condensable algebra A R unique determines a different gapped boundary R. Roughly speaking, moving bulk excitation in M to the R boundary induces a map from M to R, the mathematical description of which is a functor F R : M → R. Under such a map, the condensable algebra A R is map to the trivial excitations in R (i.e. condensed, see Fig. 4).

I. Anomalous algebraic higher symmetry
Can an algebraic higher symmetry have anomaly? How to describe its anomaly? First, an anomalous symmetry is characterized by two things: symmetry and anomaly. So an anomalous algebraic higher symmetry is characterized by a pair (R, α), where R is for symmetry and α for anomaly.
For a 0-symmetry in n-dimensional space, an anomalous symmetry is characterized by a pair (G, ω n+2 ) where ω n+2 is an (n+2)-cocycle. A more physical way to understand the anomalous symmetry (G, ω n+2 ) is to view it as the boundary symmetry of a 1-dimension-higher SPT state, which is also characterized by the pair (G, ω n+2 ). We can gauge the G-symmetry in the SPT state to get a "twisted" G-gauge theory (the Dijkgraaf-Witten theory 38 ), denoted by GT n+2 G,ωn+2 . The codimension-2 and higher excitations in such a "twisted" G-gauge theory is described by a braided fusion n-category Ω 2 GT n+2 G,ωn+2 , which is also the categorical symmetry for the anomalous G-symmetry (G, ω n+2 ). We see that we can understand the anomalous symmetry via its categorical symmetry, which is a "twisted" G-gauge theory in one higher dimension. 3,18 In fact, under the holographic point of view, a "twisted" G gauge theory in one higher dimension defines an anomalous 0-symmetry. The boundaries of the "twisted" G gauge theory give rise to all possible phases (including symmetry breaking phases), as well as all other properties, of systems with the anomalous G 0-symmetry.
Similarly, an nd bosonic system with an anomalous higher symmetry described by a higher group B(G, π 2 , · · · ) (using the notation in Ref. 39) has a larger categorical symmetry characterized by a "twisted" higher gauge theory in one-higher dimension. The different anomalies for the higher group B(G, π 2 , · · · ) are (partially) characterized by cocycles in H n+2 [B(G, π 2 , · · · ); R/Z]. The "twisted" higher gauge theory in one higher dimension defines the anomaly of the anomalous higher symmetry.
The categorical symmetry of a lattice boundary Hamiltonian with an anomaly-free algebraic higher symmetry R is defined by M = Z 1 (R). We like to ask whether M = Z 1 (R) describes the excitations in the R-gauge theory in one higher dimension. The answer is no, simply because R describes a symmetry in nd, not in one higher 5. The topological order described by a twisted ΣR gauge theory has excitations described by ΣR ⊗ ΣR rev . The twist is done by an automorphism α of Z1(ΣR), that keeps the condensable algebraic A ΣR of the dual symmetry ΣR unchanged α(A ΣR ) A ΣR .
dimension (n+1)d. So the gauge theory in one higher dimension cannot be a R-gauge theory since R lives in a different dimension. When we discuss G-gauge theory in all the dimensions, we have used the fact that the same G-symmetry can be defined in all the dimensions. For an algebraic symmetry R in nd, what the corresponding symmetry in one higher dimension (n+1)d? This turns out to be a higher nontrivial question. It turns out that an algebraic symmetry, in general, cannot be promoted to one higher dimensions. Only a special class of algebraic symmetries, described by symmetric local fusion n-categories, can be promoted to one higher dimension. This is because a symmetric fusion n-category R in nd can be viewed as a braided fusion n-category, describing 2-codimensional and higher excitations in one higher dimension (i.e. in (n+1)d). We then can do a delooping to obtain a fusion n-category ΣR. If R is a symmetry local n-category, ΣR is again a symmetric local (n+1)-category. So, ΣR describes the R-symmetry in one higher dimension. Since ΣR is also symmetric and local, we can promote further to obtain the corresponding R symmetry in all higher dimensions Σ 2 R, Σ 3 R, etc,. So in this subsection, we assume R to be symmetric local fusion n-category. Now, we can say that the categorical symmetry for an anomaly-free algebraic symmetry R is given by the excitations in the ΣR-gauge theory in one higher dimension: which follows 28 from the following identity 40 : Similarly, an anomalous algebraic higher symmetry (R, α) is defined via its categorical symmetry which corresponds to a twisted ΣR-gauge theory in one higher dimension. The twist is produced by an automorphism α of Z 1 (ΣR) (the dash-line in Fig. 5). Such a twisted ΣR gauge theory, denoted by GT n+2 ΣR,α , is characterized by its excitations described by (see Fig. 5) where we have abused the notion by denoting the invertible domain wall between Z 1 (ΣR) and Z 1 (ΣR) associated to α still by α ( see FIG. 5). The automorphism α is not arbitrary. It must satisfy α(A ΣR ) A ΣR , where A ΣR is the condensable algebra for the dual symmetry of ΣR (see Sec. VI C 5). This result allows us to classify types of anomalies that an algebraic higher symmetry can have. To summarize, Anomalous algebraic higher symmetries R are classified by the automorphisms α of Z 1 (ΣR) such that α(A ΣR ) A ΣR . Its categorical symmetry is given by Ω ΣR ⊗ As an application of the above result, we consider 1d bosonic system with an anomalous Z 3 2 symmetry. The possible anomalies are classified by H 3 (Z 3 2 , U (1)), which correspond to 2d Z 3 2 -SPT orders. The categorical symmetry of the 1d anomalous Z 3 2 symmetry is given by the 2d topological order obtained by gauging the corresponding 2d Z 3 2 -SPT states. It was found that a particular anomalous Z 3 2 symmetry, described by a so called type-III cocycle in H 3 (Z 3 2 , U (1)), has a categorical symmetry described by the 2d D 4 gauge theory GT 3 D4 . Certainly, the 1d anomaly-free D 4 symmetry also has a categorical symmetry described by the 2d D 4 gauge theory GT 3 D4 . Therefore, this particular 1d anomalous Z 3 2 symmetry is dual-equivalent to 1d D 4 symmetry. In general, two anomalous algebraic higher symmetries, (R, α) and (R , α ) are dual-equivalent if they have the same categorical symmetry: One may ask: can a categorical symmetry M be viewed as an anomalous algebraic higher symmetry? If the categorical symmetry has a form M = Ω ΣR ⊗ ΣR rev , then the categorical symmetry can indeed be viewed as an anomalous R-symmetry. But if the categorical symmetry does not have the above form, then it cannot be viewed as an anomalous algebraic higher symmetry. Categorical symmetry play a similar role as anomalous symmetry, but it is more general. In fact, after introducing categorical symmetry, we do not need to use anomalous symmetry any more. The effect of anomalous symmetry can all be covered by categorical symmetry. metry. But let us first consider an nd system with a categorical symmetry M (assuming n ≥ 1). Using boundary-bulk relation, we find that all these SET/SPT orders (which partially break the categorical symmetry spontaneously) are classified by potential anomalous nd topological orders C (i.e. nd boundary topological orders), whose bulk carries excitations described by M (see Fig. 6a). We note that the bulk is always an anomalyfree topological order, denoted by M, in one higher dimension. However, M does not uniquely determine M. There can be several M's (differ by invertible topological orders [25][26][27] ) whose excitations are described by the same braided fusion n-category M. C can be the boundaries of all those (n+1)d topological orders M. Using the Z 1 functor, we can express the above result as the following.
For an nd lattice system (i.e. an anomaly-free system) with categorical symmetry M, all its gapped liquid phases are classified by potentially anomalous nd topological orders C that satisfy where C = ΩC is the fusion n-category that describes the excitations in C (see Appendix E).
Since the fusion n-categories C (describing the excitations) partially describes an nd topological order (up to invertible topological orders [25][26][27] and SPT orders [25][26][27] ), we get a partial classification if we only use C: the gapped liquid phases (up to invertible topological orders and SPT orders) in nd lattice systems with categorical symmetry M are classified by fusion n-categories C that satisfy Z 1 (C) M (see Sec. VI B).
This partial classification cannot distinguish gapped liquid phases that differ by invertible topological orders and SPT orders.
To get a more complete classification that includes SPT orders, we need additional data to characterize the gapped liquid phases. In fact, such additional data is hidden in the above discussion: the relation (16) between two higher categories can never be simply "equal"; one needs to use some data to describe how two categories are equivalent, just like one need to use a matrix to describe how two Hilbert spaces are isomorphic. Let's denote the equivalence by γ : M γ Z 1 (C). We need to use the pair (C, γ) to describe the gapped liquid phases (see Fig. 6b). The different choices of γ can differ by automorphisms α of M. So γ is a torsor over the group of automorphisms of M. We would like to remark that the automorphisms α include the relabeling of the generators of the categorical symmetry M, which is regarded as unphysical in physics. We are interested in α's that do not relabel of the generators of the categorical symmetry M.
For details, see Sec. VI B.

K. Classification of SET orders and SPT orders with an algebraic higher symmetry
For systems with a categorical symmetry M, in the above, we classify anomaly-free gapped liquid phases C which partially break the categorical symmetry down to some algebraic higher symmetries described by R. Clearly, such a classification includes the classification of anomaly-free gapped liquid phases with a given algebraic higher symmetry R. But what is R? First, R is a subcategory of C = ΩC, or more precisely, there is a top-fully faithful functor ι : R → C (see Prop. 5). Second, R and C have the same set of bulk excitations: Z 1 (C) Z 1 (R) = M (i.e. the same categorical symmetry, see Sec. VI C). This understanding gives us a classification of anomaly-free gapped liquid phases with a given algebraic higher symmetry R. First, let us describe a simple partial classification: Given an algebraic higher symmetry described by a local fusion n-category R, anomaly-free symmetric gapped liquid phases (up to invertible topological orders and SPT orders) are classified by fusion n-categories C that (1) admit a top-fully faithful functor ι : R → C, and (2) satisfy Z 1 (C) Z 1 (R) (see Sec. VI A).
To get a more complete classification that includes SPT orders for the algebraic higher symmetry R, we need to include the equivalence γ : Z 1 (R) γ Z 1 (C) and use the data (R ι → C, γ) to classify anomaly-free symmetric gapped liquid phases. However, not every equivalence γ should be included. We know that the categorical symmetry described by Z 1 (R) or by Z 1 (C) includes the algebraic higher symmetry R. The equivalence γ should not change R that is contained in Z 1 (R) and in Z 1 (C) 40 . For details and the main classification results, see Sec. VI C.
Here, we just quote a classification of R-SPT orders, obtained by setting C = R nd SPT orders with an anomaly-free algebraic higher symmetry R are classified by the automorphisms α of Z 1 (R), that satisfy α(A R ) A R , where A R is the condensable algebra in Z 1 (R) that determines a boundary R corresponding to the dual of the symmetry R (see Fig. 4).

III. AN EXAMPLE OF ALGEBRAIC HIGHER
SYMMETRIES: G-GAUGE THEORY For a quantum system with usual symmetry, the Hamiltonian commutes with a set of operators which form a group under the operator product. In this section, we construct an example, in which the Hamiltonian commutes with a set of operators that do not form a group under the operator product. The constructed model is an exactly solvable 3d local bosonic model 5 whose ground state has a topological order described by a 3d gauge theory of a finite group G. The operators that commute with the Hamiltonian are the Wilson line operators. When G is non-Abelian, the Wilson line operators, under the operator product, form an algebra, which is not a group.
Our lattice bosonic model is defined on a 3d spatial lattice whose sites are labeled by i. Physical degrees of freedom live on the links which are labeled by ij. On an oriented link ij, the degrees of freedom are labeled by g ij ∈ G. The labels g ij 's on links with opposite orientations satisfy The many-body Hilbert space of our lattice bosonic model has the following local basis |{g ij } , g ij ∈ G, ij ∈ links of cubic lattice. (18) The Hamiltonian of the exactly solvable model is expressed in terms of string operators and point operators.

A. The string operators
The string operators B q (W 1 ) are defined on a closed loop W 1 formed by the links of the cubic lattice and are labeled by q, the irreducible representation of the gauge group G: where R q (g ij ) is the matrix of the irreducible representation. A q-string operator is given by So B q (W 1 ) is diagonal in the basis |{g ij } : B q (W 1 ) = Tr ij∈W 1 R q (g ij ) . We note that (We use ⊗ C to denote the usual tensor product of matrices or vector spaces over the complex numbers C, while ⊗ to denote the fusion of excitations.) Using we see that The ends of the strings are point-like topological excitations and the above N qs t are the fusion coefficients of those topological excitations. The quantum dimensions of those topological excitations, i.e. d q = dim(R q ), satisfy the following identity: We see that these string operators form a fusion algebra which is not a group when G is non-Abelian. Let We have Thus, B is a projection operator. In fact, it is a projection operator into the subspace with ij∈loop g ij = 1.

B. The point operators
A point operator is given by its action on the basis: Clearly they satisfy So for a non-Abelian group G, in general But we have Let us introduce where χ a is a conjugacy class labeled by a. We find that regardless if i = j or not. We note that, on a given site i, The above expression allows us to see that M ab c are nonnegative integers. Using Let (M a ) cb = M ab c , and we can rewrite the second equation in the above as For example, the permutation group of three elements Let C be a particular common eigenvector of M a whose components are all non-negative. (Such common eigenvector exists since the matrix elements of M a are all nonnegative.) The eigenvalue of such an eigenvector is λ a for M a . We choose the scaling factor of C to satisfy a λ a c a = 1.
In this case we can define Q i = a c a C a (i) that satisfy Thus Q i is a projection operator. In fact, Q i is given by where |G| is the number of elements in the group G. We can check explicitly that The red dashlines are membranes and the cross marks the boundary of the membranes. The blue thick line is the path i0 → i.

C. A commuting-projector Hamiltonian
We note that Q h (i)'s generate the local gauge transformations. Since the closed-string operators are gauge invariant, we have (for closed-string operators) Therefore, we can construct the following commuting projector Hamiltonian 5,41 where U, J > 0, and ijkl labels the loops around the squares of the cubic lattice. The ground state of the above exactly solvable Hamiltonian has a nontrivial topological order. The low energy effective theory is the G-gauge theory. 5,41

D. The point-like and string-like excitations
What are the excitations for the above Hamiltonian? There are local point-like excitations created by local operators. There are also topological point-like excitations that cannot be created by local operators. Two topological point-like excitations are said to be equivalent if they differ by local point-like excitations. The equivalent topological point-like excitations are said to have the same type.
We note that the closed string operators B q (W 1 closed ) eqn. (20) commute with the Hamiltonian (42). Thus the string operators act within the ground state subspace. We see that the ends of the open string operators create point-like excitations, which are labeled by representations R q . The types of topological point-like excitations one-to-one correspond to the irreducible representations of G. In other words, topological point-like excitations are described by representations of G in a G-gauge theory.
Similarly, there are also topological string-like excitations. They are created at the boundary of the open membrane operators. To define the membrane operators, we point out that a membrane W 2 is formed by the faces of the dual lattice, which is also a cubic lattice. The faces of the dual lattice correspond to the links in the original lattice and are also labeled by ij. Let us first assume G is Abelian. In this case, the membrane operators are defined as where the operator T ij (h) acts only on link ij and is defined as We see that C h ( W 2 ) simply shifts g ij on the membrane W 2 by h.
For non-Abelian G, the membrane operators are given by where χ a is the a th conjugacy class of G. In the ij∈ W 2 , i's are on one side of the membrane and j's are on the other side of the membrane (see Fig. 7). Last h ij is a function of h and g ij . For non-Abelian group G, h ij is complicated. But when all g ij = 1, h ij has a simple form h ij = h. For general g ij , we need to choose a base point i 0 on one side of the membrane, and a path i 0 → i on the membrane that connect the base point i 0 to any other point i on the membrane (see Fig. 7). Then we can define h ij as where (g i0i · · · g i i ) is the product of the link variables along the path i 0 → i. We note that when the closed membrane enclose only one site i (see Fig. 7), the operator C a ( W 2 ) reduces to C a (i) discussed before: Thus C a (i) can be viewed as a small membrane operator, rather than a point operator. Let us consider a loop i 0 → i → j → j 0 → i 0 . The G-flux through such a loop in the ground state |Ψ 0 is trivial: (g i0i · · · g i i )g ij (g jj · · · g j j0 )g j0i0 = 1. This is because the ground state |Ψ 0 satisfies After we apply the membrane operator (46), the G-flux through the same loop becomes which is still trivial. This allows us to conclude that for states |Ψ satisfying B|Ψ = |Ψ and for closed membrane W 2 closed , we have We can also show that For example Also where Thus in general, we have for any i, even for open membranes. The results (51) and (56) imply that closed membrane operators C a ( W 2 closed ) act within the ground state subspace of the Hamiltonian (42). Therefore, the boundary of the open membrane operators (46) create string-like excitations, which are labeled by conjugacy classes χ a .

E. Exact algebraic higher symmetry
Since the Hamiltonian (42) commutes with the closed string operators B q (W 1 closed ): we say that the Hamiltonian has an algebraic 2-symmetry generated by B q (W 1 closed ) for any closed strings. Since the composition of the symmetry transformations satisfies the fusion rule (23), which is not a group multiplication rule for non-Abelian G. Thus the B q (closed string)'s generate an exact algebraic 2-symmetry which is not a higher 2-symmetry. However, when G is Abelian, B q (W 1 closed )'s generate a higher 2-symmetry.
There is another way to describe the algebraic 2symmetry using the open string operators. 18 We note that the Hamiltonian is a sum of local operators H = i H i , where H i acts only on the degrees of freedom near site-i. We find that H i commutes with open string operators as long as the ends of the strings is a distance away from the site-i:

F. Emergent algebraic higher symmetry
We also note that the Hamiltonian (42) commutes with Thus the Hamiltonian has a 0-symmetry, i.e. a global symmetry described by symmetry group G. In fact, the Hamiltonian has a much bigger symmetry. It has a local symmetry described by group G Nv , where N v is the number of lattice sites: On the other hand, the membrane operators C a ( W 2 closed )'s do not commute with the Hamiltonian (42). Thus the Hamiltonian does not have algebraic 1symmetries. However, C a ( W 2 closed ) acts within the degenerate ground subspace. More generally, C a ( W 2 closed ) and H commute in the subspace where B ijkl = 1 (i.e. in the finite energy subspace of H when J → +∞). Therefore the Hamiltonian has an emergent low energy algebraic 1-symmetry generated by C a ( W 2 closed )'s when J → +∞. Such an emergent algebraic 1-symmetry is a (group-like) 1-symmetry only when G is Abelian.

IV. DESCRIPTION OF ALGEBRAIC HIGHER SYMMETRY IN SYMMETRIC PRODUCT STATES
A. Spontaneous broken and unbroken algebraic higher symmetry In Sec. III, we constructed a 3d lattice model that has an exact algebraic 2-symmetry generated by string operators B q (W 1 ). However, the ground state of the model eqn. (42) spontaneously breaks the algebraic 2symmetry, which gives us a topological order described by the G-gauge theory.
Here, we consider a different model by including an extra term −V δ(g ij ) and taking J → +∞ limit. Here The model also has the algebraic 2-symmetry [H, B q (W 1 closed )] = 0. If we choose the limit U V , the ground state is given by |{g ij = 1} . This ground state does not spontaneously break the algebraic 2-symmetry.
For the usual global symmetry, the spontaneous symmetry breaking is defined via non-zero order parameters. Here we would like to define the spontaneous symmetry breaking of algebraic higher symmetry in a different way: Definition 1. An algebraic higher symmetry is spontaneously broken if there exists a close space, such that the symmetry transformations are not proportional the identity operator in the nearly degenerate ground state subspace on that space.
For the Hamiltonian eqn. (61), the ground state is not degenerate on any closed spaces. Thus the algebraic 2-symmetry is not spontaneously broken. In contrast, for model (42), the ground states are degenerate is the character of the representation R q . We see that the ground state of the model (42) spontaneously breaks the algebraic 2-symmetry B q (W 1 ). In fact, the algebraic 2-symmetry is completely broken, which gives rise to the topological order described by the G-gauge theory.

B. Anomaly-free algebraic higher symmetry
In this section, we would like to discuss algebraic higher symmetry in the simplest state -symmetry unbroken state without topological order. However, some algebraic higher symmetries may not allow such a state. This leads to an important attribute of algebraic higher symmetry.
Definition 2. An algebraic higher symmetry in a lattice system is anomaly-free if the system allows a phase, which has a symmetric product state as its unique gapped ground state on each closed space.
For 0-symmetry on lattice, we can use on-siteness to define anomaly-free 0-symmetry 3 . Using this definition, we believe that all anomalous (non-on-site) 0symmetry can be realized on a boundary of a system in one higher dimension with anomaly-free (on-site) 0symmetry. 3 For finite symmetries, we believe that there is an one-to-one correspondence between anomalous 0symmetries and the SPT order in one higher dimension 3 .
(While for infinite symmetry described by a continuous compact group, we do not have the one-to-one correspondence between anomalous 0-symmetries and the SPT order in one higher dimension 3 .) As a result, the finite anomalous 0-symmetries are classified by the SPT orders in one higher dimension. Since we believe that the boundary uniquely determine the bulk 25,29 , the above result also implies that an anomalous 0-symmetry does not allow a gapped symmetric product state as the ground state 42,43 . Otherwise, the SPT order in one higher dimension must be trivial as implied by such a symmetric ground state on the boundary.
For algebraic higher symmetry, it is hard to define onsiteness. So we turn things around, and use the existence of trivial symmetric gapped ground state to define algebraic higher symmetry (where trivial means product state). In this case, algebraic higher symmetry can appear at a boundary of the trivial SPT phase for algebraic higher symmetry. The boundary of non-trivial SPT phases for an algebraic higher symmetry realize an anomalous algebraic higher symmetry. In this paper, we only consider anomaly-free algebraic higher symmetries.

C. The charge objects and charge creation operators for the exact algebraic 2-symmetry
The exact algebraic 2-symmetry in the lattice model eqn. (61) is generated by B q (W 1 closed ). The algebraic 2symmetry is anomaly-free since the model eqn. (61) allows symmetric gapped product state |{g ij = 1} as its unique ground state.
The charge objects of such a 2-symmetry live on 2-dimensional surfaces [just like the charges of a 0symmetry (the usual global symmetry) live on 0dimensional points]. To construct the 2-dimensional operators that create the charge objects of the algebraic 2-symmetry, let us review the charge creation operators for the 0-symmetry in a proper general setting.
A pair of charge and anti-charge of a 0-symmetry is created by an operator C(S 0 ) on S 0 (i.e. on two points i and j), for example where the local operator ψ a (i) form a unitary representation R ab for the 0-symmetry group G: We note that, when the two points in S 0 belong to the same connected component of the space, C(S 0 ) commutes with the algebraic 0-symmetry transformations and creates an neutral excitation. On part of S 0 , the creation operator becomes ψ a (i) which creates a non-neutral excitation.
Similarly, a neutral charge object of a k-symmetry is created by operators on closed contractible k-dimensional manifolds, such as S k . Such an operator on contractible S k commutes with the algebraic k-symmetry transformations and creates an neutral excitation. A charge object of k-symmetry is created by operators In nd, when the algebraic k-symmetry generator B q (W n−k closed ) on n − k-dimensional sub-manifold intersects with the submanifold M k at one point, we can detect the k-symmetry charge. The algebra between symmetry generators B q (W n−k closed ) and charge creation operators C(M k ) only depend on the linking between W n−k closed and ∂M k , and do not depend on the deformations of W n−k closed and ∂M k that do not change their linking. Those are key conditions for the charge creation operators C(M k ) for an algebraic higher symmetry.
For our algebraic 2-symmetry in 3d, the charge creation operator acts on 2-dimensional surfaces with or without boundary. In fact, such a charge creation operator is nothing but the membrane operator C a ( W 2 ) discussed in Sec. 8. The charge object created by C a ( W 2 ) can be detected by the 2-symmetry generator B q (W 1 closed ), when W 2 has a boundary, or when W 2 is closed and non-contractible.
In fact, on the |{g ij = 1} ground state, the creation operator can have a simpler form where ij∈ W 2 is over all the links ij of the original lattice that form the faces in W 2 of the dual lattice. Such an operator just changes g ij = 1 to g ij = h on links ij of the original lattice that form the faces in W 2 of the dual lattice. g ij = h on W 2 corresponds to a charged excitation, called a 2-charge object labeled by h, of our algebraic 2-symmetry generated by B q (W 1 closed ). If W 2 is a disk in 3d space, then the 2-charge object created by C h ( D 2 ) can be detected by the algebraic 2symmetry operator B q (W 1 closed ) if W 1 closed is linked with ∂ W 2 -the boundary of the 2-charge object (see Fig. 8). If fact, B q (W 1 closed ) = TrR q (h) in this case when acting on the 2-charge object. In comparison, for the ground state |{g ij = 1} , the 2-symmetry generator is equal to the dimension d q of the q-representation: B q (W 1 closed ) = TrR q (1) = d q . We see that the algebraic 2-symmetry cannot distinguish two 2-charge objects labeled by h and h if h and h belong to the same conjugacy class. So the distinct algebraic 2-symmetry charges are labeled by the conjugacy classes χ a of G.
We stress that the membrane operator C a ( W 2 ) that creates the 2-dimensional charge object of the algebraic 2-symmetry is an operator that acts only on the membrane W 2 . This is a very important general feature. Proposition 1. On top of a ground state that does not break the symmetry, the k-dimensional charge object of an algebraic k-symmetry is created by an operator that act only on the k-dimensional subspace that supports the charge object.
We note that in J → ∞ limit, only 2-charge objects corresponding to closed surfaces has low energy. 2-charge objects corresponding to surfaces with boundary cost energy of order J or bigger. We may consider the low energy subspace of the model in J → ∞ limit. In fact, we consider an even smaller space, the invariant sub-Hilbert space of all the 2-symmetry transformations generated by B q (W 1 closed ) operators. The collection of those created 2charge objects within the symmetric sub-Hilbert space, plus their fusion (and braiding) properties, form a higher category. The 2-charge objects are labeled by h ∈ G and created by The charged membrane-like excitations, labeled by h ∈ G, form a fusion 3-category R = 3Vec G (see Def. 16), which is also a local fusion 3-category (see Def. 3). We also refer R = 3Vec G as the representation category of the algebraic 2-symmetry. Physically, R is the fusion 3-category that describes the low energy excitations in model eqn. (61). But what is a fusion higher category and what is a local fusion higher category? Roughly speaking, a fusion higher category describes the point-like, string-like, etc excitations above a gapped liquid ground state. If an excitation can be annihilated by an operator acting on the excitations, then we say the excitation is local. Note that the operators may break any symmetry and may not be local, as long as they act on the support subspace of the excitation. The fusion higher category formed by local excitations is a local fusion higher category. Since the membrane excitations in R can all be annihilated by operators on the membranes, R is a local fusion higher category.
The precise definitions of fusion higher categories and local higher categories are difficult due to the lack of the universally accepted and well-developed model for weak n-categories. In the following, we try to give a physical definition via the notion of topological orders. Many related concepts for topological order in arbitrary dimensions and for higher categories are summarized in Appendix, which is a review and an extension of the results in Ref. 24,25,28, and 31. The following discussions use those notions extensively. Table I summarizes some related concepts in higher category and in topological order.

V. LOCAL FUSION HIGHER CATEGORY AND REPRESENTATIONS (CHARGE OBJECTS) OF ANOMALY-FREE ALGEBRAIC HIGHER SYMMETRY
Given a higher symmetry (such as a 0-symmetry), some times it can be "gauged", and other times cannot be "gauged". When a 0-symmetry cannot be gauged, we say it has a 't Hooft anomaly. For 0-symmetries and higher symmetries, "can be gauged" is a defining property of an anomaly-free symmetry. 1,14,15 But so far, gauging is defined only for higher symmetries. We do not know how to "gauge" an algebraic higher symmetry (for an algebraic way to mimic the gauging of algebraic higher symmetry, see Sec. VI C 5). This forces us to define anomalyfree algebraic higher symmetry via a different method, as described in Sec. IV B. In this section, we are going to present a categorical description of anomaly-free algebraic higher symmetry. In the next a few sections, we only discuss anomaly-free algebraic higher symmetry, and we drop "anomaly-free" for simplicity.
For a 0-symmetry G, we know that its charges are representations of G. All those representations form a symmetric fusion category RepG. Due to Tannaka duality, we can use the local fusion category RepG to fully describe the symmetry group G. 44,45 To be more precise, the charges (the representations) of G correspond to point-like excitations. Those point-like charges can condense to form other descendent excitations. All those excitation are described by a fusion n-category, if the 0-symmetry G lives in n-dimensional space. We denote such a fusion n-category as nRepG. In other words, an nd 0-symmetry G is fully characterized by a symmetric fusion n-category nRepG. We refer to nRepG as the representation category of the 0-symmetry G.
In the above, we try to use excitations (trapped by the symmetric traps) to characterize a symmetry. Here we would like to stress that the excitations described by the fusion n-category nRepG only correspond to the excitations in the symmetric sub-Hilbert space V symm of the many-body system. The fusion n-category nRepG do not include the excitations outside the symmetric sub-Hilbert space. In the thermodynamic limit, restricting to symmetric sub-Hilbert space does not affect our ability to understand the properties of a symmetric system. We would like to use the similar approach to characterize an algebraic higher symmetry (which is not characterized by groups or even higher groups). But for an algebraic higher symmetry, what are its "representations"? Here we propose that the "representations" (i.e. the charge objects) of an algebraic higher symmetry are simply the excitations above a symmetric product state, which are also described by a category -a local fusion higher cate-TABLE I. Correspondence between concepts in fusion higher category and concepts in topological order. 24,25 Concepts in higher category Concepts in physics Fusion n-category C A collection of all the types of codimension-1 and higher excitations (plus their fusion and braiding properties) in an nd (potentially anomalous) topological order. Simple objects of C The types of codimension-1 topological excitations. They can fuse. Simple 1-morphisms of C The types of codimension-2 topological excitations. They can fuse and braid. Simple (n−2)-morphisms of C The types of string-like topological excitations Simple (n−1)-morphisms of C The types of point-like topological excitations n-morphisms of C The operators acting on the Hilbert space Collection of (n−1)-morphisms, (n−2)-morphisms, etc

Collection of types of excitations trapped by local trap Hamiltonians
Local fusion n-category R The "charged" excitations (charge objects) above a trivial product ground state of a bosonic system with an algebraic higher symmetry. It is called the representation category of the algebraic higher symmetry gory.
A. The excitations in a symmetric state with no topological order To understand the above assertion, let us consider a local lattice Hamiltonian H with an algebraic higher symmetry. We assume the ground state |Ψ grnd of H has no topological order nor SPT order, i.e. can be deformed into a product state without a phase transition, via a symmetry preserving path. Then how to understand the point-like, string-like excitations, etc of the above ground state? Also similar to the 0-symmetry case, here we only consider the symmetric excitations (i.e. those trapped by symmetric traps) in the symmetric sub-Hilbert space V symm . We know that an algebraic higher symmetry is generated by many operators acting on all closed submanifolds. The symmetric sub-Hilbert space is the invariant sub-Hilbert space of all those symmetry generators.
To understand the excitations, first, let us define excitations more carefully. For example, to define stringlike excitations, we can add several trap Hamiltonians ∆H str (W 1 i ) to H such that H + i ∆H str (W 1 i ) has an energy gap. ∆H str (W 1 i ) is only non-zero along the string W 1 i and commutes with the generators of the algebraic higher symmetry. We also assume ∆H str (W 1 i ) to be stable: any small symmetric change of ∆H str (W 1 i ) does not change the ground state degeneracy in the large string W 1 i limit. The resulting string corresponds to a simple morphism in mathematics. We also define two strings labeled by ∆H str (W 1 ) and ∆ H str ( W 1 ) as equivalent, if we can deform ∆H str (W 1 ) into ∆ H str ( W 1 ) without closing the energy gap while preserving the algebraic higher symmetry. The equivalent classes of the strings define the types of the strings (see Def. 9).
In the example in Sec. IV, the 2-dimensional charge object of an algebraic 2-symmetry is created by a membrane operator. If the membrane is a closed 2dimensional subspace, then the membrane operator acts within the symmetric sub-Hilbert space V symm , and cre-ate an excitation in the fusion higher category. If the membrane has a boundary, then the membrane operator does not act within the symmetric sub-Hilbert space, and create an excitation not in the fusion higher category. When the membrane has a boundary, such a boundary is the morphism that connect the membrane excitation to the trivial excitation. In the above example, such a boundary (i.e. the morphism) is not allowed, since it breaks the algebraic 2-symmetry (i.e. the membrane with the boundary does not act within the symmetric sub-Hilbert space).

B. Local fusion higher category
Now we are ready to define a local fusion higher category, which describes the collection of excitations (i.e. the collection of types) in the system mentioned above, i.e. a system with algebraic higher symmetry whose ground state is a symmetric bosonic product state without degeneracy. Also, we only consider excitations within the symmetric sub-Hilber space V symm . For a symmetric trivial phase without topological order, it has only local excitations. From a categorical point of view, a local excitation can always be connected to the trivial excitation through a morphism as described above, if we are willing to break the symmetry. However, if we preserve the symmetry, the symmetry breaking morphism is not allowed and some excitations cannot connect to trivial excitation via symmetry preserving morphisms (i.e. symmetry preserving domain walls). This leads to the following definition: A fusion n-category R is local if we can add morphisms in a consistent way, such that all the resulting simple morphisms are isomorphic to the trivial one. Physically, this process of "adding morphisms" corresponds to explicit breaking of algebraic higher symmetry. This because, R only has morphisms that correspond to symmetric operators. Adding morphisms means including morphisms that correspond to symmetry breaking operators. If after breaking all the symmetry, R describes a trivial phase without symmetry, then R is a local fusion n-category. The above can be stated more precisely Definition 3. A fusion n-category R (see Def. 16) equipped with a top-faithful surjective monoidal functor β from R to the trivial fusion n-category: R β → nVec is called a local fusion n-category. Here, top-faithful means that the functor β is injective when acting on the top morphisms (i.e. the n-morphism in this case).
Remark 1. The top-faithful condition means that operators in R form a subset of operators in nVec, which agrees with the physical interpretation that from R to nVec we add symmetry breaking operators. The functor β may not be faithful when acting on other morphisms. In other words, every objects and morphisms in R can be viewed as (i.e. can be mapped into) objects and morphisms in nVec, but the map may not be injective.
When we are interested in fermion systems, we need to replace nVec for nsVec. More generally, the building blocks of our physical system may have even larger intrinsic symmetry (which is unbreakable or we are not willing to break) besides the fermion number parity. Let V denote the fusion n-category formed by the building blocks (V = nVec for bosons, V = nsVec for fermions, and possibly any other V for more exotic cases such as an effective theory built upon anyons). We define the notion of V-local fusion n-categories. As an example, let us consider a 1d system with degrees of freedom labeled by g i ∈ G on site i, where G is a group. The Hamiltonian of the system is given by where T i (h) is an operator The system has a symmetry G When J |V |, the ground state is a product state g |g i , that does not spontaneously break the symmetry.
Note that {|g i , g ∈ G} spans the regular representation of G. It can be further decomposed into irreducible representations. Let |a i , |b i , · · · be a basis in an irreducible representation. Under the symmetry transforma- Such a ground state plus its excitations are described by a fusion 1-category RepG whose objects correspond to the point-like excitations (i.e. the representations R of G). The 1-morphisms of RepG correspond to the symmetric local operators that act on each site. We see that the 1-morphisms directly act on the point-like excitations (the objects). If we view an excitation (an object) as a world line in spacetime, an 1-morphism that changes the excitation can be viewed as a "domain wall" on the world line. For a symmetric system, all those 1-morphisms should be symmetric operators. Respecting to those symmetric 1-morphisms, the excitations corresponding to the irreducible representations are simple objects. Different irreducible representations cannot be connected by symmetric operators, i.e. different simple objects cannot be connected by 1-morphisms.
If we add the additional 1-morphisms that correspond to local operators that break all the symmetry, then the excitations corresponding to the irreducible representations R are still allowed, but they are no longer simple object, and split into direct sum of trivial excitations: As a result, the fusion 1-category is reduced to the trivial 1-category -the category of vector spaces Vec. Thus the fusion 1-category RepG is a local fusion 1-category. Indeed, all the point-like excitations can be annihilated by local operators that may break the symmetry. Now consider a 1d system with symmetry G, whose ground state spontaneously breaks all the symmetry. In this case, the ground states are |G|-fold degenerate and are labeled by the group elements: |Ψ g , g ∈ G. The point-like excitations are domain walls, which live on the links and are labeled by the elements h of the group: |h i0,i0+1 = |Ψ g,i≤i0 Ψ hg,i≥i0+1 . Such symmetry breaking state plus its excitations are described by a fusion 1-category Vec G , whose objects correspond to the pointlike excitations (the domain walls) discussed above. We may still choose the 1-morphisms of Vec G to be the symmetric local operators acting on the sites. However, such a choice is not proper, since such 1-morphisms cannot be viewed as the "domain walls" on the world-lines of the point-like excitations (the domain walls on the links). In any case, let us proceed. If we add the 1-morphisms that correspond to local operators that break all the symmetry, then objects (the point-like domain-wall excitations) are confined (i.e. non longer allowed), since the ground state degeneracy is lifted. This appears to suggest that the fusion 1-category Vec G is not a local fusion 1-category, if we view it as describing domain walls in a spontaneous symmetry breaking state that breaks a 0-symmetry of group G. Since our choices of the 1morphisms is not proper, the above conclusion is incorrect.
In fact, Vec G can also be viewed as a fusion 1-category that describes excitations on top of a product state with an algebraic 0-symmetry. The degrees of freedom on each site i of our 1d model are labeled by group elements of a finite group G. A basis of the many-body Hilbert space is given by |{g i } , g i ∈ G. The Hamiltonian is given by where The model has an algebraic 0-symmetry generated by where q labels the representations of G. In the t → 0 limit, the ground state is a symmetric product state Above such a ground state, a point-like excitation is generated by changing g i = id to g i = h on site-i. Thus the excitations are labeled by group elements h ∈ G, with h = id corresponding to the ground state. They fuse as h ⊗ h = hh . When the algebraic 0-symmetry operators act on the excitations h, we get B q (h) = X q (h), where X q is the character for the representation q. Those point-like excitations form a local 1-fusion category Vec G .
The operators that break the algebraic 0-symmetry are given by Those operators reduce the local 1-fusion category Vec G to the trivial 1-fusion category Vec, since those operators correspond to new morphisms h → h for any h, h ∈ G. Therefore, the 1-fusion category Vec G is local. We would like to mention that the 3d generalization of the 1d model (73) was discussed in Sec. IV. Using a similar reason, we show that the 3-fusion category 3Vec G is local.

D. Representation category of algebraic higher symmetry
Let us summarize the relation between the charge objects of an algebraic higher symmetry and a local fusion higher category. Proposition 2. Consider an nd trivial ground state which is a product state with an algebraic higher symmetry. The different types of the excitations above the ground state and within the symmetric sub-Hilber space form a local fusion n-category R (i.e. with a fiber functor β : R → nVec), which is called the representation category of the algebraic higher symmetry in n-dimensional space.
We would like to conjecture that the Tannaka duality can be generalized to algebraic higher symmetries: There is an one-to-one correspondence between local fusion n-categories R and algebraic higher symmetries for bosonic systems in n-dimensional space.
In other words, the algebraic higher symmetries in nd bosonic systems are fully characterized and classified by local fusion n-categories. Since a local fusion n-category R fully characterizes an anomaly-free algebraic higher symmetry, in this paper, an algebraic higher symmetry is denoted by R.
We would like to remark that there are anomalous algebraic higher symmetries. For those symmetries, we cannot have trivial symmetric ground state, and it is difficult to define its representation category, since representation category, by definition, is formed by the charged excitations above the symmetric product state.

E. Categorical symmetry -a holographic view of symmetry
To gain a deeper understanding of algebraic higher symmetry, following Ref. 18, we would like to introduce the notion of a categorical symmetry, which is a holographic point of view of a symmetry. We know that a symmetry is simply a restriction on the Hamiltonian. But in this holographic point of view, we do not view a symmetry as a restriction on the Hamiltonian. Instead, we use a topological order without any symmetry in one higher dimension to encode a symmetry. In other words, we use long range entanglement 46 to encode a symmetry.
Let us consider an nd system with an algebraic higher symmetry R. When we restrict the system to the symmetric sub-Hilbert space of the algebraic higher symmetry V symm , the system has a non-invertible gravitational anomaly, 47 since V symm does not have a tensor product decomposition V symm = ⊗ i V i . This relates the symmetry to entanglement. Thus the system (when restricted to the symmetric sub-Hilbert space V symm ) can be viewed as a boundary of an anomaly-free topological order M in one higher dimension. The topological order M is described by an object in TO n+2 af . Which topological order M in one higher dimension gives rise to the desired algebraic higher symmetry R? We note that R is a fusion n-category. We believe that for every fusion n-category R, there is exist a unique anomaly-free topological order M in one higher dimension such that M has a boundary whose excitations realize the fusion n-category R (see eqn. (G6)). Therefore, we can find M from R via If we use M = bulk(R) to describe an algebraic higher symmetry R, then the symmetry, or more precisely, the conservation law from the symmetry is encoded in the fusion rule for the excitations with codimension-2 and higher in M. (A codimension-1 excitation in M has codimension-0 on the boundary and cannot be viewed as an excitation there.) Those excitations are described by the braided fusion n-category (see Sec. F) where M = ΩM is the fusion n-category describing the bulk excitations in M. As we move a bulk excitation in M to the boundary, it may become some boundary excitations in R, or it may condense (i.e. becomes the trivial excitation in R). So there is a monoidal functor F R : M → R. The fusion rule in M induces a fusion rule in R. Thus the bulk symmetry encoded in M becomes an algebraic symmetry in R. However, the bulk excitations of M = Z 1 (R) are more than that of R. Thus the fusion rule of excitations in M gives rise to a bigger symmetry than that from the fusion rule of excitations in R. This bigger symmetry is called categorical symmetry. 18 We know that M can have many boundaries (denoted by C ∈ TO n+1 M , see Def. 19). The excitations on the boundary is described by a fusion n-category C = Hom(C, C), which satisfies (see Prop. 37) As we move a bulk excitation in M to the boundary, it may become some boundary excitations in C, or it may condense (i.e. becomes the trivial excitation in C). So there is a forgetful functor F C : M → C. Because some excitations in M are condensed on the boundary, we say the boundary spontaneously breaks part of the categorical symmetry M. Different boundaries C's may spontaneously break different parts of the categorical symmetry M, since the forgetful functor F C may map different excitations of M into the trivial excitations in C (i.e. condense different excitations of M on the boundary). This picture motivates us to say that all the boundaries have the same categorical symmetry M, if we view the boundary as a lattice boundary Hamiltonian. If we view the boundary as a state, then the categorical symmetry M is spontaneously broken down to a smaller symmetry. The part of the categorical symmetry M, described by the excitations that condense on the boundary, is spontaneously broken. The smaller survived symmetry is an algebraic higher symmetry. We know that the bulk fusion rule only induces the fusion rule for some boundary excitations (i.e. those in the image of the forgetful functor F C ). Thus the image of F C is related to this algebraic higher symmetry -the unbroken part of the categorical symmetry.
One might expect the image of F C to be the local fusion n-category that characterizes the algebraic higher symmetry in C. But this impression is incorrect. The image of F C may not even be a fusion n-category, i.e. there may not be an anomaly-free bulk topological order M whose boundary excitations realize the image of F C .
But what is the algebraic higher symmetry in C (the unbroken part of the categorical symmetry M)? First, such an algebraic higher symmetry must be described by a local fusion n-category R. Since R is the unbroken part of the categorical symmetry M, the corresponding categorical symmetry for R should be given by the same M. Mathematically, this means that Z 1 (R) is equivalent to M: Z 1 (R) M. Since R is the algebraic higher symmetry in C, C must contain all the charge objects of R as part of excitations in it. In other words, R can be embedded into C, i.e. there exists an top-fully faithful functor ι : R ι → C. Here Definition 5. Top-fully faithful means the functor is bijective when acting on top morphisms, and is injective when acting on lower morphisms and on objects.
We know that the R-symmetry can be explicitly broken, via the functors β, β C , which changes R to nVec (see Def. 3) and changes C to C. C describes the excitations in the anomaly-free topological order C 0 ∈ TO n+1 af ≡ (n + 1)Vec that are induced from C after we explicitly break the R-symmetry in C. We note that the excitations described by C contain both the topological excitations and the symmetry-charge excitations described by R (the charge objects of the algebraic higher symmetry). One may roughly understand C as "C/R" i.e. "C mod R". More precisely, C is the pushout defined in the following diagram, Moreover, the bulk of R → nVec and C → C should coincide, which requires that γ : Z 1 (R) Z 1 (C) satisfies the condition as later illustrated in (91).
To summarize, the different boundaries of M all have the same categorical symmetry as a system. But the boundary may spontaneously breaks the categorical symmetry when viewed as a state. Because the charge objects of a categorical symmetry have non-trivial mutual statistics, the boundary that does not break the categorical symmetry M must be gapless. 18,32 For a gapped boundary, the categorical symmetry must be partially (and only partially) broken spontaneously. For the boundary C discussed above, the categorical symmetry is spontaneously broken down to the algebraic higher symmetry R. We see that Proposition 4. Categorical symmetries in ndimensional space are fully characterized and classified by non-degenerate braided fusion n-categories M (which, by definition, satisfies bulk(ΣM) = 1 n+3 , see eqn. (G6)). M describes the excitations in an (n+1)d anomaly-free topological orders M (with codimension-2 and higher). A categorical symmetry M may include several different anomaly-free algebraic higher symmetries R, where R satisfies M Z 1 (R) (see Prop. 37) On a gapped boundary C of M, the categorical symmetry is partially spontaneous broken, down to an algebraic higher symmetry R that satisfies eqn. (80).
We would like to remark that there can be several different (n+1)d anomaly-free topological orders, M's, whose excitations are described by the same anomalyfree braided fusion n-categories M. The boundary of M refers to any boundary of those (n+1)d topological orders, whose excitations are described by M.
We would like to stress that for an nd lattice boundary Hamiltonian with an algebraic higher symmetry R, the lattice boundary Hamiltonian always has a larger symmetry -a categorical symmetry, characterized by a braided fusion n-catgeory M = Z 1 (R). The categorical symmetry includes the original algebraic higher symmetry R, but also includes extra symmetries. The gapped ground state of the lattice boundary Hamiltonian must spontaneously break part of the categorical symmetry, and can only spontaneously break part of the categorical symmetry. For example, as pointed out in Ref. 18, an nd system with a 0-symmetry described by a finite group G (or a fusion n-category nRepG) actually has a larger categorical symmetry. The categorical symmetry is characterized by a braided fusion n-category Ω 2 GT n+1 G = Z 1 (nRepG) (i.e. characterized by the excitations in the G-gauge theory in one higher dimension).

VI. GAPPED LIQUID PHASES WITH ALGEBRAIC HIGHER SYMMETRY
In Sec. V, we discussed gapped liquid state with algebraic higher symmetry, but with no topological order. In this section, we are going to discuss anomaly-free gapped liquid phases with unbroken algebraic higher symmetry, which may have non-trivial topological orders. Those states are called SET states if the topological order is non-trivial (i.e. with long range entanglement), or SPT states if the topological order is trivial (i.e. with short range entanglement).
Let us first summary some previous results, which represent some systematic understanding of gapped liquid phases 48,49 for boson and fermion systems with and without symmetry (but only for 0-symmetry). In 1+1D, all gapped phases are classified by (G H , G Ψ , ω 2 ) 50,51 , where G H is the on-site symmetry group of the Hamiltonian, G Ψ the symmetry group of the ground state G Ψ ⊂ G H , and ω 2 ∈ H 2 (G Ψ , R/Z) is a group 2-cocycle for the unbroken symmetry group G Ψ .
In 2+1D, all gapped phases are classified (up to E 8 invertible topological orders and for a finite unitary on-site symmetry G Ψ ) by (G H , Rep(G Ψ ) ⊂ C ⊂ M) for bosonic systems and by (G H , Rep(G f Ψ ) ⊂ C ⊂ M) for fermionic systems 44,45,52 . Here Rep(G Ψ ) is the symmetric fusion category formed by representations of G Ψ , and Rep(G f Ψ ) is the symmetric fusion category formed by Also C is a braided fusion category and M is a minimal modular extension 44,45 .
In 3+1D, some gapped phases are liquid phases while others are non-liquid phases. The 3+1D gapped liquid phases without symmetry for bosonic systems (i.e. However, the above approaches are quite different for different dimensions and are only for 0-symmetries. In this section, we describe a classification that works for all dimensions. We also generalize the 0-symmetry in the above results to algebraic higher symmetry.
A. Partially characterize a symmetric gapped liquid phase using a pair of fusion higher categories For a gapped liquid state with unbroken algebraic higher symmetry R, there are point-like, string-like, etc excitations, that correspond to the charge objects of the symmetry. Those charge objects are described by the representation category R. In general, the gapped liquid state also has extra point-like, string-like, etc excitations, that are beyond those described by the local fusion higher category R. So the total excitations are described by a bigger fusion higher category C that includes R. This leads to the following result: Proposition 5. An nd gapped liquid phase with an unbroken algebraic higher symmetry described by a fusion n-category R, is fully described by a potentially anomalous topological order C (see Appendix E). The excitations of C are described by a fusion n-category C = ΩC which admits a top-fully faithful functor R ι → C. Thus we can use the data R ι → C to partially classify the gapped liquid phase with algebraic higher symmetry R.
One way to see the above result is to note that stacking a trivial symmetric state R and the symmetric topological order C together give rise to a fusion n-category R ⊗ C, if there is no coupling between the trivial symmetric state R and the symmetric topological order C. The R ⊗ C state has a larger algebraic higher symmetry R ⊗ R, one from the trivial symmetric state R and the other from the symmetric topological order C. However, we can add the so called "symmetric interactions" between R and C to reduce the R ⊗ R symmetry to the original symmetry R. The stacking with such symmetric interactions, which preserves the diagonal R symmetry but break the other symmetries, is denoted by ⊗ R and R ⊗ R R = R. Including the "symmetric interactions" is similar to adding the symmetry breaking morphisms in our definition of local fusion higher category (see Def. 3). Such a process can also be realized by a condensation of excitations. Since R is local, the condensation does not confine any excitations in R, and all the excitations in R become excitations in R ⊗ R C. Physically we require that R ⊗ R C = C. Therefore, all the excitations in R become excitations in C. Thus there is a functor R ι → C, which is faithful (i.e. injective) at each level of morphisms and objects. Since both R and C have the same algebraic higher symmetry R, the allowed local symmetric operators are the same. Thus the faithful functor R ι → C is fully faithful (i.e. bijective) at the top morphisms (which correspond to the local symmetric operators).
Does every pair of fusion n-categories (R, C) satisfying R ι → C describes an anomaly-free topological order with an algebraic higher symmetry? The answer is no, as implied by some counterexamples when R describes a 0-symmetry. 44,45 If the pair (R, C) does describe a symmetric topological order, does it uniquely describe the symmetric topological order? The answer is also no. For example, a pair of fusion n-categories (R, R) can describe a symmetric trivial product state. The same pair (R, R) can also describe a SPT state of the same symmetry. This is because, as we mentioned before, the fusion ncategory only describe the excitations which do not contain all the information of a topological order, and cannot distinguish different invertible topological orders. In our case here, the pair (R, R) cannot distinguish symmetric trivial product state from non-trivial SPT state with the same anomaly-free algebraic higher symmetry.
However, the R ι → C description does not miss much. In the following, we try to understand which pairs R ι → C can describe anomaly-free topological orders with an algebraic higher symmetry. We also try to seek additional information beyond R ι → C to fully characterize a symmetric topological order. One way to achieve both goals is to use the notion of categorical symmetries described in Ref. 18 and in Sec. V E, which is a holographical way to view a symmetry. This new way to view a symmetry is most suitable for algebraic higher symmetries. It gives an even more general perspective about algebraic higher symmetries. So in the next section, we first study gapped liquid phases in a bosonic system with a categorical symmetry.

B. Classification of gapped liquid phases in bosonic systems with a categorical symmetry
Let us consider an nd bosonic lattice boundary Hamiltonian with a categorical symmetry characterized by an anomaly-free braided fusion n-category M (which describes the codimension-2 and higher excitations in an (n+1)d anomaly-free topological order M ∈ TO n+2 af ). One may ask, what are the gapped liquid phases with the categorical symmetry M? The answer is that there is no such phases. This is because a gapped phase in a lattice boundary Hamiltonian with a categorical symmetry M must partially break and only partially break the categorical symmetry spontaneously. 18,32 This is because a gapped phase in an nd bosonic system with a categorical symmetry M corresponds to a gapped boundary of a (n+1)d anomaly-free topological order M with excitations described by M. The gapped boundary comes from the condensation of some of the excitations in M, and thus part of the categorical symmetry is spontaneously broken. In fact, the condensing excitations form a Lagrangian condensable algebra, which corresponds to spontaneously breaking part of the categorical symmetry. Also, the excitations that can condense (i.e. those in the Lagrangian condensable algebra) must have trivial mutual statistics between them. Therefore, we cannot condense all the excitations in M simultaneously. This is why we cannot completely break a categorical symmetry spontaneously. (Certainly, we can always partially or completely break a categorical symmetry explicitly.) This picture leads to the following result (see Fig. 6a): Proposition 6. For a system in n-dimensional space with a categorical symmetry M, all its gapped liquid phases are classified by the gapped boundaries of (n+1)d anomaly-free topological orders M with excitations described by M = Ω 2 M (see Appendix F). In other words, the gapped liquid phases are classified by (potentially anomalous) topological orders C's (objects in TO n+1 M 's, see Appendix E) satisfying the condition: The above result implies that (see Fig. 6a) Proposition 7. For a system in n-dimensional space with a categorical symmetry M, the excitations in its gapped liquid states are described by fusion n-categories C such that For every fusion n-category C satisfying Z 1 (C) M, there is one or more gapped liquid phases, C's, in nd bosonic systems with the categorical symmetry M to realize it: C = ΩC.
Let us remark the second part of the above result. Given a fusion n-category C satisfying Z 1 (C) M, can we find an nd bosonic system with the categorical symmetry M to realize it? We believe the answer is yes, provided that we have a proper definition of a fusion n-category C that is restrictive enough. We also believe that, in general, the realization is not unique. Since M is anomaly-free, there exist one or more (n+1)d topological orders, M's, that realize M = ΩM. We believe that at least one of those (n+1)d topological orders, M, has a boundary C that has excitations described by C, i.e. the boundary C satisfies C = ΩC (see Props. 35 and 39).
We remark that the boundary C that satisfies C = ΩC is not unique. One way to get different C' with the same excitations is to stack an invertible topological order to C. This does not affect C = ΩC and Z 1 (C) M.
To summarize (see Fig. 6a) The gapped liquid phases in bosonic systems with a categorical symmetry M are partially classified by the fusion n-categories C that satisfy M Z 1 (C).
Here partially means that the classification is one-tomany: the same fusion n-category C may corresponds to several different gapped liquid phases, C's, of a system with the categorical symmetry M. As we have mentioned before, here, the gapped liquid phases must break only part of the categorical symmetry M spontaneously.
To get a full one-to-one classification, we need to find extra information beyond the excitations, i.e. the fusion n-category C, to characterize the gapped liquid states. One way to get extra information is to study the boundaries of the gapped liquid states. This will be done later.
Here, we consider another type of extra information. In Prop. 7, we want Z 1 (C) (the canonical bulk of C) and M (that characterizes the categorical symmetry) to be equal. But as two braided fusion n-categories, their "equality" can only be described via a braided equivalence functor γ : M Z 1 (C). Such equivalence functor γ is the extra information that we are looking for. This is because the missing SPT orders are invertible. So we believe that they can be captured by invertible data, the braided equivalence γ. This leads to the following result (see Fig. 6b) Proposition 9. For a system in n-dimensional space with a categorical symmetry M, its gapped liquid phases (up to invertible topological orders) are described by a pair (C, γ), where C is a fusion n-category C that satisfies M γ Z 1 (C) (see Prop. 37), where γ is a braided equivalence functor.
The possible equivalence functor γ is of course not unique. One can always compose an automorphism of M with γ to obtain a different equivalence. However, when we are considering gapped liquid phases with an algebraic higher symmetry R ( instead of gapped liquid phases that may spontaneously break the categorical symmetrical M = Z 1 (R)), the different equivalences may have different physical meanings with respect to R. An A gapped liquid state with an algebraic higher symmetry R can be characterized by a pair (C, γ), or more precisely, by C ⊗ , where the R-symmetry is viewed holographically via the categorical symmetry Z1(R) γ Z1(C). The fusion n-category C on the boundary describes the excitations on top of the gapped liquid state, which include R, i.e. R ι → C. Since R is local, we also has a fiber functor β : R → nVec. The cross is a generic excitation in C. The white-circles are R-charge objects in C.
equivalence γ may either preserve the algebraic higher symmetry R, or (partially or completely) break R. To classify gapped liquid phases with an algebraic higher symmetry R, we need to select γ's that preserve the algebraic symmetry R. In the next subsection, we give several (hopefully equivalent) criteria when an equivalence γ preserves an algebraic symmetry R.

C. Classification of SET orders and SPT orders
with an algebraic higher symmetry

A simple result
Let us first give a simple partial result by ignoring γ. Given an algebraic higher symmetry R, there is an (n+1)d anomaly-free topological order M = bulk(R) (see eqn. (G6)) with excitations described by M = ΩM that has one boundary R with excitations described by R = ΩR. In other words, M = Z 1 (R). Throughout this subsection, we set M ≡ Z 1 (R). The boundary topological order R corresponds to a trivial product state with the algebraic higher symmetry R. (In other words, the trivial product state with the algebraic higher symmetry R, plus its excitations, when restricted to the symmetric sub-Hilbert space V symm correspond to the boundary topological order R.) Now consider another boundary C of M. The excitations on C are described by a fusion n-category C = Hom(C, C). However, the boundary C may spontaneously break the algebraic higher symmetry R. Here we would like to classify gapped liquid phases, C's, that do not spontaneously break the algebraic higher symmetry R.
To do so, we just need to select those C's that do not spontaneously break the symmetry R. This can be easily achieved by requiring C to contain R. We believe that all the anomaly-free topological orders with the algebraic higher symmetry R can be viewed in this way. If we re-place C by C to get a partial classification, we obtain (see Fig. 9) Proposition 10. Anomaly-free gapped liquid phases with an algebraic higher symmetry R in n-dimensional space are partially classified by fusion n-categories C, that satisfy Z 1 (C) Z 1 (R) and admit a top-fully faithful functor R ι → C.
We would like to point out that the above C classifies the excitations in the anomaly-free topological order C with an algebraic higher symmetry R (i.e. C can be realized by a bosonic lattice model in the same dimensions with the symmetry R). We know that topological orders differ by invertible gapped liquids have the same excitations. Thus the above C's cannot distinguish different invertible gapped liquids, i.e. different invertible topological orders and SPT orders. The above C's only classify anomaly-free topological orders with the algebraic higher symmetry R, up to invertible topological orders and SPT orders for symmetry R.
To obtain a more complete classification, i.e. to include the SPT orders with symmetry R, we should include the braid equivalence γ : Z 1 (R) Z 1 (C) as our data as we discussed in Sec. VI B (see Fig. 9): Proposal 1. Bosonic anomaly-free gapped liquid phases in n-dimensional space with an anomaly-free algebraic higher symmetry R are classified (up to invertible topological orders) by data (C, ι : R → C, γ : Z 1 (R) Z 1 (C)), where C is a fusion n-category that includes R (i.e. ι : R → C is a top-fully faithful functor, see Prop. 5), and γ : Z 1 (R) Z 1 (C) is a braided equivalence (a braided monoidal functor that is full and faithful for all morphisms and objects). For a special case, C = R, we get a classification of R-SPT phases.
However, the above proposal is incorrect. We cannot use an arbitrary braided equivalence γ : Z 1 (R) Z 1 (C)). The reason is that C contains the symmetry R via the embedding ι : R → C. Z 1 (R) also contains the symmetry R via the forgetful functor F R : Z 1 (R) → R. If we allow an arbitrary γ, the above two R symmetries may not be compatible (i.e. may not be joined by γ). Thus the key is to find proper conditions to select proper γ's. In the following, we describe several, hopefully equivalent, ways to do so.

A classification assuming R to be symmetric
When R is symmetric, it can be lifted to the bulk Z 1 (R) via a canonical braided embedding ι R : R → Z 1 (R). In this case, we have a simple criteria for γ to make the two R symmetries in C and Z 1 (R) compatible (i.e. to preserve the R symmetry, see Fig. 10): Proposition 11. Anomaly-free gapped liquid phases in n-dimensional space with an anomaly-free algebraic higher symmetry R (which is assumed to be symmetric) 10. Similar to Fig. 9, but now we assume R to be symmetric. In this case, the R-charges in R can be lifted into the bulk via the embedding ιR. The equivalence γ should be compatible with the liftings of R. 11. The graphic representation of eqn. (84). We assume R to be symmetric. In this case, the R-charges in R and in C can be lifted into the bulk via the embedding ιR and ιC.
are classified (up to invertible topological orders) by data (C, ι : R → C, γ : Z 1 (R) Z 1 (C)), where C is a fusion n-category that includes R (i.e. ι : R → C is a top-fully faithful functor), and γ : Z 1 (R) Z 1 (C) is a braided equivalence rendering the following diagram commutative (up to a natural isomorphism): where F C : Z 1 (C) → C is the forgetful functor.

Ref. 40 proposed this result in a slightly different manner.
An embedding ι C : R → Z 1 (C) is considered as the data for gapped liquid phases instead of ι : R → C, and it is required that F C • ι C : R → C is an embedding, thus reproduce the data ι. Then, eqn. (83) is replaced by (see Fig. 11) When C = R the above result reduces to a classification of SPT orders with symmetry R (see Fig. 12): Proposition 12. SPT phases in n-dimensional space with an anomaly-free algebraic higher symmetry R (which is assumed to be symmetric) are classified (up to invertible topological orders) by data (R, α), where α : the following diagram commutative (up to a natural isomorphism): where the central structure comes from the symmetric structure of R. By the universal property of center 29 , (85) is equivalent to

A general classification
Now we discuss a classification for more general algebraic higher symmetry where R may not be symmetric. To do so, we need a very different approach. Let us first consider the classification of bosonic SPT orders with an algebraic higher symmetry R in n-dimensional space. Those SPT orders all have excitations described by the same local fusion n-category R. To distinguish different SPT orders, we need to include extra information beyond R, and to use pairs (R, α) to describe the SPT orders, where α is an automorphism of Z 1 (R). To identify the proper α's, we notice that the physical way to distinguish different SPT orders is to include the boundary of a SPT state. Here we consider the canonical boundary that spontaneously breaks all the symmetry R.
In the following, we develop a theory for the canonical boundary that break all the symmetry R, using a holographic point of view of the symmetry R, i.e. using can be viewed as a gapped boundary of a topological order in one higher dimension with excitations Z1(R). The boundary V β = nVec β of R that breaks all the R-symmetry also has a bulk, which can be viewed as a gapped boundary of Z1(R) with excitations described a local fusion n-category R. (b) The automorphism α of Z1(R) correspond to an invertible domain wall in the bulk (the dash-line), which also has an invertible boundary (the white square). The boundary-bulk relation between Z1(R) and R is described by the bulk to boundary functor F R , which is induced by moving the bulk excitations to the boundary. Such a boundary-bulk relation is preserved by the automorphisms α, α. a topological order with excitations Z 1 (R) in one higher dimension to describe the symmetry R. In other words, we need to use the holographic point of view to describe the boundary that breaks all the symmetry R. Such a symmetry breaking boundary also has a bulk in one higher dimension. Such a bulk has a different set of excitations described by another local fusion n-category, denoted as R. In fact R can be viewed as another gapped boundary of the bulk Z 1 (R) (see Fig. 13a), therefore, Z 1 (R) Z 1 ( R). R is nothing but the local fusion ncategory that describes the dual symmetry of R (see Sec. II D). For example, when R = nRepG, the symmetry is the 0-symmetry described by the group G. The dual symmetry is an algebraic higher symmetry described by R = nVec G .
We know that the bulk Z 1 (R) and the boundaries, when viewed as systems, have a categorical symmetry Z 1 (R) that includes both the symmetry R and the dual symmetry R. The boundary R in Fig. 13 has the symmetry R but spontaneously breaks the dual symmetry R, while the boundary R has the dual symmetry R but spontaneously breaks the symmetry R. The intersection nVec β of the two boundaries breaks both the symmetry R and the dual symmetry R (see Fig. 13a).
The "bulk" of the canonical boundary nVec β of R, which is also a "boundary" of the bulk of R, gives us the criteria when the automorphism α : Z 1 (R) Z 1 (R) preserves the symmetry R and thus represents an R SPT order (see Fig. 13a). To identify the proper automorphisms, we note that α can be viewed as an invertible domain wall in the bulk Z 1 (R) (see Fig. 13b). Such an invertible domain wall has a boundary on the boundary R (the white square in Fig. 13b). Since the difference between SPT orders are invertible, the boundary of the invertible domain wall should also be invertible. This motivates us to conjecture that the boundary of the invertible domain wall α corresponds to an automorphism α of R. The automorphisms α for the bulk Z 1 (R) and α for the boundary R should preserve the whole structure of R and its boundary V β (the red-line and the blackbox in Fig. 13b). This can be achieved by requiring α, α to preserve the bulk-boundary relation described by the bulk to boundary functor F R : Z 1 (R) → R. This leads to the following result: Proposition 13. SPT orders with an algebraic higher symmetry R in n-dimensional space are classified by equivalence relations α : Z 1 (R) Z 1 (R) and α : R R, such that the following diagram is commutative (up to a natural isomorphism): The above is just for SPT orders. In the following, we use the similar approach to develop a more general categorical theory to classify both SPT and SET orders. We also allow more general algebraic higher symmetry, by allowing R to be V-local (recall Definition 4), to include at least both boson and fermion systems. When V = nVec, R describes the algebraic higher symmetry in bosonic systems. When V = nsVec, the fusion n-category of super vector spaces, R describes the algebraic higher symmetry in fermionic systems.
Remark 3. Physically, we think V as the building blocks of our system. R is built upon V with some additional symmetry that can be totally broken. β : R → V exactly describes the symmetry breaking that leaves only the intrinsic symmetry V of the building blocks which is not physically breakable (for example, fermion parity).
A gapped liquid state C with symmetry R is equipped with a top-fully faithful monoidal functor ι : R → C. There is an anomaly-free topological order C underlying C by breaking all the R symmetry. Mathematically, we may define C to be the pushout (i.e., the colimit) of V β ← − R ι − → C in the category of fusion n-categories, As a colimit, C, β C and ι 0 are uniquely determined by V β ← − R ι − → C up to isomorphisms. In particular, for SPT orders, we take C = R, ι = id R , and then C = V, β C = β, Alternatively, β can be consider as condensing some excitations (which form an algebra A β , and condensing means taking the modules over this algebra) in R. Condensing the same excitations in C (identified via ι), gives C.
C constitutes a symmetry breaking domain wall between C and C. Mathematically, C is C-C-bimodule; the left action is by fusion in C and right action is by first mapping C into C via β C and then fusion in C. To emphasize, we denote the bimodule by C β C . The bulk of C, C, as well as the domain wall C β C , can be defined via bimodule functors (see Appendix G). More precisely, • The bulk of C is Z 1 (C) := Fun C|C (C, C); • The bulk of C is Z 1 (C) := Fun C|C (C, C); • The bulk of the domain wall C β C is R C is also a symmetry breaking domain wall in the bulk, between Z 1 (C) and Z 1 (C).
• There are also bulk to wall functors In conclusion, if we view C, C with a domain wall C β C between them as a whole, the bulk is given by Z 1 (C) Fig. 14), i.e. , two non-degenerate braided fusion n-categories Z 1 (C) and Z 1 (C), a fusion ncategory R C and two bulk to wall functors F R C and F R C .
is the categorical symmetry and how it breaks down corresponding to β : R → V.
Clearly, the bulk of the product state with the Rsymmetry (and its breaking R β → V) is given by . As before, we require that the bulk of C coincide with the R-symmetric product state, which amounts to say that there should be three equivalences γ : Z 1 (R) Z 1 (C), γ : R R C , and γ 0 : Z 1 (V) Z 1 (C). The criteria for these equivalences to preserve R is given below: 14. R, V and their domain wall V β describe a trivial product state with symmetry R (as well as the symmetry breaking R β → V). C, C and their domain wall C β C describe a gapped liquid state with symmetry R. The bulk of two structures (the categorical symmetry and its breaking) must coincide. This is just the anomaly matching condition, since the categorical symmetry can be viewed as anomaly. As a result, the R-symmetric gapped liquid phase and the R-symmetric trivial phase can be two phases of the same system.

Proposition 14.
An anomaly-free gapped liquid phase in n-dimensional space with a generalized anomaly-free algebraic higher symmetry described by a V-local R is characterized by the data ι : R → C, where C is a fusion n-category, and γ, γ, γ 0 as explained above, rendering the following diagram commutative.
In particular, taking C = R and then C = V, R C = R, we obtain a classification of R-SPT orders (after renaming γ to α): The above triples of automorphisms (α, α, α 0 ), that label different R-SPT orders, can be composed, which correspond to the stacking of the SPT orders.
If we choose V = nVec, the above classifies SET/SPT orders for bosonic systems with an algebraic higher symmetry. If we choose V = nsVec, the above classifies SET/SPT orders for fermionic systems with a generalized algebraic higher symmetry.
15. An SPT state characterized by a pair (R, α), or more precisely, by the stacking of R and α through the bulk Z1(R): R ⊗ . Here the R-symmetry is viewed holographically via the categorical symmetry Z1(R).
In this formulation, there is no need to assume that R is symmetric or even braided. But assuming R, V and β : R → V are braided, we want to show that the Proposition 14 and Proposition 11 are equivalent. We sketch a tentative proof here. There is a canonical braided embedding ι R : R → Z 1 (R). Then consider the In the category of fusion n-categories, the pushout is justR.
Indeed, β can be considered as condensing some excitations A β in R. Condensing the same excitations in Z 1 (R) (identified via ι R ) gives R. Moreover, Z 1 (V) should be a full subcategory of R corresponding to the deconfined excitations and F R is the embedding. Therefore, the embedding ι R determines all the other structures Z 1 (V), R, F R , F R . Also γ : Z 1 (R) Z 1 (C) with such embedding ι R determines γ and γ 0 . Then it should be straightforword to verify that (91) is equivalent to (83).
Example 1. For R = C = RepG, V = Vec, β : RepG → Vec the forgetful functor, we have R = Vec G . Since F R corresponds to condensing the algebra Fun(G) ∈ RepG, preserving the embedding RepG → Z 1 (RepG) is the same as preserving Fun(G) and thus F R . The G = Z 2 × Z 2 case is explicitly calculated in the Sec. VI C 7.

A version of classification based on condensable algebra
In this section, we are going to describe another version of classification. Let us first consider an R-SPT state characterized by a pair (R, α), where α is a braided automorphism of Z 1 (R) (see Fig. 15). We would like to explore other equivalent ways to select proper α's. We first restrict to bosonic systems for simplicity, assuming that R is a local fusion n-category. A key feature of SPT order is that a SPT state has no topological order, i.e. it becomes a product state if we break the symmetry. How to impose such a condition, when we use the holographic point view of the symmetry R?
Here we would like to point out that if we stack R and its dual R through their common bulk M = Z 1 (R), denoted as R ⊗ M R rev , we get a trivial product state nVec (see Fig. 4). We may use this property to define a more general notion of dual symmetry.
In fact, there is a one-to-one correspondence between dual symmetries of R and the monoidal functors R → nVec. If a boundary B of Z 1 (R) satisfies R ⊗
In particular, R also defines a monoidal functor B → nVec. In other words, both R and B are local fusion n-categories. Physically, when R and R are dual to each other, every excitation in M = Z 1 (R), either condenses to Rboundary or condenses to R-boundary, or both. In this case, every excitation in M is condensed and the resulting state is trivial. We know that R can be obtain from M via a Lagrangian condensable algebra A R in M. Similarly, R can be obtained from M via another Lagrangian condensable algebra A R . Roughly speaking a condensable algebra is formed by excitations with trivial mutual statistics with each other, and those excitations can all condense simultaneously to form a gapped boundary (see Fig. 4). Thus in the R boundary, the excitations in A R condense. The non-condensing excitations become the boundary excitations that is described by R. So roughly speaking R = M/A R . (In precise mathematical language, R identifies with the category of modules over A R in M.) Similarly, R = M/A R . When R and R are dual to each other, the overlap of A R and A R is minimal and is given by the trivial excitations. Also A R and A R together generate the whole M. (More precisely, any excitation in M is contained in A R ⊗ A R .) Thus roughly speaking, A R ⊗ A R ∼ M. We see that, A R is formed by excitations in R and A R is formed by excitations in R.
describes an SPT state if it can be canceled by R.
If α keeps the AR unchanged, then (R, α) ≡ R ⊗ is also determined by the condensable algebra AR and is equivalent to R ⊗ Prop. 15 tells us that dual symmetry is an equivalent way to describe symmetry breaking. If R can be canceled by its dual: R ⊗ M R rev = nVec, then R (as well as R) is a local fusion n-category, i.e. R can be reduced to the trivial product state if we break the symmetry: R β → nVec. Since R can be viewed as (R, α = id) in Fig. 15, we see that (R, id) can be cannceled by R, which implies that (R, id) is a product state if we break the symmetry. This implies that if we do not break the symmetry, then (R, id) is a SPT state. Therefore, to see if (R, α) is a SPT state or not, we can just check if it can be cancelled by R or not. This allows us to obtain (see Fig. 16) Using the condensable algebra, we find that one class of the solutions of eqn. (96) are given by α's that keep the A R unchanged (see Fig. 17): where µ is an algebra isomorphism. In this case, Fig.  17a can be deformed into Fig. 17b.
17b with Fig. 16, we see that both R ⊗ are determined by the same condensable algebra A R , and hence are equivalent: R ⊗ . This allows us to show that also tell us that the SPT state is equivalent to the SPT state described by R ⊗

Z1(R)
, which is the trivial SPT state. In fact, α that keeps A R ∼ R unchanged may change R. Thus we believe that R ⊗ and R ⊗

Z1(R)
only differ by a relabeling of the symmetry (i.e. differ by an automorphism of R), and thus correspond to the same SPT state. Eqn. (96) has another class of solutions, given by α's that keep A R unchanged: By comparing Fig. 18b with Fig. 16, we see that ⊗ R rev . This allows us to show that, indeed, As we have mentioned, the condensable algebra A R is formed by excitations in R. So keeping A R part of M unchanged corresponds to keeping R part of M unchanged. Therefore, the automorphisms α, that satisfy eqn. (99), do not change the R symmetry. But α's generate nontrivial automorphisms of R, α : R R (see Fig. 19). . We see that different pairs of automorphisms (α, α) give rise to different boundaries (due to different α's). This leads to the following result: The automorphism α of Z1(R) correspond to an invertible domain wall in the bulk (the dash-line), which also has an invertible boundary (the white square). The boundary-bulk relation between Z1(R) and R is described by the bulk to boundary functor F R , which mapps the condesible algebra A R to the trivial excitioins on the R boundary.
Proposition 17. Bosonic SPT phases in n-dimensional space with an anomaly-free algebraic higher symmetry R are classified (up to invertible topological orders) by braided equivalences α : We would like to remark that, for a given α, different choices of µ differ by automorphisms of the condensable algebra A R . Those different µ' may lead to the same SPT order. This is because if we gauge the R-symmetry in a SPT state, the resulting topogecal order does not dependent on µ (see Remark 5). Thus, the bosonic SPT phases with a R-symmetry may actually be classified by α's, rather than the pairs (α, µ)'s.
We can generalize the above result to include SET orders with R-symmetry for bosonic or fermionic systems (see Fig. 20): Proposition 18. Anomaly-free gapped liquid phases in n-dimensional space with a generalized anomaly-free algebraic higher symmetry R are classified (up to invertible topological orders) by the data (R ι → C, γ, µ), where R is a V-local fusion n-category, C is a fusion n-category that includes R, γ : Z 1 (R) Z 1 (C) is a braided equivalence and µ : γ(A R ) A R C is an algebra isomorphism. Here R C is defined in Prop. 14 and A R , A R C are the condensable algebras in Z 1 (R), Z 1 (C) that produces the R, R C domain walls between Z 1 (R), Z 1 (C) and Z 1 (V), Z 1 (C), respectively.
Although we have included µ in the above, it is possible that different choices of µ correspond to the same gappled liquid phases, as we discussed later in Remark 5. Thus the different gapped liquid phases may actually be classified by the data (R ι → C, γ). When V = nVec, the above classifies SET/SPT orders in bosonic systems. When V = nsVec, the above classifies SET/SPT orders in fermionic systems.
Remark 4. Here we would like to sketch the reasoning that Prop. 14 and Prop. 18 are equivalent. From the condensation point of view, condensing the algebra A β in 20. Similar to Fig. 14, but here the automorphism γ is required to map the two condensable algebras, AR and A R C , into each other.
R induces the symmetry breaking β : R → V. Similarly, the symmetry breaking F R : Z 1 (R) → R in the bulk is induced by condensing the algebra Intuitively, we can think that A R replaces the role of embedding R → Z 1 (R); instead of embedding R into the bulk which is only possible when R is braided, we lift the algebra A β in R to the algebra A R in Z 1 (R). A β consists of all objects in R when V = nVec. Mathematically, R should be the category of modules over A R in Z 1 (R) while Z 1 (V) should be the full subcategory of local modules, with F R being the embedding (see Fig. 14). (Physically, Z 1 (V) corresponds the deconfined excitations after condensation while R includes both confined and deconfined excitations.) Therefore, γ together with µ : γ(A R ) A R C determines γ as an equivalence functor between the categories of modules over A R and A R C ; γ 0 is the restriction of γ to Z 1 (V). Eqn. (91) is equivalent to saying that γ preserves the lifted algebra γ(A R ) µ A R C .
5. R-gauge theory obtained by "gauging" the algebraic higher symmetry R Using the data (R ι → C, Z 1 (R) γ Z 1 (C)), we can explicitly construct the corresponding gapped liquid state with symmetry R that the data describes. This is done in Fig. 9. Since the gapped liquid state has the symmetry R, we can gauge the symmetry R to obtain a new topologically ordered stated with no symmetry. This is achieved in Fig. 21a, by stacking R and C through Z 1 (R) γ Z 1 (C), with an equivalence γ in the middle. We denote such a stacking by R ⊗ C rev . The resulting topologically ordered state is anomaly-free since it is surrounded by the trivial product state (with its codimension-2 excitations described by nVec).
Both 0-symmetries and higher symmetries have an holonomy interpretation, which allows us to gauge them via a geometric approach. In contrast, the above proposal C rev ) can be viewed as an anomaly-free topological order with no symmetry (surrounded by trivial product state), which is obtained by gauging the algebraic higher symmetry symmetry R in an SET state with symmetry R.
is the topological order obtained from an SPT state by gauging its symmetry R. (c) When C = R and γ = id, R ⊗ is the topological order obtained from a product state by gauging its symmetry R.
to "gauge" algebraic higher symmetries (which include 0symmetries and higher symmetries) is a purely algebraic approach. No geometric interpretation is used.
To understand such a proposal, let us consider a very simple case, by assuming γ = id, n = 2, and C = R = 2RepG. So the boundary R is in 2d while the bulk Z 1 (R) is in 3d. The resulting state R ⊗ Z1(R) R rev given by Fig.   21c is actually a 2d gauge theory with group G (i.e. the 2d topological order GT 3 G ). To see this, we note that the bulk Z 1 (R) is the 3d gauge theory with group G (i.e. the 3d topological order GT 4 G = bulk(2RepG)). The 2d boundary R = 2RepG is obtained from the 3d Ggauge theory GT 4 G by condensing the G-flux loops. Thus a G-flux loop in the bulk corresponds to a trivial excitation in the 2d G-gauge theory order GT 3 G . A G-flux string connecting two boundaries corresponds to a pointlike G-flux excitation in the 2d G-gauge theory GT 3 G . The point-like G-charges in the 3d G-gauge theory GT 4 G becomes the point-like G-charges in the 2d G-gauge theory GT 3 G . This suggests that, in general, R ⊗

Z1(R)
R rev in Fig. 21c is a R-gauge theory in n-dimensional space. When R = nRepG, R ⊗

Z1(R)
R rev is an nd G-gauge theory. When R describes a higher symmetry, R ⊗

Z1(R)
R rev is a higher gauge theory. But when R describes an algebraic higher symmetry, R ⊗

Z1(R)
R rev is something new, which is called a gauge theory from "gauging" the algebraic higher symmetry R in a product state. When C = R and γ = id, the resulting state R ⊗ Fig. 21b is a twisted R-gauge theory, obtained from "gauging" the algebraic higher symmetry R in a SPT state. When C = R, the resulting state R ⊗ Fig. 21a is a topological order obtained from "gauging" the algebraic higher symmetry R in the SET state characterized by data (R ι → C, γ).
Remark 5. We note that the algebra isomorphism µ : γ(A R ) A R C and the equivalence functor γ : R R C are similar data that are additional to γ : Z 1 (R) Z 1 (C). The fact that these additional data are not manifestly visible in the gauged theory, may suggest that they are fixed by γ up to certain natural higher structures (such as lower dimensional SPT or invertible phases). As an analogy, µ or γ are similar to the n-coboundaries generated by (n−1)-cochains that should be mod out when considering the nth cohomology. But the exact physical meaning of µ and γ is unclear to us for now, and will be left for future work.

A version of classification based on gauging the R-symmetry
We can also use the R-gauge theory R ⊗ Fig. 22 to obtain γ that "keeps the R part in Z 1 (R) unchanged", which leads to another version of classification. To make sense of the above statement, let us consider the natural functors from R and C to R ⊗ Here λ means mapping from the "left" boundary and ρ means mapping from the "right" boundary (ρ may not be monoidal here, but it is monoidal when restricted to a subcategory, as we will show later). The above gives two ways to map R into R ⊗ C rev , namely λ and ρ • ι. They correspond to observing the R symmetry from the left R boundary, and from the right C boundary as in Fig. 22. Thus we expect that λ and ρ • ι coincide. However, recall that in (91) γ is only required to preserve the "breakable" symmetry with respect to β : R → V.
Similarly, we only require the "breakable" symmetry to agree on left and right boundaries of the gauged theory. Let ker β be the preimage of the trivial excitation in V.
More precisely, if condensing A β gives β : R → V (A β consists of all the excitations that becomes trivial in V), ker β is the smallest fusion subcategory of R containing A β . ker β is then the "breakable" symmetry. The restriction λ| ker β and (ρ • ι)| ker β should agree. Besides, there is a natural "half-braiding" between the excitations from the left boundary and those from the right boundary. After mapped into the gauged theory: On the image λ(ker β) = ρ • ι(ker β), the above further defines a braiding: Such braiding makes ρ a monoidal functor when restricted to ι(ker β). We require the above braiding to be trivial, in the sense that there exists a braided monoidal functor λ(ker β) → nVec. These considerations lead to another version of classification (see Fig. 22): Proposition 19. Let R with β : R → V be a V-local fusion n-category. Anomaly-free gapped liquid phases in ndimensional space with an anomaly-free algebraic higher symmetry R are classified (up to invertible topological orders) by data (R ι → C, γ), where C is a fusion n-category that includes R (i.e. ι : R ι → C is a top-fully faithful functor), and γ : Z 1 (R) γ Z 1 (C) is a braided equivalence such that the following diagram is commutative (up to a natural isomorphism): and the braiding in the image λ(ker β) defined above is trivial.
When C = R, the above gives a classification of SPT phases with symmetry R. We would like to apply the above results to compute the Z 2 × Z 2 SPT phases in 1-dimensional space. This leads to new and deeper undertsanding of SPT order.
Next we examine the condensable algebras A R and A R . By definition, A R is the direct sum of anyons that maps to trivial under F R . It is easy to see that One can check that the overlap of A R and A R is (0, 0, 0, 0), which implies that R ⊗ M R rev = Vec. This result can be verified explicitly using the techniques developed in Ref. 61

and 62. Also
It is obvious that an automorphism preserving A R is the same as preserving the embedding R ι R → M, and also the same as preserving the bulk to boundary functor F R : M → R.

VII. EMERGENT LOW ENERGY EFFECTIVE ALGEBRAIC HIGHER SYMMETRY AND CATEGORICAL SYMMETRY
A. Emergent of categorical symmetry from energy scale separation In practical nd condensed matter systems, we usually have G 0-symmetry and the associated categorical symmetry M = Z 1 (nRepG) (which is also denoted as G ∨G (n−1) ). But we usually do not have higher symmetry and algebraic higher symmetries, unless we fine tune the lattice model (if we do not include dynamical electromagnetic field). 15 However, emergent algebraic higher symmetries and associated categorical symmetries can appear at low energies, if our models have energy scale separation. 15 So how to compute the emergent algebraic higher symmetries and the categorical symmetries? It turns out we just need to compute the emergence of categorical symmetries. The emergent algebraic higher symmetries can be determined from the emergent categorical symmetries directly later.
Let us consider a gapped liquid state in n-dimensional lattice. We assume the excitations in the gapped state has a large separation of energy scale. The low energy excitations (point-like, string-like, etc) are closed under fusion and form a fusion n-category C low . All other topological excitations have very high energies. Now we add interactions among those low energy excitations to drive phase transitions by condensing the low energy excitations, and to form gapless states, etc. We assume that in such process the high energy excitations remain to have higher energies. We would like to ask what are the possible phases and gapless states? Some constraints to the low energy physics come from the underlying symmetry, while other constraints come from the fusion and statistics of those low energy topological excitations. It looks hard to understand the effects of all those different constraints. But it turns out that the holographic point of view and the associated categorical symmetry can help us solve this issue.
We know that some excitations in C low are topological excitations, while others are charge objects of the underlying symmetry. To use the holographic point of view and to use categorical symmetry, we restrict to symmetric sub-Hilbert space of the underlying symmetry. In this case, every excitations in C low can be viewed as topological excitations in a hypothetical system without symmetry. However, the fusion n-category C low that describes those excitations is in general anomalous, i.e. it cannot be realized by a lattice system in the same dimension. But it can be realized as a boundary of a bTO M low = bulk(C low ) in one higher dimension (see eqn. (G6)). The bTO M low provides all the constraints to the low energy physics and solves our problem.
The excitations of the bTO M low is given by Hom(M, M) = ΣZ 1 (C low ), where Z 1 (C low ) is the emergent categorical symmetry. We see that the emergent categorical symmetry can be determined from the bTO.
We also see that the only input is the low energy excitations C low . So we do not need to have a lattice model. The above discussion remains valid for field theories without a given or known lattice regularization. (In this paper, we use the term field theory to mean theory without a given or known lattice regularization.) Thus Proposition 20. For a lattice system or a field theory with low energy excitations C low , the system has a low energy effective (i.e. emergent) categorical symmetry given by M low = Z 1 (C low ), that provides most constraints to the low energy physics. The system also has an emergent bulk-topological-order (bTO) M low = bulk(C low ), that provides all the constraints to the low energy physics.
Such a low energy effective categorical symmetry M low , or more precisely, the emergent bTO M low , is present even when low energy excitations condense, undergo phase transitions, etc, as long as all other higher energy excitations remain to have very high energies. The emergent bTO (or less precisely, categorical symmetry) controls all the low energy behaviors of the system, including allowed phases, allowed phase transitions, allowed critical points, etc. This is because the allowed phases, allowed phase transitions, allowed critical points, etc are one-toone correspond to different boundaries of the bTO. Such an emergent bTO and emergent categorical symmetry is the most practical and useful application of the notion of categorical symmetry and the holographic point of view. For example, Proposition 21. Consider a gapped liquid state in ndimensional space whose low energy energy excitations are described by a fusion n-category C low . When all other excitations remain to have higher energies, the gapped liquid phases formed by low energy energy excitations in C low must have excitations described by a fusion ncategory C that satisfy Z 1 (C) Z 1 (C low ). Actually, we have a stronger result, the gapped liquid phases formed by low energy energy excitations in C low must have excitations described by a fusion n-category C that satisfy bulk(C) bulk(C low ).
In fact, bulk(C) bulk(C low ) is nothing but the anomaly matching condition, since the bTO bulk(C) and bulk(C low ), as topological orders in one higher dimension, is the effective gravitational anomaly 3,25 , after we view the charge objects of the symmetry as topological excitations.

B. States with the full categorical symmetry
Since all the gapped liquid states in systems with an (emergent) categorical symmetry must spontaneously break part of the categorical symmetry, the states with the full unbroken categorical symmetry must be gapless. A system with a categorical symmetry M, may have many different symmetric gapless states. Those gapless states may have additional emergent categorical symmetry. So here we would like to ask, what is the minimal gapless state with the categorical symmetry M? To define the notion of " minimal gapless state" in n-dimensional space, we assume that the gapless excitations all have the same linear dispersion ω = vk. The low temperature specific heat of the gapless state has a form where (126) For a system described by a single gapless real scalar field, we find that c = 1. The minimal gapless state has minimal c.
From the above discussions, we see that minimal gapless states with the categorical symmetry M are actually minimal gapless boundary of topological order with excitations described by M in one higher dimension. Ref. 33,34, and 47 discussed how to obtain gapless boundaries for 2d topological orders, using modular covariant partition functions or topological Wick rotation. Those gapless boundaries do not break the categorical symmetry M. Those approach also allow us to obtain the minimal gapless boundaries with minimal central charge. However, for a given categorical symmetry, it is not clear whether its minimal gapless state is unique or not. 18

VIII. EXAMPLES
In the section, we discuss some gapped phases. In particular, we identify their algebraic higher symmetry and categorical symmetry. We also discuss low energy effective categorical symmetry when some topological excitations have low energies.

A. The category of 0d topological orders
The category of 0d topological orders TO 1 is the category of 0d gapped phases with no symmetry. In 0d, a stable gapped phase has non-degenerate ground state, which corresponds to a simple object in the category of 0d gapped phases, denoted as TO 1 . This is the only simple object in TO 1 , and is the unit object of stacking operation ⊗, which is the tensor product of vector spaces. We denote this unit object as 1. There are accidental degenerate ground states, which corresponds to a composite object 1 ⊕ 1 ⊕ · · · ⊕ 1 m copies = m1. In TO 1 , an 1-morphism from m1 to n1 is an n × m complex matrix M : m1 M − → n1. Such a fusion 1-category happen to be 1Vec. We see that TO 1 = 1Vec ≡ Vec.
B. 2d topological order described by Z2 gauge theory The 2d Z 2 topological order described by the Z 2 gauge theory is denoted by GT 3 Z2 . Codimension-2 excitations are described by the following braided fusion 1-category Ω 2 GT 3 Z2 , which has four simple objects (the point-like excitations): 1, e, m, f with the following Z 2 fusion rule (127) 1 is the trivial excitation. e, m, f are topological excitations which have mutual π-statistics between them. e, m are bosons, and f is a fermion. Such a topological order GT 3 Z2 can be realized by lattice models in the same dimension (see Ref. 5,63,and 64). Therefore, the bulk of GT 3 Z2 is the 3d product state, i.e. Bulk(GT 3 Z2 ) = 1 4 (see eqn. (G4)). This implies that Z 1 (ΩGT 3 Z2 ) = 2Vec, where ΩGT 3 Z2 is a fusion 2-category that describes the codimension-1 and codimension-2 excitations in the topological order GT 3 Z2 . The 2d topological order GT 3 Z2 has no categorical symmetry since Z 1 (ΩGT 3 Z2 ) = 2Vec.
Next, we consider the situation when e particles have low energies, and m, f particles have very high energies. The low energy excitations form a fusion 2-category 2RepZ 2 (after condensation completion), which simply describes 2d bosons with mod-2 conservation. In this limit, we have a low energy effective categorical symmetry characterized by the 3d Z 2 gauge theory GT 4 Z2 = bulk(2RepZ 2 ). 2RepZ 2 (a Z 2 symmetric phase) describes the excitations in one of the gapped boundary of 3d Z 2 gauge theory GT 4 Z2 , obtained by condensing the Z 2 -flux lines in GT 4 Z2 at the boundary. The 3d Z 2 gauge theory GT 4

Z2
has another gapped boundary whose excitations are described by the fusion 2-category 2Vec Z2 (a Z 2 symmetry breaking phase with e boson condensation), obtained from condensing the Z 2 charges in GT 4 Z2 at the boundary. When e bosons have low energies, our system has only this two gapped phases. The continuous phase transition between the two gapped phases is described by a critical point which has the full categorical symmetry characterized by Ω 2 GT 4 Z2 (the braided fusion 2-category that describes the excitations in the 3d Z 2 gauge theory). This critical point is the same as the critical point of 2d quantum Ising model (or 3d statistical Ising model), which has the same categorical symmetry Ω 2 GT 4 Z2 , as discussed in Ref. 18.
Last, we consider the situation when f particles have low energies, and e, m particles have very high energies. The low energy excitations form a fusion 2-category 2sVec, which simply describes 2d fermions with mod-2 conservation. There is a Z f 2 symmetry from the mod-2 conservation of the fermions. In this limit, we have a low energy effective categorical symmetry characterized by 3d twisted Z 2 gauge theory with fermionic Z 2 charge, denoted by GT 4 is different from the categorical symmetry Ω 2 GT 4 Z2 discussed above. So when f fermions have low energies, our system has different properties from when e bosons have low energies.
When f fermions have low energies, our system can have 16 gapped phases (up to E 8 2d bosonic invertible topological order) labeled by α ∈ Z 16 , which correspond to 2d fermionic invertible topological orders. The continuous transition between α and α + 1 phases is described by the following 2d non-interacting Majorana fermion theory: 65,66 The transition happens when m change sign, which change the chiral central charge of the edge state by 1/2. 65,66 The gapless state at m = 0 have the full categorical symmetry Ω 2 GT 4 C. 3d topological order described by Z2 gauge theory The 3d Z 2 topological order GT 4 Z2 (described by the Z 2 gauge theory) has codimension-2 and codimension-3 excitations described by the braided fusion 2-category Ω 2 GT 4 Z2 : The simple objects (the string-like excitations) are labeled by 1 s , m s , e s , m s ⊗e s , with the following symmetric fusion 1 s is the trivial string. m s is a bosonic topological stringlike excitation, that corresponds to the Z 2 -flux string. The simple 1-morphisms (the point-like excitations), that connect 1 s → 1 s , are labeled by 1 p , e p , with the following Z 2 fusion e p ⊗ e p = 1 p .
1 p is the trivial particle. e p is a bosonic topological excitation with trivial mutual statistics. However, e p and m s has a non-trivial mutual π-statistics between them. We also have simple 1-morphisms that connect m s → m s , which are labeled by 1 ms , e ms with the following Z 2 fusion e ms ⊗ e ms = 1 ms .
They correspond to the point-like excitations on the string m s . The e s string mentioned above is a descendent excitation, formed by condensing e p point-like excitations along the string. Since e p has a mod 2 conservation, the e p condensed state is a spontaneously Z 2 symmetry breaking state. This leads to the fusion role e s ⊗ e s = 2e s . Such a GT 4 Z2 topological order has a trivial categorical symmetry since Bulk(GT 4 Z2 ) = 1 5 or Z 1 (ΩGT 4 Z2 ) = 3Vec (where ΩGT 4 Z2 describes the excitations in GT 4 Z2 ). However, when some excitations have low energy and other have high energies, the system may have a low energy effective categorical symmetry.
When e p particles have low energies and m s strings have very high energies, the low energy excitations are described by a fusion 3-category 3RepZ 2 generated by e p particles. In this limit, the low energy effective categorical symmetry is Z 1 (3RepZ 2 ) = Ω 2 GT 5 Z2 , which is nothing but the (codimension-2 excitations of) 4d Z 2 gauge theory. Such a categorical symmetry has two gapped phases: The 3d Z 2 gauge theory GT 4 Z2 with excitations Ω 2 GT 4

Z2
and the e p condensed phase or the trivial phase with excitations 2Vec. The transition between the phases is the Higgs transition. The critical point for the Higgs transition has the full categorical symmetry Ω 2 GT 5 Z2 . Such a critical point is the same as the critical point in 3d quantum Ising model or 4d statistical Ising model, which is described by non-interacting massless real scaler field When m s strings have low energies and e p particles have very high energies, the low energy excitations are described by a fusion 3-category 3Rep(Z 2 ) has only a single trivial object, two simple 1morphisms: trivial string 1 s and Z 2 flux string m s , and a single trivial 2-morphism. In this limit, the low energy effective categorical symmetry is describes the excitations in the 4d Z 2 2-gauge theory obtained by gauging Z (1) 2

1symmetry. The 4d Z
(1) 2 2-gauge theory has a string-like Z 2 charge and string-like Z 2 flux. The Z 2 string-charge and the Z 2 string-flux has mutual π-statistics. Such a categorical symmetry has two gapped phases: the 3d Z 2 gauge theory with excitations Ω 2 GT 4 Z2 and the m s condensed phase or the trivial phase with excitations 2Vec. The transition between the two phases is the confinement transition. The critical point for the confinement transition has the full categorical symmetry Ω 2 GT 5 . Such a critical point is different from the Higgs transition critical point which has a categorical symmetry Ω 2 GT 5 Z2 (for details, see Ref. 18).

D. 3d topological order described by twisted Z2
gauge theory The 3d topological order described by the twisted Z 2 gauge theory (i.e. 3d Z 2 gauge theory with fermionic point-like Z 2 charge) is characterized by the braided fusion 2-category Ω 2 GT 4 Z f 2 , which is similar to Ω 2 GT 4 Z2 , except now e p is a fermion. Many results discussed above remain unchanged. In particular, when the Z 2 flux strings have low energies and Z 2 point charges have high energies, the system have an low energy effective categorical symmetry Ω 2 GT 5 Z (1) 2 as discussed above. But when Z 2 point charges have low energies and the Z 2 flux strings have high energies, the system has a very different behavior since the Z 2 point charges are fermions. In this limit, the low energy excitations are described by fusion 3-category 3Rep(Z f 2 ) (with trivial object, trivial 1morphism, and Z f 2 2-morphisms which contain fermions). The categorical symmetry is Ω 2 GT 5 2 )) (describing excitations in the 4d Z 2 gauge theory with fermionic Z 2 point charge).
There is no 3d fermionic invertible topological order. So there is only one minimal gapped state that break the categorical symmetry Ω 2 GT 5 (2) two gapped states differ by a stacking of product state are equivalent, where the degrees of freedom of the product state may have a non-uniform but bounded density. If there is no symmetry, the local unitary transformation has no symmetry constraint, and the corresponding gapped liquid phases of local bosonic or fermionic systems are topological orders [67][68][69] . In the presence of finite internal symmetry, the local unitary transformation is required to commute with the symmetry transformations, and the corresponding gapped liquid phases include spontaneous symmetry breaking orders, symmetry protected trivial (SPT) orders, 2,54,70 symmetry enriched topological (SET) orders 41,46,[71][72][73][74][75][76][77][78] .
In this paper, we only consider bosonic systems with finite internal symmetries. We do not consider spacetime symmetries (such as time reversal and translation symmetries), nor continuous symmetry (such as U (1) symmetry). So in this paper, when we refer symmetry, we only mean finite internal symmetry.
We would like to remark the above definition has an important feature. A gapped liquid phase with some noninvertible topological excitations on top of it is not a gapped liquid phase according to the definition. (The notion of non-invertible topological excitations is defined in the next section.) We note that the Hamiltonian here may not have translation symmetry. Thus it is hard to tell if the ground of a Hamiltonian has excitation in it or not. Using our above definition, a gapped liquid state is a ground state of a Hamiltonian that has no non-invertible topological excitations. However, a gapped liquid state may contain invertible topological excitations. In fact, two gapped liquid states differ by invertible topological excitations are very similar, and both can be viewed as proper ground states.
To see the above point, let us start with a gapped state of N sites with a topological excitation in the middle. We may double the system size by stacking a product state of N sites to the left half of the system, or to the right half of the system. Both operations are equivalence relations, and the resulting states of 2N sites should be equivalent, i.e. be connected by a finite-depth local unitary transformation. However, in the presence of the non-invertible topological excitation, the excitation appears at left 1/4 or right 1/4 of the enlarged systems (see Fig. 23). Such two enlarged systems are not connected by finite-depth local unitary transformations, which can only move the non-invertible topological excitation by a finite distance. Thus a gapped liquid state with some non-invertible topological excitations can non-longer be viewed as a gapped liquid state.
However, a gapped liquid state with some invertible topological excitations can still be viewed as a gapped liquid state. This is because finite-depth local unitary transformations can move invertible topological excitations by a large distance across the whole system. Thus the gapped liquid phases defined above may contain some invertible topological excitations.
Ref. 24 and 25 outline a description of topological orders (i.e. without symmetry) in any dimensions, via braided fusion higher categories. Here, we would like to review and expand the discussions in Ref. 24 and 25. We would like to remark that the needed higher category theory is still not fully developed. So our discussion here is just an outline. We hope that it can be a blue print for further development. However, our discussions become rigorous at low dimensions (such as 1d and 2d).

Appendix B: Trivial, local, and topological excitations
The reason that gapped liquid phases can be described by higher categories is that higher category is the natural language to describe excitations within a topological order, as well as domain walls between different gapped liquid phases. To understand this connection, let us define excitation more carefully. We consider a gapped liquid state, which is the ground state of a local Hamiltonian H. As discussed in last section, a gapped liquid state does not contain any non-invertible topological excitations. To define excitations in H, for example, to define string-like excitations, we can add several trap Hamiltonians δH str (W 1 a ), labeled by a, to H such that H + a δH str (W 1 a ) has an energy gap. δH str (W 1 a ) is only nonzero along the string W 1 a . Here we require δH str (W 1 a ) to be local along the string W 1 a . δH str (W 1 a ) can be any operator, as long as its acts on the degrees of freedom near the string W 1 a . The ground state subspace V fus (W 1 1 , W 1 2 , · · · ) of H + a δH str (W 1 a ) (where is also called the fusion space) corresponds to string-like excitations located at W 1 1 , W 1 2 , etc(see Fig. 24). If the ground state subspace is stable (i.e. unchanged) against any small change of δH str (W 1 a ), then we say the correspond string on W 1 a is a simple excitation (or a simple morphism in mathematics). If the ground state subspace has accidental degeneracy (i.e. can be split by some small change of δH str (W 1 a ), see Fig. 25), then we say the correspond string on W 1 a is a composite excitation (or a composite morphism in mathematics). A composite excitation I is a direct sum of several simple excitations (B1) In other words, I can be viewed as an accidental degeneracy of excitations i, j, · · · . We see that different stringlike excitations can be labeled by different trap Hamiltonians δH str (W But the above definition give us too many different strings, and many of different strings actually have similar properties. So we would like to introduce a equivalence relation to simplify the types of strings. We define two strings labeled by δH str (W 1 ) and δ H str ( W 1 ) as equivalent, if we can deform δH str (W 1 ) into δ H str ( W 1 ) without closing the energy gap. The equivalence classes of the strings define the types of the strings. We would like to point out that if W 1 is an open segment, the corresponding string is equivalent to the trivial string 1 s described by δH str (W 1 ) = 0, since we can shrink the string along W 1 to a point without closing the gap. We would like to remark that if the two excitations are defined on a closed sub-manifold, then we can define their equivalence by deforming their trap Hamiltonians into each other in the space of local trap Hamiltonians without closing the energy gap. The above definition is more general, since the local-unitary transformations and stacking of product states can be applied to a part of the sub-manifold that support the excitations, and we can examin if the two excitations turn into each other on the part of the sub-manifold.
Definition 10. An excitation is trivial if it is equivalent to the null excitation defined by a vanishing trap Hamiltonian.
Definition 11. An excitation is invertible if there exists another excitation such that the fusion of the two excitations is equivalent to a trivial excitation.
The above equivalence relation can also be phrased in a way similar to Def. 7: Proposition 22. A type of excitations is an equivalence class of gapped ground states with added trap Hamiltonian acting on a m-dimensional subspace W m , under the following two equivalence relations: (1) two gapped states connected by a finite-depth local unitary transformation acting on the subspace W m are equivalent; (2) two gapped states differ by a stacking of product states located on the subspace W m are equivalent.
We see that, when m > 0, the excitations defined above are gapped liquid state on the sub space W m , and there is no lower dimensional non-invertible excitations on W m .
We also would like to introduce the notion of non-local equivalence and non-local type: Definition 12. Two excitations are non-locally equivalent (i.e. are of the same nl-type) if they can be connected by non-local-unitary transformations and by stacking of product states.
Definition 13. An excitation is local if it has the same nl-type as the null excitation.
We see that a trivial excitation is always a local excitation. But a local excitation may not be a trivial excitation.

Definition 14. An excitation is topological if it is nonlocal.
Again, the above non-local equivalence relation can also be phrased in a way similar to Def. 7: Proposition 23. A nl-type of excitations is an equivalence class of gapped states with added trap Hamiltonian acting on a m-dimensional subspace W m , under the following two equivalence relations: (1) two gapped states connected by a non-local unitary transformation acting on the subspace W m are equivalent; (2) two gapped states differ by a stacking of product states located on the subspace W m are equivalent.
We also believe that Proposition 24. Two excitations have the same nl-type if and only if they can be connected by gapped or gapless domain walls. We note that the morphisms in higher category only correspond to gapped domain walls.
We would like to remark that for point-like excitations the notion of type and nl-type coincide.
Those different concepts of excitations were discussed in Ref. 25, where the nl-type was called elementary type, and topological excitation was called elementary topological excitation. The local excitation was called descendant excitation in Ref. 25.
To illustrate the above concepts that we just introduced, let us consider a 2d Z 2 topological order 63,64 for bosons described by 2+1D Z 2 gauge theory.
Example 2. The Z 2 topological order has four types of point-like excitations, labeled by 1, e, m, f , where e is the Z 2 charge, m is the Z 2 vortex, and f is a fermion -the bound state of e and m. 1 is a trivial point-like excitation. The Z 2 topological order also has four nltypes of point-like excitations, labeled by 1, e, m, f . 1 is a local point-like excitation, and e, m, f are topological point-like excitations.
The Z 2 topological order has only one nl-type of string-like excitations, which is a local string-like excitation. The Z 2 topological order has six types of string-like excitations, generated by 1 s , e s , m s , f s , with two additional types given by f s ⊗ m s = e s ⊗ f s and m s ⊗ f s = f s ⊗ e s : The e s -type of string-like excitation is formed by the eparticles, condensing into a 1d phase of spontaneous Z 2 symmetry breaking state. We note that the e-particles have a mod-2 conservation, and an emergent Z 2 symmetry. Similarly, the m s -type of string-like excitation is formed by the m-particles, condensing into a 1d phase of spontaneous Z 2 symmetry breaking state. The f stype of string-like excitation is formed by the f -particles, condensing into a 1d topological superconducting phase Next we consider a 3d trivial product state of bosons.
Example 3. Such a state has trivial nl-type of point-like, string-like, and membrane-like excitations, i.e. all excitations are local. It also has trivial type of point-like and string-like, but it has non-trivial types of membrane-like excitations. In fact those non-trivial types of membranelike excitations corresponding to 2d topological orders. Thus there are infinite types of membrane-like excitations corresponding to infinite different 2d topological orders, even though the 3d state has trivial topological order and is a trivial product state of bosons. All those membrane-like excitations are local but not trivial.
We see that to have a complete macroscopic description of trivial product state of bosons in n-dimensional space without symmetry, we need to classify (n − 1)d topological orders of bosons, which corespond to types codimension-1 excitations. We also see that to have a complete macroscopic description of nd topological order of bosons without symmetry, we need to classify (n − 1)d SET orders of bosons/fermions with symmetries (i.e. the emergent symmetry).
Appendix C: Trivial topological order (the product states) and its excitations In the last section, we see that the types of dimensionk excitations in a trivial product state in n-dimensional space correspond to topological orders (gapped liquid phases) in k-dimensional space. Thus the study of the trivial topological order and its excitations of various dimensions allows us to understand topological orders in spatial dimensions less then n. This motivated us to develop a comprehensive theory of trivial topological order.
All trivial topological orders are product states and all product states belong to one phase, if there is no symmetry. We denote the trivial topological order in ndimensional space as 1 n+1 (n + 1 is the spacetime dimension). 1 n+1 is also referred as an object. Once we have the trivial topological order 1 n+1 , we also have accidental degeneracy of several 1 n+1 's (i.e. several product states). We denote a gapped liquid phase formed by m degenerate product states as 1 n+1 ⊕ · · · ⊕ 1 n+1 m copies = m1 n+1 . So, after the completion, the collection of trivial topological orders has objects m1 n+1 . We refer 1 n+1 as simple object, while m1 n+1 (m > 1) as composite object. We see that the composite object does not correspond to a stable phase, since the accidental m-fold degeneracy can be easily split by local perturbations in the Hamiltonian.
The collection of trivial topological orders in (n+1)D spacetime, {m1 n+1 }, is a set. However, the objects in the set have many relations. Two objects can be connected by a gapped codimension-1 domain wall a : m 1 1 n+1 → m 2 1 n+1 , which is called an 1-morphism. For example, an 1-morphism a : 21 n+1 → 31 n+1 can be represented as Physically, it means that there is a gapped domain wall between the first product state in 21 n+1 = 1 n+1 1 ⊕ 1 n+1 2 and the second product state in 31 n+1 = 1 n+1 , and such a gapped domain wall is not degenerate. We denote such a gapped domain wall as ( 1 n+1 ). All other domain walls between different product states have higher energy density or gapless. In this paper, we do not consider gapless domain walls and we always assume gapless domain walls have infinite energy density.
We can have another 1-morphism b : Physically, it means that there is a gapped domain wall between the first product state in 21 n+1 and the second product state in 31 n+1 , and such a gapped domain wall are 2-fold degenerate. So we express the gapped domain wall as ( 1 n+1 ). The most general 1-morphism c : 21 n+1 → 31 n+1 has a form where m k ij ∈ N. Here, for example, ( 1 n+1 ) denote a gapped domain wall between the first product state in 21 n+1 and the second product state in 31 n+1 , and k labels different types of accidentally degeneracte gapped domain wall between the two product states. m k ij is the accidental degeneracy of the domain walls of the same type k. We see that an 1-morphism is like a matrix that can also be added.
In particular, a 1-morphism k : 1 n+1 → 1 n+1 , denoted by ( 1 n+1 k 1 n+1 ), is the codimension-1 excitation discussed in the last section, where k labels the different types of the excitations, as defined in Def. 9. Such an excitation corresponds to a topological order in (n−1)-dimensional space. We use Hom(1 n+1 , 1 n+1 ) denote the collection of all morphisms from 1 n+1 to 1 n+1 , which happen to the collection of all topological orders in (n−1)-dimensional space. We would like to remark that Hom(1 n+1 , 1 n+1 ) is also complete in the sense that it not only contain stable topological orders, its also contain accidental degeneracy of topological orders. In other words, if a, b ∈ Hom(1 n+1 , 1 n+1 ), then the accidental degeneracy of a and b, a ⊕ b, is also in Hom(1 n+1 , 1 n+1 ). Thus just like the collection of trivial topological orders {m1 n+1 }, Hom(1 n+1 , 1 n+1 ) is also closed under the "degeneracy" operation ⊕.
Two codimension-1 topological orders a, b may also be connected by a gapped domain wall of codimension-2: k : a → b (see Fig. 26). We call k a 2-morphism. To be precise, here, the "domain wall" really means types of domain walls. We regard two domain walls as equivalent if they differ only by local unitary transformations and local addition of product states on the wall. The collection of 2-morphisms from a to b is denoted as Hom(a, b). We see that the collection of 2-morphisms from 1 n to 1 n , Hom(1 n , 1 n ), is the collection of condimension-2 excitations, which are also topological orders in (n−2)dimensional space. Such a collection also contain a product state (trivial topological order) in (n−2)-dimensional space, denoted as 1 n−1 ∈ Hom(1 n , 1 n ). Also, the collection of 2-morphisms from a to a, Hom(a, a), is the collection of condimension-2 excitations on the codimension-1 excitation a. We would like to remark that it is possible that Hom(a, b) = 0, which means that there is no gapped domain wall between a and b. Here 0 denotes the "zero" category, which can be roughly thought as the linearized and categorified version of the empty set.
The above discussion can be continued. This allows us to define 3-morphisms, 4-morphisms, etc. The nmorphisms correspond to codimension-n or dimension-0 (i.e. point-like) excitations. The point-like excitations are world lines in spacetime. The domain wall on world-lines are (n+1) mophisms. In general, a pointlike excitation p (an n-morphism) may have degenerate ground states (of the Hamiltonian with traps). We denote the vector space of the degenerate ground states as V fus (p, · · · ), where · · · represent other excitations which are fixed. Then a (n+1)-morphism o from one pointlike excitations p 1 to the other p 2 (where the two excitations are near each other) is a linear operator acting near p 1 and p 2 from V fus (p 1 , · · · ) and V fus (p 2 , · · · ): V fus (p 1 , · · · ) o → V fus (p 2 , · · · ). We denote such a (n+1)morphism as o : Just like the objects (also called 0-morphisms), the morphisms also can be divided into two classes: the simple morphisms (which correspond to stable excitations whose ground state cannot be split by any local perturbations near the excitations) and composite morphisms (which correspond to unstable excitations with accidental degeneracy).
The objects (i.e. the 0-morphisms), as well as mmorphisms can also fuse or compose. Let a, b, c be three (m − 1)-morphisms, and k ∈ Hom(a, b) and l ∈ Hom(b, c) are two m-morphisms. Then, a composition of k and l is given by a m-morphism from a to c: Hom(a, c). The subscript b indicates that k and l are fused together via the "glue" b (see Fig. 26). The picture Fig. 26 also has a dual representation Fig.  27 In the above, we discussed excitations of various dimensions in a trival topological order. We may reverse the thinking and use all the excitations to characterize the trival topological order, or more generally, a nontrivial topological order. This is equivalent to using higher categories to characterize topological orders or trivial orders. However, in order to use excitations to describe topological orders or trivial orders, the first issue one faces is weather to use type or use nl-type of excitations to construct higher categories. The notions of type and nl-type were discussed in Ref. 24 and 25. In physics, when we refer topological excitations, we usually mean the nl-types of excitations, which seems to suggest using nl-type to construct higher category. However, in mathematics, it is more natural to use types of excitations to build the higher categories that describe topological orders. 31 In some sense, nl-types are like the basis vectors in a vector space. The completion under "+" give rise to all the types which form the whole vector space. In higher category theory, such a completion corresponds to condensing the nl-types of exciations to construct all the types of excitations. The process of adding all the types of excitations in a category is called condensation completion which is discussed in Ref. 31. (Note that the condensation completion in Ref. 31 only includes defects that correspond to gapped liquid phases that have gappable boundaries.) In this paper, we do a more general condensation completion that includes all the descendent excitations that correspond to all possible gapped liquid phases. In other words, we use types of all excitations to build the higher categories.
In n-dimensional space, the trivial topological order has dimension-(n − 1) excitations, dimension-(n−2) excitations, etc. Those excitations can fuse (the ⊗ operation) and can have accidental degeneracy (the ⊕ operation). The excitations can also have domaon walls between them (the morphisms). The collection of excitations, plus those extra structures form a fusion ncategory, which is denoted as Hom(1 n+1 , 1 n+1 ).
The precise definition of a fusion n-category is difficult to write down due to the lack of a universally accepted and well developed model of weak n-categories. But this is not the only problem. Recently, by ignoring this problem, Johnson-Freyd managed to solve other important problems and provided a workable definition in Ref. 28. Due to its complexity, we choose to not to give Johnson-Freyd's definition, but to provide a rough and physical intuition underlying the definition instead.
Definition 16. A fusion n-category is an n-category, which is • C-linear: the n-morphisms are required to form complex vector spaces, • additive (with ⊕ operation); • monoidal: with fusion ⊗ operation, which is compatible with the C-linear and additive structures; • semi-simple (all composite object x has a unique decomposition x = a ⊕ b · · · ) and the tensor unit is simple; • condensation complete 31,40 : the 0-morphisms (the objects), 1-morphisms, 2-morphisms, etc include all the decedent excitations; and satisfies certain fully dualizable condition amounts to the invariance of the physical reality by deforming and folding of the associated topological order.
Since the excitations in a trivial topological order is surrounded by product states, we can add more product states to form a higher dimensional trivial topological order, and view the same excitations as excitations in a higher dimension trivial topological order. In fact, we can view any excitations in a trivial topological order as excitations in an infinite dimensional trivial topological order. So we can always braid the excitations in a trivial topological order by viewing the excitations as in an infinite dimensional trivial topological order. Since the excitations in an infinite dimensional trivial topological order have trivial braiding and exchange properties, those excitations form a symmetric fusion higher category with trivial exchange properties. Thus

25.
The fusion n-category Hom(1 n+1 , 1 n+1 ), that describes the excitations in a trivial topological order 1 n+1 in n-dimensional space, can always be promoted into a braided fusion n-category.
In fact, such a braided fusion n-category is a symmetric fusion n-category with trivial exchange properties.
Appendix D: The category of anomaly-free topological orders Definition 17. An anomaly-free topological order is a gapped liquid phase that can be realized by a local bosonic lattice models in the same dimension.
In a trivial topological order 1 n+2 (n+1)-dimensional space, a type of codimension-1 excitation correspond to an anomaly-free topological order in n-dimensional space. This because we can remove the product state around a codimension-1 excitation and view it as an anomaly-free topological order. Thus Hom(1 n+2 , 1 n+2 ) is the set of anomaly-free nd topological orders. Those nd topological orders have excitations on them and have domain walls between them, which correspond to morphisms. They can also fuse ⊗ and have accidental degeneracies

Concepts in higher category
Concepts in physics (n + 1)Vec (a symmetric fusion (n+1)category, also denoted as TO n+1 af ) The collection of all nd anomaly-free topological orders, which can all be realized by bosonic lattice model in the same dimension. A simple object (0-morphism) of TO n+1 af A topological order (with non-degenerate ground state on S n ). ⊕. Besides, we include all descendant excitations (condensation completion). Adding those structures to the set Hom(1 n+2 , 1 n+2 ), we make it into a fusion (n+1)category (see Table II).
Definition 18. The category of anomaly-free topological orders in n-dimensional space, is the fusion (n+1)-category given by Hom(1 n+2 , 1 n+2 ) where 1 n+1 is a product state in (n+1)-dimensional space. We denote the category of anomaly-free topological orders as TO n+1 af . In the above, we have defined anomaly-free topological orders via a microscopic approach, since we used the notion of product states and condensing low dimensional excitations to construct descendant excitations. Can we define anomaly-free topological orders using only the macroscopic notions? Here we would like to point out that the anomaly-free topological orders have a defining macroscopic property called the principle of remote detectability: Proposition 26. A topological order is anomaly-free if and only if any excitations of non-trivial nl-type can be detected remotely (such as via braiding) by some other excitations.
Here the nl-type also has a defining macroscopic property Proposition 27. Two excitations have the same nl-type if and only if they can be connected by gapped or gapless domain walls.
The gapped domain walls are the morphisms that we have discussed, while the gapless domain walls are not included in our discussion. Also the notion of "detecting remotely" is not defined carefully. This reveals the difficulty to define anomaly-free topological order macroscopically beyond 2-dimensional space. The above just points out a possible direction.
The stacking of two (stable) anomaly-free topological orders M n+1 : Since a commutative monoid may have a submonoid which is actually an Abelian group, anomaly-free topological orders have a subset of invertible topological orders, [25][26][27] which form an Abelian group under the stacking ⊗. All the invertible topological orders in ndimensional space also form a fusion (n+1)-category denoted as TO n+1 inv .
Definition 19. The category of anomalous topological orders with a fixed anomaly in ndimensional space, is the linear (n+1)-category given by Hom(1 n+2 , M n+2 ) where 1 n+2 is the unit object in TO n+2 af and M n+2 is a simple object in TO n+2 af . It is a right module over fusion (n+1)-category Hom(1 n+2 , 1 n+2 ), a left module over fusion (n+1)category M n+1 = Hom(M n+2 , M n+2 ) and thus a bimodule. Here M n+2 is an anomaly-free topological order in (n+1)-dimensional space, which determine the anomaly. We denote the category of anomalous topological orders with anomaly M n+2 as TO n+1 M n+2 . Such a category describes all the gapped boundaries of the anomalyfree topological order M n+2 .
In fact, the category of anomalous topological orders with anomaly M n+2 is the category of all gapped boundaries of an anomaly-free topological order M n+2 . Physically, a boundary includes its neighbor which is part of bulk. We see that the data that describes the category of all gapped boundaries include a linear (n+1)-category B n+1 and a fusion n + 1-category M n+1 that acts on B n+1 .

Appendix F: Looping and delooping
From an n-category we can construct a fusion (n−1)category via a process called looping (see Fig. 27 where the morphisms are viewed as paths and loops): Definition 20. Given an n-category C, we choose an object a (the "base point"). We can construct a fusion (n−1)-category, denoted as Ω a C, whose objects are given by the morphisms k : a → a. In other words Ω a C = Hom(a, a). If C is a fusion n-category, we usually choose the base point to be the unit of fusion 1 C : ΩC = Hom(1 C , 1 C ), and ΩC becomes a braided fusion ncategory.
To apply the looping to a physical situation, let us consider a single simple object C n+1 in TO n+1 M n+2 , which corresponds to an nd gapped boundary of an anomalyfree topological order M n+2 in (n+1)-dimensional space. C n+1 is also an nd anomalous topological order. For special case M n+2 = 1 n+2 , C n+1 becomes an nd anomalyfree topological order. Hom(C n+1 , C n+1 ) is the collection of (n − 1)d excitations on the boundary. Here we include the morphisms, as well as the ⊗ and ⊕ operations to view Hom(C n+1 , C n+1 ) as a fusion n-category, denoted by C n = Hom(C n+1 , C n+1 ). Thus C n describe all the codimension-1, codimension-2, etc excitations on the nd boundary C n+1 The unit object under ⊗ in C n is denoted as 1 C n = id C n+1 , which is the trivial codimension-1 excitations in C n+1 . Then the looping ΩC n = Hom(1 C n , 1 C n ) is a fusion (n−1)-category, which describes the codimension-2 excitations on the nd boundary C n+1 . Those excitations can also braid and ΩC n is in fact a braided fusion (n−1)category.
We see that the looping of a fusion category C n is obtained by dropping the objects (the codimension-1 excitations) and include only the morphisms of the trivial object (the codimension-2 excitations). The looping operation can be continued, and the commutativity increases. For example ΩΩC n ≡ Ω 2 C n is a symmetric fusion (n−2)-category, etc.
There is reverse process of looping, called delooping (see Fig. 27). From a fusion n-category, we can construct a (n+1)-category via delooping: Definition 21. Given an fusion n-category C, we can construct a (n+1)-category, denoted as Σ * C, which has only one object * and whose morphisms are given by the objects of C. In other words, Hom( * , * ) = C. We can complete Σ * C by adding the composite objects * ⊕ * · · · , to obtain a linear (n+1)-category with ⊕ operation. We can also do a condensation completion, by adding objects (the gapped liquid phases) formed by the codimenson-1 excitations (i.e. the 1-morphisms of Σ * C), the codimenson-2 excitations (i.e. the 2-morphisms of Σ * C), etc. The resulting (n+1)-category is denoted as ΣC.
As an application to physics, let us consider a braided fusion (n−1)-category C n−1 that describes the codimension-2 and higher excitations in n-dimensional space. Then the delooping Σ 1 n C n−1 is the fusion ncategory with only one object 1 n , which correspond to the trivial codimension-1 excitation in the n-dimensional space. We can do a condensation completion by adding 1 n ⊕ 1 n · · · , as well as all the descendant codimension-1 excitations, obtained from condensing codimension-2 and higher excitations. The resulting fusion n-category is ΣC n−1 . If we can add a braiding structure to ΣC n−1 , making it a braided fusion n-category, then the delooping plus condensation completion can be continued. ΣΣC n−1 = Σ 2 C n−1 is a fusion (n+1)-category.
We note that excitations in a trivial topological order 1 n+1 in n-dimensional space are described by a fusion ncategory Hom(1 n+1 , 1 n+1 ). It contains (n − 1)d, (n−2)d, · · · , 0d excitations. If we drop the (n − 1)d excitations, the remaining (n−2)d, · · · , 0d excitations correspond to excitations in trivial topological order 1 n in (n−1)dimensional space, and are described by Hom(1 n , 1 n ). This way we find ΩHom(1 n+1 , 1 n+1 ) = Hom(1 n , 1 n ). (F1) All the excitations in trivial topological order are descendent excitations. Thus if we add one layer of descendent excitations in one higher dimension, we obtain excitations of a trivial topological order in one higher dimension. Therefore we have ΣHom(1 n , 1 n ) = Hom(1 n+1 , 1 n+1 ). (F2) We note that the codimension-1 excitations in a trivial topological order is embeded in a product state in 1 higher dimension. We can also view the same excitation as embeded in a product state in 2 higher dimension.
In this case, the excitation becomes codimension-2 and can braid. Thus Hom(1 n+1 , 1 n+1 ) can also be viewed as a braided fusion n-category, and we can perform delooping. In fact, the braiding is trivial, and Hom(1 n+1 , 1 n+1 ) can be viewed as a symmetric fusion n-category.
Since Hom(1 n+2 , 1 n+2 ) = TO n+1 af is the category of anomaly-free topological orders in n-dimensional space, we find that We note that in 0-dimensional space, the category of anomaly-free topological order has only one simple object 1 1 , which corresponds to a single quantum state |ψ . The set of 1-morphisms Hom(1 1 , 1 1 ) = C is the set of 1-by-1 complex matrices. We see that the category of anomalyfree topological orders in 1-dimensional space is given by Vec -the category of complex vector spaces: Since the higher category of the vector spaces is given by delooping plus condensation completion we see that the category of anomaly-free topological orders is given by We would like to remark that the fusion (n+1)-category TO n+1 af = Hom(1 n+2 , 1 n+2 ) = (n + 1)Vec can also be promoted into a braided fusion (n+1)-category, which is actually, a symmetric fusion (n+1)-category with trivial exchange property (see Prop. 25). After we promote (n+1)Vec to a braided fusion (n+1)-category, we can denote it as (n+1)Vec. In other words, (n+1)Vec is the braided fusion (n+1)-category obtained from the fusion (n+1)-category (n+1)Vec by adding the trivial braiding structure (which is always doable).
Consider an anomaly-free topological order M n+1 ∈ TO n+1 af . Its excitations are described by a fusion ncategory M n = Hom(M n+1 , M n+1 ). The objects in M n are codimension-1 excitations, which cannot braid and cannot be remotely detected by any excitations. Thus, according to Prop. 26, those codimension-1 excitations must all have the trivial nl-type. We believe that Proposition 29. all the excitations with the trivial nltype are descendant excitations, coming from the condensations of lower dimensional excitations.
Thus, the codimension-1 excitations in an anomaly-free topological order are all descendant excitations. Droping those codimension-1 excitations gives us the looping ΩM n . The delooping of ΩM n addes back those descendant codimension-1 excitations. We find Proposition 30. The excitations in an anomaly-free topological order described by fusion n-category M n satisfy the following relation The reverse may not be true: a fusion n-category M n satisfying M n = ΣΩM n may not describe the excitations in an anomaly-free topological order.
that the pair (C n+1 , M n ) uniquely determines the bulk topological order and how the bulk topological order is connected to the boundary. If we ignore the information about how the bulk is connected to the boundary, we believe that C n+1 ∈ Hom(1 n+2 , M n+2 ) uniquely determines the bulk topological order: Proposition 33. There is only one anomaly-free topological order M n+2 in TO n+2 af , which gives rise to the boundary We denote such boundary-bulk relation as The above is the accurate meaning of boundary uniquely determines bulk. C n can determine the boundary topological order C n+1 up to an invertible topological order. Since we believe that all invertible topological orders are anomaly-free, the excitations C n = Hom(C n+1 , C n+1 ) in the boundary topological order C n+1 can already determine the bulk topological order M n+2 . We obtain Proposition 34. For any fusion n-category C n , there is only one anomaly-free topological order M n+2 in TO n+2 af admitting a boundary C n+1 ∈ Hom(1 n+2 , M n+2 ) such that We denote such boundary-bulk relation as Here, we have assumed the following.
Proposition 35. A fusion n-category C n can always be realized by the excitations in a potentially anomalous topological order C n+1 such that C n = ΩC n+1 .
The above result was shown for n = 1 case. Given a fusion category C, we can explicitly construct a 2d stringnet liquid state, 80 that has a boundary realizing the fusion category C. 81 For n > 1, the general construction is sketched in Prop. 39.
In the third version, we only consider the excitations on a particular gapped boundary C n+1 ∈ B n+1 , and instead of determining bulk topological orders, we ask only whether boundary excitations can determine bulk excitations. The boundary excitations are described by a fusion n-category C n = Hom(C n+1 , C n+1 ). Again Hom(M n+2 , M n+2 ) (nd excitations in M n+2 ) does not act within C n . However, the (n −1)d excitations in M n+2 act within C n . The braided fusion n-category of all the (n − 1)d excitations is given by M n = ΩHom(M n+2 , M n+2 ), which acts on C n . In other words, C n is a left module over M n . It is also a right module over nVec = Hom(1 n+1 , 1 n+1 ). A gapped boundary, up to an invertible topological order, is described by a pair (C n , M n ). In fact, the data (C n , M n ) determines the boundary excitations, which in turn determine the gapped boundary, up to an invertible topological order. We believe that the data (C n , M n ) can determine the category of the bulk excitations (i.e. determine the bulk topological order up to an invertible topological order). In fact, we believe that C n alone can uniquely determines the category of the bulk excitations.
Proposition 36. There is only one anomaly-free topological order M n+2 in TO n+2 af , up to invertible topological orders, which gives rise to the category of boundary excitations: The above result can be rephrased. Let us denote the fusion (n+1)-category of the bulk excitations as M n+1 (which is given by Hom(M n+2 , M n+2 )). Then M n+1 is uniquely determined by a braided fusion n-category M n = ΩM n+1 , via delooping: M n+1 = ΣM n , since the bulk topological order is anomaly-free.
Proposition 37. The braided fusion n-category M n is uniquely determined by C n where boundary-bulk relation Z 1 is called Z 1 center (the Drinfeld center when n = 1). Thus C n uniquely determines M n+1 via M n+1 = ΣZ 1 (C n ).
Mathematically, the above result is phrased as Proposition 38. From any fusion n-category C n , we can always construct a unique braided fusion n-category Z 1 (C n ), which is the maximal one equipped with a central monoidal functor M n F C n −−→ C n , i.e. for any x in M n and y in C n F C n (x) ⊗ y y ⊗ F C n (x), such that Z 1 (C n ) is the category of codimension-2 excitations in the bulk of C n M n = Z 1 (C n ).
Such central functor F C n : Z 1 (C n ) → C n is also referred to as the forgetful functor, since by construction, objects in Z 1 (C n ) can be viewed as objects in C n equipped with additional half braiding structures. We now explain an explicit construction of Z 1 . To do this, consider a slightly more complicated configuration as in Fig. 28, where C n+1 ∈ Hom(1 n+2 , M n+2 ), D n+1 ∈ Hom(1 n+2 , N n+2 ), K n+1 ∈ Hom(M n+2 , N n+2 ), and V n = Hom(D n+1 , K n+1 ⊗ M n+2 C n+1 ). We view V n as a collection of domain walls between the boundary C n+1 and D n+1 and it uniquely determines the "bulk" K n+1 , which is a domain wall between the bulk of C n+1 , and the bulk of D n+1 , namely between M n+2 and N n+2 . Observe that all three fusion n-categories, K n = Hom(K n+1 , K n+1 ), C n = Hom(C n+1 , C n+1 ), and D n = Hom(D n+1 , D n+1 ), act on V n . Moreover, the three actions commute with each other. Here we want to separate the action of K n from those of C n and D n .
Let us introduce Fun(V n , V n ) as a collection of endofunctors of the linear n-category V n , or more precisely, a category of linear functors f : V → V. In other words, for objects v, w ∈ V, the functor f satisfies Note that these functors are higher functors between higher categories, and consist of many structures at different levels of morphisms. In this paper, we are not giving rigorous descriptions, but only listing the structures at the object level for illustration. The structures on higher morphisms are similar. Fun(V n , V n ) is naturally a linear monoidal category since, for f, g ∈ Fun(V n , V n ), we can define Now we can see that an action of K n on V n is the same as a monoidal functor K n → Fun(V n , V n ), in other words an object k ∈ K n corresponds to a functor f k ∈ Fun(V n , V n ): where k ⊗ K n v describes the fusion of an object k ∈ K n to an object v ∈ V n along the domain wall K n+1 . Similarly, we have the actions of C n and D n on V n , which commute with each other and make V n into a C n -D n -bimodule. Thus the action of K n , that commutes with both actions of C n and D n , identifies K n with the bimodule endofunctors of V n , i.e. , all the linear functors that commute with both actions. More precisely, denote the left action of C n by C ⊗ C n v and right action of D n by v ⊗ D n d for v ∈ V n , C ∈ C n , d ∈ D n , a bimodule functor is a functor f : V n → V n together with natural isomorphisms and other appropriate higher structures.
We note that the above are addional structures rather than conditions.
Denote the category of all bimodule functors by Fun C n |D n (V n , V n ). The monoidal functor K n → Fun(V n , V n ) can be promoted to K n → Fun D n |C n (V n , V n ). Following the holographic principle, we expect such functor to be an equivalence. Thus we have the following boundary-bulk relation, extended to domain walls on the boundary and in the bulk: Proposition 39. Let C n+1 ∈ Hom(1 n+2 , M n+2 ), D n+1 ∈ Hom(1 n+2 , N n+2 ), and their excitations C n = Hom(C n+1 , C n+1 ), D n = Hom(D n+1 , D n+1 ). A C n -D n -bimodule V n , viewed as a collection of domain walls between C n+1 and D n+1 , uniquely determines a domain wall K n+1 in the bulk, i.e. , K n+1 ∈ Hom(M n+2 , N n+2 ). In other words, there is a unique K n+1 ∈ Hom(M n+2 , N n+2 ) such that V n = Hom(D n+1 , K n+1 ⊗ M n+2 C n+1 ). The excitations on K n+1 is given by K n = Hom(K n+1 , K n+1 ) = Fun C n |D n (V n , V n ). (G16) Objects in K n correspond to functors in Fun C n |D n (V n , V n ).
As a special case, take V n = D n = C n , i.e. , we view C n as a collection of domain walls between C n+1 and itself. The "bulk" of C n is the trivial domain wall in the bulk of C n+1 and the excitations on the trivial domain wall are just the codimension-2 excitations in the bulk of C n+1 . We obtain the explicit construction M n = Z 1 (C n ) := Fun C n |C n (C n , C n ). (G17) For a bimodule functor f ∈ Z 1 (C n ), and any y ∈ C n f (1 C n ) ⊗ y f (y) y ⊗ f (1 C n ).
We see that a bimodule funcor f is the same as an object f (1 C n ) in C n together with the half braiding f (1 C n )⊗y y ⊗ f (1 C n ). The forgetful functor is thus F C n : Fun C n |C n (C n , C n ) = Z 1 (C n ) → C n , In this paper, we mainly use the third version of boundary-bulk relation Z 1 (C n ) = M n : the codimension-1 boundary excitations described by a fusion n-category C n uniquely determines the codimension-2 bulk excitations described by a braided fusion n-category M n . In contrast, eqn. (G4) is a relation between a boundary topological order (i.e. an anomalous nd topological orderan object in Hom(1 n+2 , M n+2 )) and a bulk topological order (i.e. an anomaly-free (n+1)d topological orderan object in TO n+2 af ).
Appendix H: Example of topological orders and the corresponding higher categories
By definition, the invertible topological orders form Abelian groups under the stacking ⊗. In different dimensions, those groups are given by [25][26][27] (n + 1)D : 0 + 1 1 + 1 2 + 1 3 + 1 4 + 1 TO n+1 inv : The generator of TO 3 inv is the E 8 bosonic quantum Hall state described by the wave function where the K-matrix is given by which satisfies det(K) = 1. The generator of TO 5 inv is given following 4d bosonic system (described by path integral for cochain fields 82 ) where a Z2 is a Z 2 -valued 1-cochain, b Z2 is a Z 2 -valued 2-cochain, and w n is the n th Stiefel-Whitney class of the tangent bundle of the closed spacetime manifold M 4+1 . The path integral only depends on the cohomology classes of w 2 and w 3 , since the path integral is invariant under the following gauge transformation w 2 → w 2 + dγ, w 3 → w 3 + dλ, The path integral can be calculated exactly Z = a Z 2 ,b Z 2 e i π M 4+1 (w2+ da Z 2 )(w3+ db Z 2 ) , = 2 N l +Nt e i π M 4+1 w2w3 (H6) where N l is the number of the links and N t the number of the triangles in the triangulated spacetime M 4+1 . The non-trivial topological invariant e i π M 4+1 w2w3 implies that eqn. (H4) realize a non-trivial 4d invertible topological order. The invertible topological order have no non-trivial nltype of excitations, i.e. no non-trivial topological excitations. All the excisions are local. The different types of local excitations are described by the trivial fusion ncategory nVec for an nd invertible topological order.
For example, in 2-dimensional space, the objects in the category of invertible topological orders TO 3 inv form an Abelian group Z. The morphisms on each object form a trivial fusion 2-category 2Vec. Since the E 8 quantum Hall state has no gapped boundary, it is not an exact topological order, but is a closed (i.e. anomaly-free) topological order. Therefore, TO 3 inv has no 1-morphisms between different objects. All domain walls between different objects are gapless. The 1-morphism that connect the same object is also trivial. This is because such a 1morphism corresponds to an 1 + 1D excitation and there is no non-trivial 1 + 1D anomaly-free topological order.
In our attempt to use higher categories to characterize topological orders, the invertible topological orders are the most difficult ones. This is because higher categories mainly describes the excitations, but the excitations on top of invertible topological orders are identical to those on top of of trivial product state. Fortunately, in the category of topological orders, we also have information on the stacking operation ⊗ and the gapped domain walls between topological orders. This allows us to distinguish invertible topological orders. The invertible topological orders in 2d are particularly difficult, since we do not even have any gapped domain walls (i.e. no 1-morphisms). Only the stacking operation ⊗ allows us to distinguish 2d invertible topological orders.

G-topological orders
Another class of topological orders for bosonic systems are called G-topological orders (see Sec. III), which are described by gauge theories with a finite group G. We use GT n+1 G ∈ TO n+1 af to denote G-topological order in n-dimensional space. We use ΩGT n+1 G to denote the fusion n-category that describes the excitations in GT n+1 G and use Ω 2 GT n+1 G to denote the braided fusion (n−1)category that describes the excitations with codimension-2 and higher in GT n+1 G . It is known that GT n+1 G is anomaly-free and has gapped boundary. An example of GT 3 Z2 is given by Example 2.
Let us describe 3d Z 2 -topological order GT 4 Z2 in more details. Such a state has two nl-types of point-like excitations 1, e, two nl-types of string-like excitation 1 s , m s , and one trivial nl-type membrane-like excitations. The e-particle has a fusion e⊗e = 1 and the m s -loop has a fusion m s ⊗m s = 1 s . The Z 2 -topological order also has two types of point-like excitations 1, e, four types of stringlike excitation 1 s , m s , e s , e s ⊗m s . The string e s is formed by e-particles condensing into the Z 2 symmetry breaking state. The e s -loop has a fusion e s ⊗ e s = 2e s . Those point-like and string-like excitations form the braided fusion 2-category Ω 2 GT 4 Z2 . The 3d Z 2 -topological order GT 4 Z2 has infinite types of membrane-like excitations corresponding to infinite different 2d topological orders formed by trivial pointlike excitations 1's. GT 4 Z2 also has infinite types of membrane-like excitations corresponding to infinite different 2d SET orders with Z 2 symmetry, formed by eparticles with mod-2 conservation. There are third types of membrane-like excitations corresponding to 2d topological orders formed m s -loops. The m s -loops has a mod-2 conservation that corresponds to a Z 2 higher symmetry. Thus, this kind of 2d topological orders can be viewed as having a spontaneous breaking of Z 2 1symmetry. Those point-like, string-like, and membranelike excitations form the fusion 3-category ΩGT 4 Z2 . The point-like and string-like excitations form the braided fusion 2-category Ω 2 GT 4 Z2 . The above 3d Z 2 -topological order GT 4 Z2 is anomalyfree, which means that it can be realized by a bosonic lattice model, as shown in Sec. III. Another way to realize GT 4 Z2 is via the path integral of Z 2 -valued 1-cochain fields, a Z2 : 82 where da Z 2 =0 is a summation over Z 2 -valued 1cocycles. One can also realize GT 4 Z2 via the path integral of Z 2 -valued 2-cochain fields, b Z2 : 82 where db Z 2 =0 is a summation over Z 2 -valued 2cocycles.
But the above boundary-bulk relation between fusion higher categories and braided fusion higher categories only tell us that GT 4 Z2 is either anomaly-free or has invertible anomaly. The stronger boundary-bulk relation is given by This boundary-bulk relation tells us that GT 4 Z2 is anomaly-free.
We would like to mention that there is also a 3d twisted Z 2 -topological order where the point-like Z 2 -charges are fermions. We denote such a twisted Z 2 -topological order as GT 4 Z f 2 . The twisted Z 2 -topological order GT 4 Z f 2 is also anomaly-free and can be realized by the path integral of Z 2 -valued 2-cochain fields, b Z2 : 58,82,83 where db Z 2 =0 is a summation over Z 2 -valued 2cocycles, M 3+1 is a (3+1)-dimensional closed spacetime (with a triangulation), and w 2 is the second Stiefel-Whitney class of the tangent bundle of M 3+1 . Here we used a fact that b Z2 b Z2 +b Z2 w 2 is a Z 2 -valued coboundary. The topological term e i π M 3+1 b Z 2 b Z 2 = e i π M 3+1 b Z 2 w2 makes the point-like Z 2 -charges to be fermions.

A 2d anomalous topological order
Now, let us consider an anomalous topological order in 2d, denoted as C 3 Z2 , which has two nl-types of point-like excitations, labeled by 1, e, where 1 is a trivial point-like excitation and e has a Z 2 fusion e⊗e = 1. The anomalous topological order has two types of point-like excitations, which are also given by 1, e. The anomalous topological order has only one nl-type of string-like excitations, which is a local string-like excitation. But it has two types of string-like excitations, labeled by 1 s , e s . The e stype of string-like excitation is formed by the e-particles, condensing into a 1d phase of spontaneous Z 2 symmetry breaking state. The e s loop has a fusion e s ⊗ e s = 2e s . 1 s , e s are local string-like excitations, i.e. belong to the trivial nl-type of string-like excitations.
The excitations in the anomalous topological order C 3

Z2
are described by a fusion 2-category C 2 Z2 = ΩC 3 Z2 = 2Rep(Z 2 ). C 2 Z2 has two simple objects 1 s , e s . On 1 s , there are two simple 1-morphisms 1, e. On e s , there are also two simple 1-morphisms 1 es , d es , with a fusion rule d es ⊗ d es = 1 es . There is one simple 1-morphisms σ ∈ Hom(1 s , e s ) and one simpleσ ∈ Hom(e s , 1 s ), with fusion rules σ ⊗ (H12) The bulk of the anomalous topological order C 3 Z2 is the Z 2 -topological order in 3-dimensional space GT 4 Z2 : Since GT 4 Z2 is non-trivial, C 3 Z2 is anomalous. In fact C 3

Z2
is a 2d gapped boundary of the 3d Z 2 topological order GT 4 Z2 obtained via condensation of Z 2 -flux strings. We have a similar relation for excitations where C 2 Z2 = ΩC 3 Z2 is the fusion 2-category describing the excitations in C 3 Z2 . The relation eqn. (H13) carries more information than eqn. (H14). We would like to remark that when we stack the two anomalous topological orders, both the boundaries and the bulks are stacked: 4. Anomalous 3d Z2 topological order The anomaly-free 3d Z 2 -topological order GT 4 Z2 discussed above can also be realized via the path integral of Z 2 -valued 1-cochain and 2-cochain fields, a Z2 and b Z2 : 82 where a Z 2 ,b Z 2 is a summation over Z 2 -valued 1-cochain and 2-cochain. The above path integral has a gauge invariance for closed M 3+1 In this formulation, the twisted 3d Z 2 -topological order GT 4 Z f 2 is realized by the path integral Z = a Z 2 ,b Z 2 e i π M 3+1 b Z 2 da Z 2 +b Z 2 w2 . (H18) The above path integral is also gauge invariant for closed M 3+1 a Z2 → a Z2 + γ, b Z2 → b Z2 + dβ, w 2 → w 2 + dγ.
The path integral only depends on the cohomology classes of w 2 , So it describes an anomaly-free theory.
In this section, we are going to study an anomalous 3d Z 2 topological order, realized by the following path integral Under the gauge transformation a Z2 → a Z2 + γ, b Z2 → b Z2 + λ, w 2 → w 2 + dγ, w 3 → w 3 + dλ, the above path integral is not invariant. The gauge non-invariance can be fixed by adding a bulk term e i π N 5 w2w3 in one higher dimension, where ∂N 5 = M 3+1 . The resulting path integral Z = a Z 2 ,b Z 2 e i π M 3+1 b Z 2 da Z 2 +a Z 2 w3+b Z 2 w2 e i π N 5 w2w3 (H22) is gauge invariant, i.e. only depends on the cohomology classes of w 2 and w 3 . Since, the path integral requires a bulk in one higher dimension to be gauge invariant (i.e. only depends on the cohomology classes of w 2 and w 3 ), so it describes an anomalous theory. We denote such a 3d anomalous Z 2 -topological order as GT 4,w2w3 Such a 3d anomalous Z 2 -topological order GT 4,w2w3 Z f 2 has a fermionic point-like Z 2 charge. If the world-sheet for the Z 2 flux loop is unorientable, there is a worldline that marks the reversal of the orientation. Such an orientation-reversal world-line corresponds to a fermion world-line. In other words, the anomalous Z 2 topological order has a special property that a un-orientable worldsheet of the Z 2 -flux must bind with a world-line of the fermionic point-like Z 2 charge. Such a fermionic worldline corresponds to the orientation reversal loop on the unorientable worldsheet. The 3d anomalous Z 2 -topological order GT 4,w2w3 where I 5 w2w3 is the 4d invertible topological order characterized by the topological invariant e i π N 5 w2w3 . [25][26][27] The boundary-bulk relation (H23) implies the following boundary-bulk relation for the excitations since the excitations in an invertible topological order are described by a trivial braided fusion higher category. Despite the right-hand-side of Z 1 (ΩGT 4,w2w3