Unifying constructions of non-invertible symmetries

In the past year several constructions of non-invertible symmetries in Quantum Field Theory in d ≥ 3 have appeared. In this paper we provide a unified perspective on these constructions. Central to this framework are so-called theta defects , which generalize the notion of theta-angles, and allow the construction of universal and non-universal topological symmetry defects. We complement this physical analysis by proposing a mathematical framework (based on higher-fusion categories) that converts the physical construction of non-invertible symmetries into a concrete computational scheme

In this paper we will show that these constructions are all special instances of a general, unified framework.
The key point is that defects after gauging an invertible symmetry Γ can be obtained from defects before gauging, possibly stacked with TQFTs. We call defects produced this way (twisted) theta defects. A defect D may lead to multiple defects after gauging, if there are multiple ways of 'coupling' D to Γ gauge fields in the bulk spacetime. On the other hand, a defect D may not lead to any defect after gauging if there is an obstruction to couple D to bulk Γ gauge fields (which may sometimes be curable by stacking a TQFT on top of the defect). These defects are a generalization of the theta 1 defects constructed in [7] by coupling decoupled TQFTs (of various codimensions) to bulk Γ gauge fields.
A mathematical study of different kinds of Γ-couplings for different kinds of defects at the level of symmetry categories 2 leads to familiar concepts in higher-category theory like higher-vector spaces and higher-representations of (higher-)groups. We cleanly describe the connection of the physical study of Γ-couplings to these categorical notions, along with the various subtleties related to Karoubi completions (physically known as condensations/gaugings) that arise for 3-categories and higher.
The paper has two main parts: a physics proposal for the construction of non-invertibles in section 2, and in section 3 a mathematical proposal for the relevant structures that are needed to capture the physical proposal, using concepts related to higher fusion categories.
We conclude with a program for the classification of non-invertible symmetries in section 4.

Non-Invertible Symmetries From Invertibles: A Unified Perspective
The question we would like to answer in this paper is whether there is a unified construction of the non-invertible symmetry defects [1][2][3][4][5][6][7][8][9] in QFTs. In this section, we answer the question in the affirmative and present such a unified framework.

Theta Angles
The construction is a generalization of the construction appearing in [7], which we now review. The construction of [7] can be understood as generalizing the notion of theta angle to higher-codimensions, and so we refer to the symmetries arising via this construction as theta symmetries.
Higher-Group Symmetric QFTs. Consider a d-dimensional QFT T with a non-anomalous higher-group symmetry 3 Γ. We can convert such a QFT to a 'Γ-symmetric d-dimensional QFT' by choosing a gauge-invariant coupling 4 S of T to background gauge fields for Γ. Such a coupling S allows one to define partition functions of T in the presence of a background gauge field for Γ, and the fact that S is gauge-invariant means that these partition functions are the same for two background gauge fields related by a background gauge transformation.
After choosing S, we can label the Γ-symmetric d-dimensional QFT as T S . We also say that T is the QFT underlying the Γ-symmetric QFT T S .
Gauging a Higher-Group. Given a Γ-symmetric d-dimensional QFT T S , we can sum over the background gauge fields for Γ consistently, or in other words we can gauge the Γ symmetry. This produces a new d-dimensional QFT that we denote by T S /Γ. Note that, if S ′ is some other gauge invariant coupling of T to background gauge fields for Γ, then the corresponding gauged theory T S ′ /Γ is apriori different from T S /Γ, though there might exist a duality/isomorphism between the two for specific T. 3 Throughout this paper, except for a short paragraph discussing the example of theta angle in U (1) gauge theories, Γ will be taken to be a completely finite higher-group, meaning that every p-form group Γ (p) inside Γ is taken to be finite. 4 We will give a precise mathematical definition of "coupling" in certain situations in next section. We hope that the usage of this word and concepts surrounding it in this section will be clear at an intuitive level.
Product of Higher-Group Symmetric QFTs. Now, given two Γ-symmetric d-dimensional QFTs T S and T ′ S ′ , we can stack them to obtain a new Γ-symmetric d-dimensional QFT that can be denoted as The d-dimensional QFT underlying T S ⊗ Γ T ′ S ′ is the QFT T ⊗ T ′ obtained by stacking T and T ′ . To promote the underlying QFT T ⊗ T ′ to a Γ-symmetric QFT T S ⊗ Γ T ′ S ′ , we need to first choose a Γ symmetry of T ⊗ T ′ and then describe a coupling for this Γ symmetry.
The Γ symmetry of T ⊗ T ′ is chosen to be the diagonal of the Γ × Γ symmetry of T ⊗ T ′ descending from Γ symmetry of T and Γ symmetry of T ′ . The precise coupling of Γ is then simply obtained by "adding" the couplings S and S ′ .

SPT Phases Protected by Higher-Group Symmetry. The simplest d-dimensional
QFT is the trivial d-dimensional TQFT I. The trivial theory I is trivially symmetric under a non-anomalous higher-group symmetry Γ. But there can be various gauge-invariant couplings S of I to background gauge fields for Γ. There is always a trivial coupling that we denote by The corresponding Γ-symmetric d-dimensional TQFTs I S are often called as 'SPT phases protected by higher-group Γ'. That is, the underlying d-dimensional QFT for an SPT phase is the trivial theory I.
The SPT phases I S form a group SPT Γ d under the above product operation (2.1). The identity element of the group is the SPT phase I S 0 which is also referred to as the trivial SPT phase.
Monoid of Higher-Group Symmetric QFTs. In fact, we have for any arbitrary Γ-symmetric d-dimensional QFT T S . Thus, d-dimensional Γ-symmetric QFTs form a monoid which the product being (2.1) and the identity element being I S 0 .
Theta Angles. On the other hand, if we use a non-trivial SPT phase in the above stacking, we obtain where T S ′ is a Γ-symmetric QFT obtained from T by using a coupling S ′ , which is closely related but different from the coupling S. As a consequence, the set of all Γ-symmetric QFTs obtainable from the same underlying QFT T admits a group action by the group SPT Γ d . The action is free without any fixed points. Let O T Γ ⊆ T Γ be an orbit of the above group action. Pick two Γ-symmetric d-dimensional QFTs T S and T S ′ in the orbit O T Γ . We have for a unique I S ′′ ∈ SPT Γ d . Correspondingly, it is often said that the d-dimensional QFT T S ′ /Γ is related to the d-dimensional QFT T S /Γ by the 'theta angle' I S ′′ . See figure 1.
Example: Theta Angle in U (1) Gauge Theory. Consider the 4d trivial theory I and consider gauging its U (1) 0-form symmetry. This leads to pure Maxwell theories T θ in 4d, which is a family of theories differentiated by a circle valued parameter θ, known as the theta angle. Two Maxwell theories T θ ′ and T θ+θ ′ are related by a theta angle θ, for which the corresponding U (1) SPT phase has effective action (here F denotes the field strength for U (1) 0-form background).
Example: Discrete Theta Angle in SO(3) Gauge Theory. It was pointed out in [56] that there are two versions of SO(3) gauge theories in 4d, which are usually distinguished by labeling the gauge group as SO(3) ± where the subscript ± labels a discrete Z 2 valued theta angle.
This discrete theta angle is realized in terms of the above general construction as follows.
Let T be a 4d SU (2) gauge theory whose matter content is such that the Z 2 center of the SU (2) gauge group survives as a Z 2 1-form symmetry of T. Now, there are (on a spin manifold) two 4d Z 2 1-form symmetric SPT phases, one whose effective action is trivial, and the other whose effective action is where P(B 2 ) is the Pontryagin square of the 1-form symmetry background field B 2 . Gauging the Z 2 1-form symmetry of T converts the gauge group to SU (2)/Z 2 ∼ = SO (3), resulting in a 4d SO(3) gauge theory. Depending on whether the invertible theory (2.7) is included or not in this gauging process, one either lands on the SO(3) + theory or the SO(3) − theory.

Theta Symmetries
In this subsection we generalize the considerations of the previous subsection to (topological and non-topological) defects of a QFT.
Defects From QFT Stacking. We begin our discussion with the general phenomenon that a p-dimensional QFT can always be treated as a p-dimensional defect of a d-dimensional This can be understood by generalizing the stacking procedure above. We can stack a Action of QFTs on Defects. We can actually generalize the stacking procedure to obtain an action of p-dimensional QFTs on p-dimensional defects of T. Stacking T with a p-dimensional defect D p of T gives rise to a new p-dimensional defect If T is a p-dimensional TQFT and D p is a topological defect, then D (T) p ⊗ D p is another topological defect of T. This is related to the 'TQFT coefficients' of [4] as will become clear later.
A special case arises if we take T to be a p-dimensional TQFT with n trivial vacua and keep D p to be an arbitrary defect. Then, we have the equivalence where the right hand side denotes a direct sum of n copies of D p .
Action of Topological Defects on General Defects. There is another action on general (topological or non-topological) p-dimensional defects of T via topological p-dimensional defects of T. Stacking a p-dimensional topological defect D p of T on top of a general p- In the special case that D p = D (T) p arises from a p-dimensional TQFT T, we have defined two actions (2.8) and (2.10) of it on general p-dimensional defects of T. Both of these actions coincide.
Fusion of Topological Defects. If we take D ′ p to also be topological in (2.10), then the action of D p on D ′ p is simply the fusion of D p and D ′ p .
Higher-Group Symmetric Defects. Consider again a d-dimensional QFT T with nonanomalous higher-group symmetry Γ. Let S be a gauge-invariant coupling of T to Γ background fields, and let T S be the corresponding Γ-symmetric d-dimensional QFT. A p-dimensional (topological or non-topological) defect D p of T can be converted into a 'Γ-symmetric pdimensional defect of T S ' by choosing a gauge-invariant coupling J of D p to background gauge fields for Γ living in the d-dimensional bulk spacetime. Combining such a defect coupling J with the bulk coupling S allows one to define correlation functions of D p in the presence of a background gauge field for Γ, and the fact that J is gauge-invariant means that these correlation functions are the same for two background gauge fields related by a background gauge transformation. Note that a choice of defect coupling J may be inconsistent for a choice of bulk coupling S, but consistent for another bulk coupling S ′ . After choosing J , we can label the Γ-symmetric p-dimensional defect of T S as D (J ) p . We also say that D p is the defect underlying the Γ-symmetric defect D (J ) p . Note that there might not exist a gauge-invariant coupling J to Γ backgrounds, or even a coupling afflicted with a 't Hooft anomaly, for some p-dimensional defects D p of T. On the other hand, for some p-dimensional defects D p , there might exist multiple couplings J leading to multiple Γ-symmetric defects D In what follows, we will treat both cases p < d − 1 and p = d − 1 together, and in the latter case the coupling J will stand for the tuple (J L , J R ).
Defects Surviving the Gauging. As we discussed above, we can gauge the Γ symmetry of T with coupling S to obtain a d-dimensional QFT T S /Γ. A Γ-symmetric p-dimensional defect D (J ) p of T S survives the gauging procedure due to gauge invariance of the coupling J . p /Γ of T S /Γ is also topological. In this case, the above procedure describes how to deduce the (invertible or non-invertible) symmetries of T S /Γ from the information regarding the symmetries of T.
Γ-Symmetric Defects By Stacking Γ-Symmetric QFTs. Let T S ′ be a Γ-symmetric p-dimensional QFT with T being the underlying p-dimensional QFT. Then, rather similarly to figure 2, we can stack T S ′ in the spacetime occupied by T S to obtain a Γ-symmetric is obtained canonically from the coupling S ′ , and hence we omit it.
If T is a TQFT, i.e. if T S ′ is a Γ-symmetric TQFT, then D Theta Symmetries. Thus, in the gauged d-dimensional QFT T S /Γ, we obtain a universal of (generically non-topological) p-dimensional defects, parametrized by Γ-symmetric p-dimensional QFTs.
Restricting attention to those T that are TQFTs, we obtain a universal sector of generically non-invertible symmetries, parametrized by Γ-symmetric p-dimensional TQFTs, in any d-dimensional QFT T S /Γ that can be obtained by gauging non-anomalous Γ higher-group symmetry of another d-dimensional QFT T.
These symmetries were discussed in [7], and we refer to them as theta symmetries as their construction is a generalization of the construction of theta angles discussed above.
Action of Γ-Symmetric QFTs on Γ-Symmetric Defects. We can generalize the above stacking procedure to obtain an action of p-dimensional Γ-symmetric QFTs on p-dimensional Γ-symmetric defects of the d-dimensional Γ-symmetric QFT T S . Stacking T S ′ with a p- of T S , whose underlying p-dimensional defect is obtained by the action of the underlying QFT T of T S ′ on the underlying defect D p of D (2.15) of T S , whose underlying p-dimensional defect is p+1 of T obtained by stacking a (p + 1)-dimensional 5 We are suppressing the coupling S of T to Γ (p) backgrounds as it does not play any role in what follows.   As described in [7] all such condensation surface defects can themselves be obtained as theta symmetries associated to the gauging procedure T → T/Γ (0) , by stacking T with 2d TQFTs with Γ (0) non-anomalous 0-form symmetry, and then performing the Γ (0) gauging in the whole d-dimensional spacetime.
Example: Condensation Defects Arising from Duality Defects. Consider the example studied by the paper [2] involving a 4d spin-QFT T with a Z where A 1 is the background field for 0-form symmetry and B 2 is the background field for 1-form symmetry. The paper [2] constructs a non-invertible 3-dimensional topological defect D (S) 3 (known as a duality defect [3]) in the 4d theory T/Z 1-form symmetry, which has the fusion rule is a non-invertible 3-dimensional condensation defect that can be obtained by gauging the dual Z  can be understood as a theta defect. The defect D (S) 3 , on the other hand, will be discussed later as an example of a 'twisted' theta defect defined in the next subsection.
As remarked in the previous paragraph, the condensation defect D can be realized as a theta defect. This defect is obtained by stacking T SS , which is the 3d Dijkgraaf-Witten TQFT based on a Z 2 gauge group and a non-trivial twist, on T before gauging the Z (1) 2 1-form symmetry. The action of the Dijkgraaf-Witten theory can be written as where a 1 is the Z 2 gauge field. This theory T SS can be identified with the double semion model, which contains a bosonic Z 2 line operator (namely the Deligne product of semion and anti-semion) generating a non-anomalous Z 2 1-form symmetry, which is used to convert the above double semion TQFT into a Z 2N symmetries having similar mixed 't Hooft anomaly as (2.22), such as 4d pure Super-Yang-Mills [10,19]. These theories also contain twisted theta defects whose fusion gives rise to condensation defects that can be understood as theta defects.

Twisted Theta Symmetries
We can incorporate a 'twist' in the construction of theta symmetries discussed above to construct a generalized version of theta symmetries that we refer to as twisted theta symmetries.
Unlike theta symmetries, which are universal and exist in any theory that can be obtained by gauging an invertible higher-group symmetry of some other theory, the twisted theta symmetries are theory-dependent and hence non-universal.
Consider a situation in which we have a topological defect D   Figure 6. Two different gauge-invariant couplings J 0 (on the left) and J 0,d−2 (on the right) of the identity 2-dimensional defect D Relationship Between Theta and Twisted Theta. Note that a theta defect is a twisted theta defect with a trivial twist, i.e. the twist given by the identity p-dimensional defect D (id) p of T; but converse is not true, as a twisted theta defect with a trivial twist might still involve a coupling J that is intrinsic to the QFT T and cannot be decoupled to a coupling J for the stacked TQFT T. An example is provided in the upcoming paper [9] which we reproduce in a generalized form below.
, without any 't Hooft anomalies for the two symmetries. There are then two possible couplings J for making the identity 2-dimensional defect D   2 , which is a trivial theta defect. However, on the other hand, the topological defect 2 is a twisted theta defect which is not a theta defect. It can be recognized as the condensation surface defect obtained by gauging the Z 2 × Z 2 1-form symmetry of T/Z (0) 2 on a two-dimensional worldvolume in spacetime along with a discrete torsion specified by the Obstruction: Localized 't Hooft Anomaly. It is interesting to classify the various kinds of obstructions for coupling D then admits a gauge-invariant coupling "J +S ′ " to bulk Γ backgrounds. The anomaly for the coupling J is canceled by the anomaly for the coupling S ′ . We call the resulting Γ-symmetric Gauging the Γ symmetry, we obtain a p-dimensional twisted theta defect 3 . Now we look for a 3d TQFT with Z 2 1-form symmetry that carries the anomaly (2.33).
One of the candidates, that was used in [2], is the semion model, or U (1) 2 Chern-Simons theory T S . 7 This 3d TQFT carries a line operator, namely the semion, which generates a Z 2 1-form symmetry with anomaly (2.33).
2 1-form symmetry, we obtain a 3d twisted theta defect 2 . The fusion of this defect with itself was discussed in (2.22). This fusion can now be derived using (2.18). First of all, we could have used the 3d spin-TQFT TS given by the antisemion model, for which the anti-semion line generates an anomalous Z 2 1-form symmetry, to construct another 3d twisted theta defect Since T S is isomorphic to TS as spin 3d TQFTs, the resulting twisted theta defects D (S) 3 and D (S) 3 are also isomorphic (or in other words dual). Thus, The right hand side is the twisted theta defect with trivial twist and hence is a theta defect. The 3d TQFT used for stacking is the double semion model T SS discussed earlier since To begin with, S is a 3d topological boundary condition of a 4d invertible TQFT I S rather than a 3d TQFT. The partition function of IS on a 4d manifold M4 is where σ(M4) is the signature of M4. Consequently, I S is invisible on a spin 4-manifold, because the signature of such a manifold is a multiple of 16. This fact allows us to treat S as a spin 3d TQFT, which is the right context here as the 4d QFT T is a spin theory. We thank Jingxiang Wu for related discussions.
The Z (1) 2 1-form symmetry is implemented on T S by the semion, on TS by the anti-semion, and hence must be implemented on T SS by the bosonic semion times anti-semion line. Thus, we see that the fusion is precisely the theta defect D (2.41) Again there is a generalization to Z (2.42) Let us note that many other examples of twisted theta defects generalizing the above construction of [2]  p . We can also have "worse" obstructions, where even an anomalous coupling J cannot be found.
One of the simplest such obstructions is symmetry fractionalization of Γ on the worldvol- p , which is easiest to understand for a 0-form symmetry group Γ (0) . This occurs when symmetry action of Γ (0) on D p does not close and in fact gives rise to a larger 0-form symmetry group Γ (0) E symmetry on D p which is an extension of the group Γ (0) leading to a short exact sequence with the key property being that the above sequence does not split. In such a situation, we say that Γ (0) 0-form symmetry is fractionalized to Γ 2. The coupling J found in the previous step is actually non-anomalous/gauge-invariant.
If the above two conditions are satisfied, then we obtain a twisted theta defect (2.44) in the gauged QFT T S /Γ.
Example: Gauging Z 4 by gauging two Z 2 s sequentially. We encounter an example of such an obstruction in the upcoming paper [9]. The context is the study of a d-dimensional QFT T with Z 4 non-anomalous 0-form symmetry. First gauge the Z 2 subgroup of Z 4 to pass on to the theory T/Z 2 . This theory has a residual Z (d − 2)-form symmetry, with a mixed 't Hooft anomaly symmetry is generated by topological line operators that can be condensed on a surface to give rise to a condensation surface defect that we label as D . We show in [9], . See figure 7. Moreover, since that is P squares to the identity line on D , as shown in figure 7, we learn that implying that the Z where T is a 2d TQFT with two trivial vacua. The line operators on D . The (ij)-th entry in the matrix  Since P is order 2, as shown in the figure, we learn that J is order 4, and hence the Z . Then the matrix

Symmetries from Topological Interfaces
Interfaces. Interfaces are (topological or non-topological) (d−1)-dimensional defects living between two d-dimensional QFTs T (L) and T (R) . We will say that an interface D d−1 is 'from Action of QFTs on Interfaces. We can stack (d − 1)-dimensional QFTs on interfaces from T (L) to T (R) to obtain new interfaces from T (L) to T (R) . Stacking a (d − 1)-dimensional QFT T with an interface I d−1 creates a new interface that we denote as This process is rather similar to the one shown in figure 3. If T is a (d−1)-dimensional TQFT and I d−1 is a topological interface, then I (T) d−1 is another topological interface. As for defects, a special case arises if we take T to be a (d − 1)-dimensional TQFT with n trivial vacua and keep I d−1 to be an arbitrary defect. Then, we have the equivalence where the right hand side denotes a direct sum of n copies of I d−1 .
Actions of Topological Interfaces on General Interfaces. Consider a topological interface I d−1 from T 1 to T 2 , and a general (topological or non-topological) interface I ′ d−1 from T 2 to T 3 . We can act from the left by I d−1 on I ′ d−1 to obtain an interface is also topological. In this case, the topological interface I d−1 ⊗ I ′ d−1 is referred to the fusion of the topological interfaces I d−1 and I ′ d−1 . If T 1 = T 2 ≡ T, then the above is a left action of topological defects of T on interfaces from T to T 3 .
Similarly, if I d−1 is a general interface from T 1 to T 2 and I ′ d−1 is a topological interface from T 2 to T 3 , we can act from the right by I ′ d−1 on I d−1 to obtain an interface from T 1 to T 3 . If I d−1 is topological, then I d−1 ⊗ I ′ d−1 is also topological. In this case, the topological interface I d−1 ⊗ I ′ d−1 is referred to the fusion of the topological interfaces I d−1 and I ′ d−1 . If T 2 = T 3 ≡ T, then the above is a right action of topological defects of T on interfaces from T 1 to T.
survives the procedure of gauging both Actions of Topological Symmetric Interfaces on General Symmetric Interfaces.
S 2 can act from the left on a general (topological or non-topological) ( S 3 , whose underlying interface is on the left is given by J 1 and on the right is given by J ′ 3 .
Similarly, a topological ( S 3 can act from the right on a general (topological or non-topological) (Γ 1 , Γ 2 )-symmetric interface Fusion of Topological Interfaces After Gauging. Let I above be topological interfaces. After gauging, we obtain a topological interface S 2 /Γ 2 , and a topological interface S 3 /Γ 3 . Their fusion is given by where the right hand side is the topological interface from T (1) Example: Non-invertible Symmetries from ABJ Anomalies. [6] used this method to construct non-invertible codimension-1 topological defects in 4d gauge theories with ABJ anomalies. A continuous symmetry afflicted with an ABJ anomaly acts on a gauge theory by shifting the theta angle. Thus a topological codimension-1 defect implementing such an anomalous symmetry transformation provides an invertible interface I 3 , or in other words a duality, between gauge theories with different values of theta angles.
Let us illustrate using one of the simplest examples appearing in [6]. Let T (1) be 4d U (1) gauge theory with θ = 0 and T (2) be 4d U (1) gauge theory with θ = π. Assume T (1) has a U (1) global symmetry with an ABJ anomaly, providing an interface I 3 from T (1) to T (2) . As discussed in [6], the theory T (2) can also be obtained from T (1) by gauging a Z We can construct another topological interface from T (2) to T (1) as follows. Take T to be a 3d TQFT with Z

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8 Note that to make sense of this effective action, we need to restrict to spin 4d U (1) gauge theories. 9 This requirement is there to make sure that the 4d TQFT attached to T, capturing the gravitational anomaly of T, vanishes on spin 4-manifolds, allowing us to treat T as an absolute (rather than relative) spin TQFT.
to the 4d QFT Composing this topological interface with I 3 , we obtain a topological defect of the 4d QFT T (1) , which generates the non-invertible symmetry discussed in [6].

Condensations
A final generalization of the above considerations arises by noticing that we can replace d-dimensional QFTs everywhere by d-dimensional defects of a larger D-dimensional QFT.
The higher-group symmetries will be localized along d-dimensional worldvolumes of these ddimensional defects, and the whole machinery (about their gauging etc.) will carry through.
Such localized symmetries were discussed in detail by [5].
The above machinery then allows us to produce new d-dimensional defects of the D- The machinery discussed in this section then allows us to study sub-defects and subinterfaces of condensation defects.
Higher-Categorical Structure of Symmetries. In fact, we can iterate the above procedure. We can replace D-dimensional QFTs by D-dimensional defects of larger D ′ -dimensional QFTs. The machinery of this section then studies the symmetries localized along sub-defects of defects of a QFT, the gauging of such symmetries and the fate of sub-sub-defects and sub-sub-interfaces between such sub-defects.
Of course, we can keep iterating the above procedure. Turning it around, this means that we could apply all of the machinery discussed in this section not only to defects of a ddimensional QFT T, but also to sub-defects of defects of T, and sub-sub-defects of sub-defects of T and so on.
If we restrict ourselves to the study only of topological defects and topological sub-defects etc, then this iterative structure of defects inside defects, and their various properties is all expected to be captured in the information of a (d − 1)-category C T associated to the QFT T, known as the symmetry category of T. See [5,7] for more detailed discussions of the higher-categorical aspects of non-invertible symmetries.
Starting from next section, we will assume that the readers have familiarized themselves with this higher-categorical structure.

Distinction Between Condensation and Theta Defects
In this section, we discussed theta defects and condensation defects, both of which provide classes of non-invertible symmetries. In this box, we discuss the distinction between the two classes. First of all, the two classes are defined differently. The theta defects are obtained by inserting a lower p-dimensional TQFT T inside a bulk d-dimensional QFT T and then gauging a combined symmetry S of T and T. On the other hand, the condensation defects are obtained by gauging a symmetry S of a d-dimensional QFT T on a pdimensional submanifold of the d-dimensional spacetime.
However, even though the definitions are different, one may wonder whether the two classes actually turn out to be the same. This was discussed in some detail in [7], where it was concluded that the two classes are different. The reference [7] studied the two classes for p = 2. If S is a finite abelian 0-form symmetry group Γ (0) , then it was shown concretely in section 4 of [7] that the two classes of defects in fact coincide. Reference [7] also provided an abstract argument that the two classes of defects coincide even if Γ (0) is a non-abelian finite 0-form symmetry group. However, as pointed out in [7], if S is a 2-group symmetry in which 0-form and 1-form symmetries mix non-trivially, there exist theta defects that are not condensation defects, and hence the two classes of defects start to differ.
For higher values of p, the two classes differ already for S = Γ (0) as not all Γ (0) symmetric p-dimensional TQFTs for p ≥ 3 admit topological boundary conditions. The existence of a Γ (0) -symmetric topological boundary condition would imply that the resulting theta defect can also be constructed as a condensation defect.
Finally, if we consider twisted theta defects, then they are distinct from condensation defects already for p = 2 and S = Γ (0) . Also note that the twisted theta defect (2.36) is not a condensation defect.

Mathematical, Higher-Categorical Structure
In this section, we translate various special cases of the physical construction described above into well-known mathematical concepts in category theory. It should be noted that what is discussed below is only physicists' attempt at giving a mathematical definition to the physical concepts encountered in the discussion of symmetries in QFTs, but there might be various adjectives and subtleties missing from the mathematical discussion. Our purpose is to point out the relevant mathematical objects in an effort to motivate the precise mathematical treatment of the physical concepts encountered in the study of symmetries.

Higher Vector Spaces and Non-Anomalous Topological Orders
Higher-Categories of Universal Topological Defects. As we discussed in the previous section, one of the ways of constructing p-dimensional topological defects of any d-dimensional QFT T is to simply stack a p-dimensional TQFT T on a p-dimensional locus inside the spacetime occupied by T. Thus any d-dimensional theory T carries a universal sector of topological defects described by (d−1)-dimensional TQFTs and topological defects/interfaces of (d − 1)-dimensional TQFTs 10 .
This universal sector is expected to be described by a monoidal (d − 1)-category T d−1 whose objects are (d−1)-dimensional TQFTs and morphisms are topological defects/interfaces of (d−1)-dimensional TQFTs. Note that T d−2 is contained inside T d−1 as the (d−2)-category 10 Note that this automatically includes all lower dimensional TQFTs. For example, (d − 2)-dimensional TQFTs can be viewed as topological codimension-1 defects of the completely trivial (d − 1)-dimensional TQFT.
formed by endomorphisms (i.e. the defects) of the identity object (i.e. the trivial (d − 1)dimensional TQFT) of T d−1 . Continuing iteratively T p for every p < d is contained in T d−1 .

1-Category of Universal Line Defects and Vector
Spaces. Let us study special cases of T p higher-categories. For p = 1, we have that is the category T 1 of 1d TQFTs can be identified with the category Vec of finitedimensional vector spaces. The identification is made by a bulk-boundary correspondence: A 1d TQFT T ∈ T 1 is mapped to the vector space V T of local operators living at the 0d boundary of T. We can also identify V T with the space of states assigned to a point by T. The fusion rule of T with an arbitrary line defect that is the TQFT T reduces in each vacuum to an Euler number counterterm.
2-Vector Spaces. Instead of studying 2d TQFTs, one can study 2d non-anomalous 11 topological orders, which are defined as 2d TQFTs modulo invertible 2d TQFTs [50], and form a There is a canonical inclusion whose image contains 2d TQFTs having n vacua such that restricting to any vacuum we obtain the trivial TQFT with λ = 0. Thus O 2 is a fusion 2-category with a single simple object (upto isomorphism) corresponding to the completely trivial 2d TQFT. Physically, this is just the well-known fact that there are no non-trivial topological orders in 2d.
We can identify where 2-Vec is the fusion 2-category formed by '2-vector spaces', which are by definition finite 12 semi-simple abelian 1-categories. The identification is made by a bulk-boundary correspondence: A 2d TQFT T ∈ O 2 ⊂ T 2 is mapped to the 1-category C T of line operators living on the boundary of T. If T has n vacua, then C T has n simple objects and can be identified as where on right hand side we have a direct sum of n copies of the 1-category Vec of finite dimensional vector spaces. The fusion rule of T with an arbitrary surface defect D 2 of T is where n D 2 denotes a direct sum of n copies of D 2 .

2-Category of Universal Surface Defects.
We can express T 2 as where 2-Vec R + describes the Euler counter-terms, and is simply the monoidal 2-category of R + -graded 2-vector spaces. 11 Here anomaly refers to gravitational anomaly. The presence of this anomaly means that we are studying 2d theories that are relative, and should be properly understood as boundary conditions of 3d invertible TQFTs. 12 All categories we discuss are C-linear unless otherwise stated. Note that the categories C Γ q discussed later are non-linear.
Universal 3d Defects and 3-Vector Spaces. 13 For p = 3, the 3-category O 3 formed by 3d non-anomalous topological orders is (3.13) where the right hand side is the fusion 3-category formed by '3-vector spaces', which is by definition the 3-category formed by multi-fusion 1-categories.
The identification Miraculously, the category of boundary lines C B T completely determines the TQFT T via the well-known Turaev-Viro construction based on C B T . In particular, the modular multi-tensor category M T formed by topological line defects of T is recovered as where Z(C B T ) denotes the Drinfeld center of C B T . A different topological boundary condition B ′ T of T carries a different multi-fusion category C B ′ T but Turaev-Viro construction based on it produces the same 3d TQFT T. At the level of topological line defects of T, we have an of Drinfeld centers. 13 We thank Thibault Décoppet and David Jordan for discussions regarding various points appearing in this subsection from this point onward. 14 Such a boundary condition does not exist when we are dealing with anomalous 3d TQFTs (calling such systems as TQFTs is a misnomer, as such 3d theories are topological boundary conditions of 4d TQFTs rather than properly defined 3d theories). In particular if c − ̸ = 0, then the corresponding modular tensor category M T cannot be expressed as a Drinfeld center and so is not included in 3-Vec.

Is 3-Vec the Simplest Fusion 3-Category?
The final issue that we would like to discuss is the following puzzle: 3-Vec is often referred to as the simplest fusion 3-category. However, as we have seen, this includes many non-trivial 3d topological orders, e.g. topological orders characterized by non-trivial modular tensor categories (albeit only those that can be expressed as Drinfeld centers). Physically, extending the 1d and 2d examples discussed above, it seems that we could study a closed set of 3d topological orders which is simpler than the above set.
This simpler set comprises of the trivial 3d TQFT and its direct sums. A 3d TQFT in this simpler set has n vacua such that in each vacuum the TQFT reduces to the trivial theory.
Clearly, such 3d TQFTs form a rather simple monoidal 3-category that we refer to as where the subscript 0 stands for trivial, as this 3-category. In fact, just like Vec and 2-Vec, the 3-category 3-Vec 0 has a single simple object upto isomorphism corresponding to the trivial 3d TQFT.
The crucial part of the definition of a fusion 3-category [50] violated by the monoidal 3-category 3-Vec 0 is that a fusion 3-category C has to be Karoubi complete, which physically means that any 3d topological defect obtained by gauging a (possibly non-invertible) symmetry localized along the worldvolume of a 3d topological defect corresponding to an object of C should also correspond to an object of C [49]. 15 Recall that Morita equivalence of fusion categories C1 and C2 is equivalent to the statement that their Drinfeld centers are same Z(C1) ∼ = Z(C2).
This condition clearly fails for 3-Vec 0 . For example, consider gauging the Z 2 0-form symmetry of the trivial 3d TQFT. This produces the 3d Z 2 Dijkgraaf-Witten gauge theory without twist, also known as the toric code, which has a single vacuum but carries a modular tensor category of lines containing more than one simple object. This makes it clear that toric code lies outside 3-Vec 0 .
In fact, upon Karoubi completing 3-Vec 0 , that is upon adding objects corresponding to 3d topological orders that can be produced by gauging 3d TQFTs contained in 3-Vec 0 , we land on a fusion 3-category equivalent to 3-Vec, i.e. 16 We refer to 3-Vec 0 as a pre-fusion 3-category 17 .
On the other hand, Vec and 2-Vec are Karoubi complete. For example, 2d Z 2 gauge theory has two vacua and, upto an Euler counterterm, can be identified with the element obtained as the direct sum of two copies of Vec.
Physically Motivated Definition of p-Vec. Going along the above lines, we would like to define p-Vec as the simplest fusion p-category. Physically, we would want p-Vec to describe the simplest p-dimensional non-anomalous topological orders that are closed under condensations.
For this purpose, let us begin by defining a monoidal p-category p-Vec 0 (3.21) as the category describing the trivial p-dimensional TQFT and its direct sums. This category has a single simple object (upto isomorphism) corresponding to the identity codimension-1 defect.
Recall that, for a monoidal p-category C, Ω(C) := End 1 (C) is a monoidal (p − 1)-category obtained by restricting to endomorphisms of the identity object of C. We denote by Ω p (C) the monoidal category obtained by applying Ω p−1 to the monoidal category Ω(C). 16 To see this, first of all note that 3-Vec comprises of all topological orders admitting topological/gapped boundaries. On the other hand, 3-Vec 0 comprises only of (direct sums of) trivial topological order. Now, a topological order O1 is obtained by (generalized) gauging another topological order O2 if and only if there exists a topological interface from O1 to O2. If O2 is trivial, then such a topological interface is the same as having a topological boundary for O1. Combining these statements together, we see that Kar(3-Vec 0 ) ∼ = 3-Vec. 17 That is, we define a pre-fusion p-category C to be a category satisfying all the nice properties required to be a fusion category except Karoubi completion. The Karoubi completion Kar(C) of a pre-fusion category C is a fusion p-category.
We then have Thus, p-Vec 0 has a single simple 1-endomorphism (upto isomorphism) of the identity object corresponding to the identity codimension-2 defect, and so on because Note that p-Vec 0 is always a pre-fusion category, but may or may not be a fusion category.
For low-dimensional cases we have and so it is fusion, but as we saw above 3-Vec 0 is not fusion. with p-Vec 0 being pre-fusion, and p-Vec and O p being fusion p-categories.

Higher Representations
Categories Associated to Higher-Groups. A p-group Γ can be converted into a monoidal q-category C Γ q for q ≥ p − 1. These categories capture the topological properties of the classifying space of the p-group Γ. Note that the categories C Γ q are non-C-linear. Let us discuss some simple cases: To a 1-group (i.e. an ordinary group, or a 0-form symmetry group) Γ = Γ (0) , we can first of all associate a 0-category which is the group itself. The p-category C Γ (0) p associated to Γ (0) has isomorphism classes of simple objects labeled by elements of Γ (0) , whose fusion is controlled by group multiplication.
Consider a (p+1)-group with only non-trivial component being a p-form symmetry group Γ (p) . We label the corresponding monoidal categories as C Γ (p) q for q ≥ p. The q-category C Γ (p) q has a single simple object (upto isomorphism), Ω n (C Γ (p) p ) has a single simple object (upto isomorphism) for n < p, and Ω p (C Γ (p) where we have used the categories C Γ (0) * defined above.
Consider now a 2-group Γ, which contains a 0-form group Γ (0) , a 1-form group Γ (1) and a Postnikov class valued in the group cohomology 18 [ω] ∈ H 3 (Γ (0) , Γ (1) ) . (3.28) The associated 1-category C Γ 1 has simple objects labeled by elements of Γ (0) with their fusion controlled by group multiplication of Γ (0) , and morphisms of the identity object labeled by elements of Γ (1) with their fusion controlled by group multiplication of Γ (1) . The information of the Postnikov class is captured in the associator of simple objects of C Γ 1 .
Higher-Group Graded Higher Vector Spaces. We can linearize the p-category C Γ p by allowing the p-morphisms to be valued in C. Let us call the resulting pre-fusion p-category as p-Vec 0 Γ , (3.29) which can be understood as Γ-graded version of the pre-fusion category p-Vec 0 discussed earlier.
We can now Karoubi complete to define p-Vec Γ := Kar p-Vec 0 Γ , which we refer to as the fusion p-category of Γ-graded p-vector spaces.
Making a QFT Higher-Group Symmetric. A d-dimensional QFT T is converted into a Γ-symmetric d-dimensional QFT T S for a higher-group Γ by choosing a monoidal functor

31)
18 Note that we could also have an action ρ of Γ (0) on Γ (1) , in which case the Postnikov class is valued in the twisted group cohomology H 3 ρ (Γ (0) , Γ (1) ). We are choosing the action ρ to be trivial for simplicity.
where the target category C T is the (d − 1)-category capturing all topological defects of T, known as the symmetry category 19 of T. The functor S is what we referred to as the 'gaugeinvariant coupling of T to Γ background gauge fields' in the previous section.
Higher-Categories of Universal Γ-Symmetric Topological Defects. In the same way as p-dimensional TQFTs provide topological defects for any d-dimensional QFT T, Γsymmetric p-dimensional TQFTs provide Γ-symmetric topological defects for any Γ-symmetric This universal sector of Γ-symmetric topological defects is expected to be described by a monoidal (d−1)-category T Γ d−1 whose objects are Γ-symmetric (d−1)-dimensional TQFTs and morphisms are Γ-symmetric topological defects/interfaces of Γ-symmetric (d−1)-dimensional TQFTs. Note that T Γ d−2 is contained inside T Γ d−1 as the (d − 2)-category formed by endomorphisms (i.e. the Γ-symmetric topological defects) of the identity object (i.e. the trivial We can recognize T Γ p as the monoidal p-category of functors Here BC is a p-category built from a monoidal (p − 1)-category C as follows: BC contains a single object and the q-morphisms of BC are (q − 1)-morphisms of C (with 0-morphisms being objects).
This is easy to see: From our previous definition, a p-dimensional TQFT described by an object T ∈ T p is made Γ-symmetric by choosing a monoidal functor where End T (T p ) is the monoidal (p − 1)-category formed by endomorphisms of the object T of T p . But such a functor is equivalent to a functor of the form (3.32).
We can similarly define monoidal p-category O Γ p of Γ-protected non-anomalous p-dimensional topological orders as the monoidal p-category of functors Higher-Representations of Higher-Groups. Recall that a finite-dimensional represen- where V is a finite-dimensional vector space and End(V ) is the space of linear maps from V to itself. Such a map is equivalent to a functor where Vec is the 1-category of finite-dimensional vector spaces 20 . Such monoidal functors form the category Rep(Γ (0) ) of finite-dimensional representations of Γ (0) .
We can extend the above definition by changing the target category involved in (3.36).
with M being a monoidal category, can be referred to as representations of Γ (0) valued in M and generate a monoidal category Rep M (Γ (0) ). p-Rep p-Vec 0 (Γ (0) ) to capture p-dimensional Γ (0) -protected topological orders in which part of Γ (0) symmetry is spontaneously broken leading to multiple vacua permuted by the Γ (0) action, such that in each vacuum a subgroup Γ (0) ′ of Γ (0) is spontaneously preserved and that vacuum carries additionally a p-dimensional SPT phase protected by Γ (0) ′ 0-form symmetry. In this case, the other two p-categories p-Rep p-Vec (Γ (0) ) and p-Rep Op (Γ (0) ) capture more general Γ (0) -protected topological orders including SET phases. which are known to be classified (upto isomorphism) by the group cohomology H 2 (Γ (0) , U (1)) recovering the well-known classification of such SPT phases.

Now we can perform the higher-categorical generalization. Functors
Projective Higher-Representations. Just like we defined higher-representations above, we can define projective higher-representations of a q-group Γ as follows. To define projective p-representations, we need to first of all choose a target monoidal (p + 1)-category M p+1 and an object I p+1 ∈ (p + 1)-Rep M p+1 (Γ) such that I p+1 is a monoidal functor of the form Γ-Anomalous TQFTs. A p-dimensional Γ-anomalous TQFT T S , or in other words a pdimensional (gravitationally non-anomalous) TQFT T with a coupling S of T to Γ background fields afflicted with a 't Hooft anomaly, is defined as a projective representation where the SPT phase I p+1 ∈ T Γ p+1 is known as the (p + 1)-dimensional anomaly theory capturing the 't Hooft anomaly associated to the p-dimensional Γ-anomalous TQFT T S .

Modules and Bimodules
In this subsection, we very roughly sketch how non-universal Γ-symmetric (topological or non-topological) defects of a Γ-symmetric d-dimensional QFT T S can be constructed.  Figure 9. The figure depicts one of the possible conditions that topological operators in S have to satisfy for T S to be a Γ-symmetric d-dimensional QFT. In fact, any two string diagrams related by a topological rearrangement of topological operators in S (such that the topological move leaves the boundary of the string diagram invariant) need to be equal. Figure 10. To define a coupling J of D p to Γ background gauge fields, we need to choose topological operators lying at the junctions of D p and topological operators generating Γ higher-group symmetry. In the figure we have illustrated such a junction operator J (γq) p−q−1 lying at the junction of D p with a bulk topological codimension-(q + 1) operator labeled by element γ q ∈ Γ (q) , namely the q-form symmetry component of Γ. This is some of the most basic data of J . We also need to choose junctions of D p with the junctions (shown in green in figure 8) of topological operators D  to be a Γ-symmetric p-dimensional defect of T S . In fact, any two string diagrams related by a topological rearrangement of topological operators in J and S (such that the topological move leaves the boundary of the string diagram invariant) need to be equal.
Let us note that the above information is not sufficient to specify a codimension-1 Γsymmetric defect. In this case the coupling J needs to be refined into left and right couplings J L and J R . We discuss this refinement later in this subsection.
In favorable situations, when there are no associators (coherence relations) for Γ topological defects in the presence of D p , the choice of coupling amounts to the choice of a monoidal functor J : where C Dp is the monoidal (p − 1)-category describing symmetries localized along D p . If D p is topological, it is an object of the p-category Ω d−p (C T ) and we have is the endomorphism (p − 1)-category of D p ∈ Ω d−p (C T ). The map from topological interfaces described in the previous paragraph to the topological sub-defects of D p chosen by the above functor J is obtained by a folding operation, see figure 12.
One can always implement the folding operation to convert information about J into a map of the form (3.48), but it will in general not be a monoidal functor. There will be obstructions for example of the type shown in figure 13. Finally, for the combined coupling J = (J L , J R ) to be fully gauge-invariant, we have to In fact, above we did not describe full information for converting a codimension-1 defect Implementing the Conditions in 2d. The implementation of the above set of conditions is best understood for 2d QFTs and Γ = Γ (0) a 0-form symmetry group. This was reviewed using the modern language of symmetries in [41].   Similarly, coupling J R shown in figure 14 satisfying the condition shown in figure 15 converts (I d−1 , J R ) into a right module for the algebra A R . That is, the category of Γ Higher-Dimensions. In a similar fashion, [8,9] implemented the various conditions discussed in this subsection for d = 3 QFTs T with very special choices of symmetry 2-categories C T and special classes of 2-group symmetries. A systematic exploration of the conditions discussed here in various dimensions with various symmetry categories and for various types of higher-groups would be very interesting to tackle in future works.

A Program for Classification of Non-Invertible Symmetries
In section 2, we provided a rather general physical formalism, based on gauging of invertible symmetries, that can be used in a variety of ways to construct non-invertible symmetries. In fact, we showed that several constructions of non-invertible symmetries of higher-dimensional QFTs appearing in recent literature describe special examples of the overarching structure presented here.
In section 3, we attempted to formalize the physical construction of section 2 into precise mathematical objects. We were successful at formalizing parts of the structure, while for the remaining parts we provided an intuitive approach in subsection 3.3 that can be made mathematically precise using the machinery of higher-category theory.
Thus, section 3 should be viewed as providing all the essential mathematical ideas required to make the physical construction of section 2 concrete and amenable to computations. Using these ideas, one should be able to concretely construct many different kinds of non-invertible symmetries carrying out the procedures detailed in section 2, as discussed below: • First of all, one can consider understanding the universal symmetries that every Some examples of theta symmetries were concretely discussed in great computational detail in [7,8]. They analyzed the categories 2-Rep T 1 (Γ) = 2-Rep Vec (Γ) = 2-Rep(Γ) for Γ a 2-group symmetry, which includes purely 0-form symmetry and purely 1-form symmetry. Extension to the computation of (d − 1)-Rep T d−1 (Γ) for other values of d and various types of higher-groups Γ would be an interesting problem to tackle in future works.
• The next level of complexity is the construction of non-universal symmetries of T S /Γ, namely those symmetries that arise from those topological defects of T that cannot be constructed by stacking TQFTs on top of T.
A well-known example of such symmetries are provided by the duality defects of 4d QFTs discussed in [2,3]. A systematic exploration of various kinds of possible duality defects utilizing all the ingredients described in section 2 would be an interesting problem to tackle in future works.
Such symmetries are also systematically studied in the upcoming paper [9] for Γ = Γ (0) a 0-form symmetry group and d = 3. Recall that we did not provide a precise mathematical recipe for the computation of such non-universal symmetries, but rather sketched some mathematical ideas in subsection 3.3. Thus, the upcoming paper [9] should be viewed as evidence that the ideas of subsection 3.3 can be converted into precise mathematical computations that, despite the occurrence of many subtleties, can be concretely carried out. We provide numerous checks for the validity of these computations in [9] and encounter interesting phenomena like symmetry fractionalization on top of condensation defects.
Generalizing the analysis of [9] to arbitrary 2-groups Γ in d = 3, and/or to higher d would be a very interesting direction for future research.
• A generalization of the above non-universal symmetries involves the understanding of topological interfaces from T S /Γ to T starting from codimension-1 topological defects of T. Such interfaces can be composed with known interfaces from T to T S /Γ to construct new codimension-1 topological defects of T.
An example of this procedure was used by [6] to construct non-invertible symmetries of 4d QFTs using ABJ anomalies. A systematic exploration of such topological interfaces has not been undertaken yet even for simple examples of Γ, d and C T . It would be a very interesting problem to tackle in future works. It should be noted that such topological interfaces provide examples of "non-invertible dualities" between T S /Γ and T, so would be interesting to explore on their own as generalizations of the standard invertible dualities.
• It should be noted that all of the above methods are also applicable to the construction of condensation defects by simply replacing QFT T by the identity defect (of some dimension) in a QFT T and more generally to the construction of new topological defects by gauging symmetries localized on an arbitrary topological defect in a QFT.
A systematic analysis of condensation defects was performed in [4,7] for d = 3 and gauging of 1-form symmetries on surfaces. Extensions of these works to higher dimensions and gauging of other kinds of higher-group symmetries, and also extensions to gauging of symmetries localized on non-identity defects would be interesting problems to tackle in future works.