Categorification of algebraic quantum field theories

This paper develops a novel concept of 2-categorical algebraic quantum field theories (2AQFTs) that assign locally presentable linear categories to spacetimes. It is proven that ordinary AQFTs embed as a coreflective full 2-subcategory into the 2-category of 2AQFTs. Examples of 2AQFTs that do not come from ordinary AQFTs via this embedding are constructed by a local gauging construction for finite groups, which admits a physical interpretation in terms of orbifold theories. A categorification of Fredenhagen's universal algebra is developed and also computed for simple examples of 2AQFTs.


Introduction and summary
An algebraic quantum field theory (AQFT) is a functor A : C → Alg K from a category of spacetimes C to the category of associative and unital algebras over a field K, which satisfies certain physically motivated axioms such as Einstein causality and the time-slice axiom [HK64,BFV03,FV15]. The (non-commutative) algebra A(c) that is assigned to an object c ∈ C is interpreted as the algebra of quantum observables of the theory on the spacetime c.
Describing quantum observables in terms of ordinary algebras in Alg K is however insufficient to capture the important, but rather subtle, higher categorical structures that feature in gauge theories. For instance, in the context of the BRST/BV formalism [FR12,FR13], the quantum observables of a gauge theory are described by differential graded algebras (dg-algebras) and the latter contain in general more information than their 0-th cohomology, which is the ordinary algebra of gauge invariant quantum observables. An axiomatic framework for homotopy-coherent AQFTs with values in dg-algebras was developed in [BSW19], see also [BS19] for a review and [BBS19] for concrete examples. In these works it was also shown that the higher structures encoded by dg-algebras are crucial for formalizing descent (i.e. local-to-global) properties of quantum gauge theories.
The main aim of this paper is to develop a novel kind of higher categorical AQFTs, which describe quantum observables in terms of locally presentable K-linear categories. As we explain in detail below, such AQFTs are more sensitive to global aspects of quantum gauge theories, e.g. finite gauge transformations, than those assigning dg-algebras.
In order to motivate why it is reasonable to describe quantum observables of gauge theories by locally presentable K-linear categories, let us first recall why ordinary AQFTs assign associative and unital K-algebras. From the point of view of quantum theory, a (non-commutative) algebra A ∈ Alg K is interpreted as a quantized function algebra on the phase space X of a physical system, i.e. A arises as a (deformation) quantization of the commutative algebra O(X) of Kvalued functions on X. If X is a sufficiently "nice" space (in technical terms, X is an affine scheme over K), there is no loss of information when passing from X to its function algebra O(X). This explains why it is justified to quantize the space X by quantizing its function algebra O(X).
However, many important examples of phase spaces that feature in physics are not of this "nice" kind. For instance, if the phase space X is a stack, as it happens to be in a gauge theory, it is in general not true that X is faithfully encoded by its function algebra O(X), which in this case is a dg-algebra, see e.g. [Toe06]. As an illustrative example, let G be a finite group and consider the quotient stack BG := { * }//G, which is a non-trivial stack, namely the classifying stack of principal G-bundles. The corresponding function dg-algebra O(BG) = C • (G, K) is then given by the group cochains with values in the trivial G-representation K. Taking for example G = Z 2 , the cyclic group of order 2, all cohomology groups H n (Z 2 , K) = 0, for n = 0, are trivial if K has characteristic zero. It then follows that O(BZ 2 ) ≃ H 0 (Z 2 , K) ≃ K = O({ * }) is quasi-isomorphic to the function algebra of the point { * }, i.e. all information about the group G = Z 2 is lost when passing from the stack BG to its function algebra. As a consequence, it is in general not reasonable to quantize a stack X by quantizing its function dg-algebra O(X). We would like to emphasize that these issues arise for finite gauge transformations and not for infinitesimal gauge transformations. In particular, every Lie algebroid X = Y //g is completely determined by its function dg-algebra, which in this case is given by the Chevalley-Eilenberg cochains O(X) = CE • (g, O(Y )) on the Lie algebra g with values in the ordinary function algebra O(Y ) of the affine scheme Y .
The feature that stacks are in general not completely determined by their function dg-algebras is well-known to algebraic geometers, see e.g. [Bra14] for an excellent overview, who have also proposed the following interesting solution: Instead of assigning a function dg-algebra O(X) to a space or stack X, it is better to assign the category QCoh(X) of quasi-coherent sheaves on X. The latter is a locally presentable symmetric monoidal K-linear category that should be interpreted roughly as the category of vector bundles over X. This is indeed a better choice because, by a theorem of Lurie [Lur04], every geometric stack X can be reconstructed from its quasi-coherent sheaf category QCoh(X). This fact becomes evident in our illustrative example BG from above: We observe that QCoh(BG) = Rep K (G) is the symmetric monoidal category of K-linear representations of G. By Tannakian reconstruction, Rep K (G) encodes the full information about the group G, hence QCoh(BG) is indeed much richer than the function dg-algebra O(BG) considered in the previous paragraph (think of G = Z 2 for example). Furthermore, for a "nice" space X, i.e. an affine scheme over K, the usual function algebra O(X) can be recovered as follows: One finds that in this case QCoh(X) ≃ Mod O(X) is the symmetric monoidal K-linear category of (right) modules over the commutative algebra O(X), hence O(X) can be reconstructed from QCoh(X) as the endomorphism algebra End(O(X)) ∼ = O(X) of the rank 1 free module O(X) ∈ Mod O(X) , i.e. as the endomorphism algebra of the monoidal unit of QCoh(X). This means that for "nice" spaces the function algebra perspective and the quasi-coherent sheaf category perspective are compatible and carry the same information.
The previous paragraph explained the need to move from the function algebra perspective to the quasi-coherent sheaf category perspective. This, however, raises another question: What does it mean to quantize a quasi-coherent sheaf category? As an illustrative example, let us start with the case where the space X is "nice", i.e. affine, and assume that we already have a non-commutative algebra A ∈ Alg K that quantizes the function algebra O(X). We may then form the locally presentable K-linear category Mod A of right A-modules and interpret it as a quantization of the quasi-coherent sheaf category QCoh(X) ≃ Mod O(X) . It is important to observe the following structural difference between Mod A and Mod O(X) : The tensor product ⊗ O(X) on Mod O(X) is only well-defined because O(X) is a commutative algebra and hence every right O(X)-module is automatically an O(X)-bimodule. Since the quantized algebra A is noncommutative, there is no counterpart on Mod A of the tensor product structure on Mod O(X) . However, there is a counterpart on Mod A of the monoidal unit O(X) ∈ Mod O(X) , which is given by the pointing A ∈ Mod A of the K-linear category Mod A , see also [JF15]. This suggests that the quantization of the symmetric monoidal K-linear category QCoh(X) should be a pointed K-linear category. We would like to emphasize that this idea was made precise in the framework of derived algebraic geometry, see [PTVV13,CPTVV17] and also Toën's ICM 2014 contribution [Toe14]. In this context, the quantization of a derived stack X endowed with an n-shifted symplectic structure is described by a quantization of the symmetric monoidal ∞-category QCoh(X) of quasi-coherent sheaves as an E n -monoidal ∞-category. Since the phase space of a physical system is 0-shifted symplectic, we recover our intuition that one should quantize QCoh(X) as an E 0 -monoidal, i.e. pointed, K-linear category.
Let us explain in more detail the content of the present paper and the results we obtain: In Section 2 we introduce an equivalent perspective on ordinary AQFTs as prefactorization algebras [CG17] with values in the symmetric monoidal category Alg K of associative and unital K-algebras. This perspective will be used in Section 3 to introduce our novel concept of categorified AQFTs (called 2AQFTs) that describe quantum observables by locally presentable K-linear categories, in contrast to associative and unital algebras. In more detail, we define a 2AQFT as a (weak) prefactorization algebra on an orthogonal category C (cf. Definition 2.1) with values in the symmetric monoidal 2-category Pr K of locally presentable K-linear categories. In Section 4 we explore the relationship between ordinary AQFTs and our new concept of 2AQFTs. We construct a biadjunction (cf. Theorem 4.3) that exhibits the 1-category of ordinary AQFTs as a coreflective full 2-subcategory of the 2-category of 2AQFTs, hence our framework for 2AQFTs includes ordinary AQFTs faithfully. In Section 5 we develop a local gauging construction for AQFTs with finite group actions, which allows us to construct concrete examples of 2AQFTs that admit an interpretation as categorified orbifold theories. In particular, we exhibit examples of 2AQFTs (cf. Example 5.7) that are not truncated, i.e. not equivalent to ordinary AQFTs, thus showing that our concept of 2AQFT is a genuine generalization of ordinary AQFTs. The main result of this section is Theorem 5.11: We prove that a categorified orbifold theory is truncated, i.e. equivalent to an ordinary AQFT, if and only if a suitable Hopf-Galois condition is fulfilled, which can be interpreted as a non-commutative analog of the condition that a G-action on a space is free. This matches with the intuition that an orbifold σ-model with a quotient stack X//G as target boils down to an ordinary σ-model with the quotient space X/G as target whenever G acts freely on the space X (cf. Remark 5.12). In Section 6 we study a local-to-global extension for 2AQFTs (called Fredenhagen's universal category), which is a higher categorical analog of Fredenhagen's universal algebra [Fre90,Fre93,FRS92]. We develop concrete models for computing Fredenhagen's universal category and provide simple examples for extensions of 2AQFTs from intervals to the circle S 1 . Appendix A introduces the relevant formalism for 2categorical operad theory that we use throughout this paper.

AQFTs and algebra-valued prefactorization algebras
Let us fix once and for all a field K of characteristic zero. We briefly review the definition of algebraic quantum field theories (AQFTs) on an orthogonal category C in the sense of [BSW17]. We then prove that such theories admit an equivalent description in terms of prefactorization algebras on C with values in the symmetric monoidal category Alg K of associative and unital K-algebras. The latter perspective will be particularly useful for developing a categorification of AQFTs in Section 3.
Definition 2.1. An orthogonal category C := (C, ⊥) is a small category C together with a subset ⊥ ⊆ MorC t × t MorC of the set of pairs of morphisms with a common target, such that: (ii) If (f 1 , f 2 ) ∈ ⊥, then (g f 1 h 1 , g f 2 h 2 ) ∈ ⊥, for all composable C-morphisms g, h 1 and h 2 .
Example 2.2. Let Open(M ) be the category of non-empty open subsets U ⊆ M of a manifold M with morphisms U → V given by subset inclusions U ⊆ V ⊆ M . We introduce an orthogonality relation ⊥ M by declaring two morphisms U 1 , U 2 ⊆ V ⊆ M to be orthogonal if and only if U 1 ∩ U 2 = ∅. The orthogonal category Open(M ) := (Open(M ), ⊥ M ) features in factorization algebras [CG17] and, for M = S 1 the circle, also in chiral conformal AQFT [Kaw15]. 4. An algebraic quantum field theory (AQFT) on an orthogonal category C is a functor A : C → Alg K to the category of associative and unital K-algebras that satisfies the ⊥-commutativity property: For all orthogonal pairs (f 1 : c 1 → t) ⊥ (f 2 : c 2 → t), the induced commutator is zero. The category of AQFTs on C is the full subcategory of the functor category that consists of all ⊥-commutative functors.
In preparation for our definition of categorified AQFTs in Section 3, we prove that the category AQFT(C) is equivalent to the category of Alg K -valued prefactorization algebras on C. The following definition introduces a colored operad P C that generalizes the prefactorization operad of Costello and Gwilliam [CG17] to an arbitrary orthogonal category C. The operad of Costello and Gwilliam is recovered by taking C = Open(M ) for a manifold M , see Example 2.2. For the relevant background and notations for colored operads we refer the reader to [Yau16,BSW17] and also to Appendix A.
Definition 2.5. The prefactorization operad P C associated to an orthogonal category C is the Set-valued colored operad defined by the following data: (1) The objects of P C are the objects of the category C.
(2) The sets of operations are for each object t ∈ C and each tuple of objects c := (c 1 , . . . , c n ) ∈ C n . For the empty tuple c = ∅, we set P C t ∅ := { * t } to be a singleton.
(2.4) (4) The identity operations are 1 := id t ∈ P C t t . (5) The permutation actions P C (σ) : P C t c → P C t cσ are given by P C (σ)(f ) := f σ := (f σ(1) , . . . , f σ(n) ) . (2.5) Let us endow the category Alg K of associative and unital K-algebras with its standard symmetric monoidal structure. The tensor product of two algebras A, B ∈ Alg K is given by the tensor product algebra A ⊗ B. Concretely, that is the tensor product of vector spaces with multiplication given by ( The monoidal unit is K ∈ Alg K and the symmetric braiding is given by the Alg K -morphisms The symmetric monoidal category Alg K has an underlying Set-valued colored operad (see e.g. [EM09]) that we denote by the same symbol Alg K . Concretely, the objects are the objects of Alg K and the sets of operations are given by . (2.6) The composition maps are determined by the monoidal structure, the identity operations are the identity morphisms, and the permutation actions are obtained from the symmetric braiding.
Definition 2.6. The category of Alg K -valued prefactorization algebras on C is defined by where Alg K is regarded as a colored operad (as explained above) and [−, −] denotes the Homcategory from Remark A.6.
Remark 2.7. Let us unpack this definition by using the definitions from Appendix A. (These definitions simplify drastically in the present case because both P C and Alg K are Set-valued colored operads. Hence, all coherence data are necessarily trivial. Non-trivial coherence data will be needed to describe categorified AQFTs in Remark 3.4.) An Alg K -valued prefactorization algebra F ∈ Alg P C (Alg K ) is given by the following data: (1) For each c ∈ C, an associative and unital K-algebra F(c) ∈ Alg K .
(2) For each tuple f = (f 1 , . . . , f n ) ∈ P C t c of mutually orthogonal C-morphisms, an Alg Kmorphism (called factorization product) from the tensor product algebra n i=1 F(c i ). For the empty tuple c = ∅, the Alg K -morphism F( * t ) : K → F(t) associated to the only element * t ∈ P C t ∅ is necessarily the unit of F(t), because K is the initial object in Alg K .
These data are required to satisfy the following axioms: In the last diagram, we have denoted by τ σ : n i=1 F(c σ(i) ) → n i=1 F(c i ) , a σ(1) ⊗ · · · ⊗ a σ(n) → a 1 ⊗ · · · ⊗ a n the Alg K -morphism that permutes the factors of the tensor product algebra.
It is easy to see that every A ∈ AQFT(C) defines an Alg K -valued prefactorization algebra on C by introducing the factorization products (2.11) where µ n A(t) : A(t) ⊗n → A(t) , a 1 ⊗ · · · ⊗ a n → a 1 · · · a n denotes the n-ary multiplication in the associative and unital algebra A(t) ∈ Alg K . Using ⊥-commutativity and f i ⊥ f j , for all i = j, one shows that A(f ) is indeed an Alg K -morphism on the tensor product algebra. Furthermore, every AQFT(C)-morphism ζ : A → B defines an Alg P C (Alg K )-morphism between the corresponding prefactorization algebras, hence we obtain a functor AQFT(C) → Alg P C (Alg K ). Note that this functor is fully faithful.
Conversely, we have the following lemma showing that every Alg K -valued prefactorization algebra is completely determined by an underlying AQFT. (2.12) where µ n F(t) denotes the n-ary multiplication in the associative and unital algebra F(t) ∈ Alg K . In particular, F is completely determined by its underlying functor F : C → Alg K , which satisfies the ⊥-commutativity property from Definition 2.4 and hence defines an AQFT.
Proof. Using the composition maps from Definition 2.5, we compute for all k = 1, . . . , n, where * t ∈ P C t ∅ denotes the unique arity zero operation. The corresponding commutative diagram in (2.9a) then reads as (2.14) Using further that F(id c k ) = id F(c k ) (cf. (2.9b)) and that F( * c i ) : K → F(c i ) is the unit of F(c i ) ∈ Alg K (cf. Remark 2.7), the commutative diagram (2.14) implies that for all a k ∈ F(c k ). By definition of the product of a tensor product algebra, it then follows that for all a 1 ⊗ · · · ⊗ a n ∈ n i=1 F(c i ), which proves (2.12). Using further that every two elements of the form a ⊗ 1 B and 1 A ⊗ b commute in a tensor product algebra A ⊗ B, it follows that the underlying functor F : C → Alg K is ⊥-commutative.
Summing up, we have proven the following Theorem 2.9. For every orthogonal category C, there exists a canonical isomorphism between the category of AQFTs on C and the category of Alg K -valued prefactorization algebras on C.
Remark 2.10. The equivalent description of AQFTs in terms of Alg K -valued prefactorization algebras provides an interesting conceptual interpretation of the ⊥-commutativity property from Definition 2.4. From the prefactorization algebra point of view, every quantum field theory comes with two different kinds of "multiplications", namely the object-wise products µ F(c) : F(c)⊗F(c) → F(c), for every c ∈ C, and the factorization products for every tuple f of mutually orthogonal C-morphisms. These two kinds of "multiplications" are compatible with each other because the factorization products F(f ) are Alg K -morphisms. The ⊥-commutativity property is thus a consequence of an Eckmann-Hilton argument. △

Definition of 2AQFTs
The aim of this section is to introduce a categorification of the concept of AQFTs, which we shall call 2-categorical algebraic quantum field theories (2AQFTs). While ordinary AQFTs assign associative and unital K-algebras to the objects of an orthogonal category C, our novel concept of 2AQFTs will assign locally presentable K-linear categories, cf. [AR94,BCJF15].
Recall that a K-linear category is a category D that is enriched over the symmetric monoidal category Vec K of vector spaces over K. Concretely, this means that we have a vector space D(d, d ′ ) ∈ Vec K of morphisms, for every pair of objects d, d ′ ∈ D, and that the composition Given two Klinear categories D and E, a K-linear functor F : D → E is a functor such that the maps A K-linear category D is called locally presentable if it is 1.) cocomplete, i.e. has all small colimits, and 2.) generated under small colimits by a set Γ ⊂ D 0 of objects that are λ-presentable for some infinite cardinal λ, see e.g. [BCJF15] for a recollection of the relevant material on locally presentable categories. The natural concept of functors F : D → E between two locally presentable K-linear categories D and E is given by co-continuous K-linear functors, i.e. K-linear functors that preserve all small colimits. Natural transformations in this context are just ordinary natural transformations.
Definition 3.1. The operad Pr K of locally presentable K-linear categories is the Cat-enriched colored operad (cf. Definition A.1) defined by the following data: (1) The objects are all locally presentable K-linear categories.
(2) For T and D = (D 1 , . . . , D n ) locally presentable K-linear categories, the category of operations is the full subcategory of the functor category that consists of all functors F : n i=1 D i → T that are K-linear and co-continuous in each variable. For the empty tuple D = ∅, we set Pr K T ∅ := Fun(1, T), where 1 is the category with only one object and its identity morphism.
(3) The composition functors γ :  where flip σ : n i=1 D σ(i) → n i=1 D i is the permutation functor and Id flip σ : flip σ ⇒ flip σ its identity natural transformation.
Remark 3.2. With some abuse of notation, we will sometimes denote by the same symbol Pr K the underlying 2-category of 1-ary operations of the Cat-enriched colored operad from Definition 3.1. It should be clear from the context whether we mean by the symbol Pr K a Cat-enriched colored operad or a 2-category. The underlying 2-category Pr K is known to be (closed) symmetric monoidal with respect to the Kelly-Deligne tensor product D ⊠ E of locally presentable K-linear categories, whose monoidal unit is given by the K-linear category Vec K of vector spaces, see [Kel05] and also [BCJF15] for a review. This symmetric monoidal structure is linked as follows to our Cat-enriched colored operad from Definition 3.1: By the universal property of the Kelly-Deligne tensor product, the categories of operations are equivalent to the categories of co-continuous K-linear functors out of the Kelly-Deligne tensor product. Hence, the Cat-enriched colored operad Pr K can also be understood as the underlying operad of the symmetric monoidal 2-category (Pr K , ⊠, Vec K ). This alternative perspective will become useful in some of our computations in Sections 4, 5 and 6. △ Recalling Theorem 2.9, ordinary AQFTs on C are equivalently Alg K -valued prefactorization algebras, i.e. AQFT(C) ∼ = Alg P C (Alg K ). Replacing the target Alg K with Pr K suggests the following where P C is the prefactorization operad from Definition 2.5 and Pr K is the Cat-enriched colored operad from Definition 3.1.
Remark 3.4. Let us unpack this definition by using the definitions from Appendix A. An object A ∈ 2AQFT(C) is given by the following data: (1) For each c ∈ C, a locally presentable K-linear category A(c) ∈ Pr K .
(2) For each tuple f = (f 1 , . . . , f n ) ∈ P C t c of mutually orthogonal C-morphisms, a functor (called factorization product) that is K-linear and co-continuous in each variable. For the empty tuple c = ∅, this is an object a t := A( * t ) ∈ A(t) (called pointing) that is associated to the only element t | q q q q q q q q q q q q q q These data are required to satisfy the axioms from Definition A.2.
A 1-morphism ζ : A → B in 2AQFT(C) is given by the following data: (1) For each c ∈ C, a co-continuous K-linear functor ζ c : A(c) → B(c). ( Note that, for f = * t ∈ P C t ∅ , this amounts to an isomorphism These data are required to satisfy the axioms from Definition A.4. A 2-morphism Γ : ζ ⇒ κ between two 1-morphisms ζ, κ : A → B in 2AQFT(C) is given by the following data: (1) For each c ∈ C, a natural transformation These data are required to satisfy the axioms from Definition A.5. △ Remark 3.5. Category-valued prefactorization algebras were studied before in the context of factorization homology of 2-manifolds [BZBJ18a,BZBJ18b]. Our framework for 2AQFTs allows us to interpret the examples studied in these papers as 2-dimensional topological AQFTs. This is achieved by considering the orthogonal category Man 2 of 2-dimensional (oriented or framed) manifolds, with orthogonality relation given by disjointness, and restricting to topological theories by considering locally constant prefactorization algebras. △

Inclusion-truncation biadjunction
In this section we explore the relationship between ordinary AQFTs and our novel concept of 2AQFTs from Definition 3.3. We shall show that every A ∈ 2AQFT(C) has an underlying ordinary AQFT π(A) ∈ AQFT(C), which we call the truncation of A. Our truncation construction is given by a 2-functor π : 2AQFT(C) → AQFT(C). We shall also define, for every A ∈ AQFT(C), a 2AQFT ι(A) ∈ 2AQFT(C) that assigns to each object c ∈ C the locally presentable K-linear category ι(A)(c) = Mod A(c) of right A(c)-modules. This construction is given by an inclusion pseudo-functor ι : AQFT(C) → 2AQFT(C). Inclusion and truncation are compatible with each other in the sense that they determine a biadjunction ι ⊣ π, see e.g. [Str80,Str87] and also [LN16] for the relevant bicategorical background. We prove that this biadjunction exhibits AQFT(C) as a coreflective full 2-subcategory of 2AQFT(C). The conceptual meaning and relevance of this result is as follows: On the one hand, ordinary AQFTs can be studied equally well inside the 2-category of 2AQFTs by applying the fully faithful inclusion pseudo-functor ι : There is no loss of information in doing so, because the unit η : id ⇒ π ι of the biadjunction is a natural isomorphism and hence one can recover every A ∈ AQFT(C) from its corresponding 2AQFT ι(A) by applying the truncation 2-functor. On the other hand, the 2-category 2AQFT(C) has in general also objects that do not lie in the essential image of the inclusion pseudo-functor ι. These are the genuine 2AQFTs that are not fully determined by their truncation to an ordinary AQFT. We refer to Section 5 for concrete examples.

Truncation
Given any A ∈ 2AQFT(C), we define its truncation π(A) ∈ AQFT(C) by providing the required data listed in Remark 2.7: (1) For each c ∈ C, we set to be the endomorphism algebra of the pointing a c ∈ A(c). (Note that this is an associative and unital K-algebra, because A(c) is a K-linear category.) (2) For each non-empty tuple f = (f 1 , . . . , f n ) ∈ P C t c of mutually orthogonal C-morphisms, the given functor where • denotes composition in A(t). As noted in Remark 2.7, the Alg K -morphism π(A)( * t ) : K → π(A)(t) associated to the empty tuple * t ∈ P C t ∅ is the unit id at of π(A)(t).
Using the axioms of 2AQFTs from Remark 3.4, it is easy to check that π(A) satisfies the axioms of Alg K -valued prefactorization algebras from Remark 2.7. Hence, π(A) ∈ AQFT(C) is an AQFT by Theorem 2.9.
Let us consider now a 1-morphism ζ : A → B in 2AQFT(C). For each c ∈ C, the K-linear functor ζ c : A(c) → B(c) restricts to endomorphism algebras as ζ c : End(a c ) → End(ζ c (a c )). The coherence map in (3.10) that is associated to * c provides an isomorphism ζ * c : b c → ζ c (a c ) in the category B(c), which we use to define the Alg K -morphism Using the axioms of 1-morphisms of 2AQFTs from Remark 3.4, it is easy to check that π(ζ) : π(A) → π(B) is a morphism of Alg K -valued prefactorization algebras in the sense of Remark 2.7, and hence by Theorem 2.9 a morphism of AQFTs.
Proposition 4.1. For every orthogonal category C, the construction above defines a 2-functor which we call the truncation 2-functor.

Inclusion
Let A ∈ AQFT(C) be an ordinary AQFT, regarded as an Alg K -valued prefactorization algebra via Theorem 2.9. We define its inclusion ι(A) ∈ 2AQFT(C) by providing the data listed in Remark 3.4: (1) For each c ∈ C, we set to be the K-linear category of right A(c)-modules. This is a locally presentable K-linear category, see e.g. [BCJF15].
(2) For each non-empty tuple f = (f 1 , . . . , f n ) ∈ P C t c of mutually orthogonal C-morphisms, the given

which admits a left adjoint functor (called the induced module functor)
The latter functor is clearly K-linear and co-continuous. Observe further that the functor ⊗ n : n i=1 Vec K → Vec K , (V 1 , . . . V n ) → V 1 ⊗ · · · ⊗ V n taking n-ary tensor products of vector spaces induces a functor that is K-linear and co-continuous in each variable. We define by composition (4.10) For the empty tuple c = ∅, we set the pointing ι(A)( * t ) := A(t) ∈ Mod A(t) to be the rank 1 free A(t)-module.
(3) The coherence natural isomorphisms in (3.7) are given by pasting of The natural isomorphisms (⋆) and (⋆⋆) are canonically determined by the coherence isomorphisms for tensor products. (Recall that the induced module functor (4.8) is given by a relative tensor product.) The natural isomorphism (⋆⋆⋆) is canonically determined by uniqueness (up to a unique natural isomorphism) of left adjoint functors and the strict composition property see also (2.9a).
(4) The coherence natural isomorphisms in (3.8) are canonically determined by uniqueness of left adjoint functors and the strict identity property A(id t ) * = id * A(t) = id Mod A(t) of the right adjoints, see also (2.9b).
(5) The coherence natural isomorphisms in (3.9) are given by pasting of The natural isomorphism (⋆) is canonically determined by the coherence isomorphisms for tensor products and the natural isomorphism (⋆⋆) is canonically determined by uniqueness of left adjoint functors and the strict permutation property (τ σ ) A(f σ) * of the right adjoints, see also (2.9c).
Since the coherences in (3-5) are canonically given by coherence isomorphisms, one confirms that ι(A) ∈ 2AQFT(C) satisfies the axioms of 2AQFTs from Remark 3.4.
Let us consider now a morphism ζ : A → B in AQFT(C). Then the following data defines a 1-morphism ι(ζ) : ι(A) → ι(B) in 2AQFT(C), see also Remark 3.4: (1) For each c ∈ C, we set to be the K-linear and co-continuous induced module functor along the given Alg K -morphism (2) The coherence natural isomorphisms in (3.10) are given by pasting of (4.14) where (⋆) is canonically determined by the coherence isomorphisms for tensor products and (⋆⋆) is canonically determined by uniqueness of left adjoint functors and the strict naturality right adjoints, see also (2.10).
Proposition 4.2. For every orthogonal category C, the construction above defines a pseudofunctor which we call the inclusion pseudo-functor.

Biadjunction
We now prove that the pseudo-functors in Propositions 4.1 and 4.2 determine a biadjunction, with the inclusion ι : AQFT(C) → 2AQFT(C) as the left adjoint and the truncation π : 2AQFT(C) → AQFT(C) as the right adjoint. We describe first the unit η : id ⇒ π ι of this biadjunction, which is easier than the counit ǫ : ι π ⇒ id because AQFT(C) is just a 1-category, hence η is a natural transformation between ordinary functors. Let A ∈ AQFT(C) be an ordinary AQFT. From the explicit descriptions of π and ι in Sections 4.1 and 4.2, we observe that is the endomorphism algebra of the rank 1 free module A(c) ∈ Mod A(c) , for every c ∈ C. We define the A-component η A : A → π ι(A) of the unit η as the AQFT(C)-morphism determined by the component Alg K -morphisms is obvious, hence we have constructed the desired natural transformation η : id ⇒ π ι. We further observe that η is a natural isomorphism because each of its components (4.17) is an isomorphism, with inverse given by the ) that evaluates an endomorphism h on the unit element 1 A(c) ∈ A(c).
Using the natural transformation η : id ⇒ π ι, we can define, for every A ∈ AQFT(C) and B ∈ 2AQFT(C), a functor between Hom-categories where we note that the target is a set, which we regard as a category with only identity morphisms.
Theorem 4.3. Let C be any orthogonal category. Then the functor (4.18) is an equivalence of categories, for every A ∈ AQFT(C) and B ∈ 2AQFT(C). As a consequence, we obtain a biadjunction ι : whose left adjoint is the inclusion pseudo-functor from Proposition 4.2 and whose right adjoint is the truncation 2-functor from Proposition 4.1. Because the unit η : id ⇒ π ι is a natural isomorphism, this biadjunction exhibits the category AQFT(C) of ordinary AQFTs as a coreflective full 2-subcategory of the 2-category 2AQFT(C).
Proof. Let us recall from [BCJF15] the following fact: For any associative and unital K-algebra A ∈ Alg K , denote by BEnd(A) the full K-linear subcategory of Mod A ∈ Pr K on the object A ∈ Mod A . (Note that BEnd(A) is just the endomorphism algebra End(A) regarded as a K-linear category with only one object.) Then, for any locally presentable K-linear category D ∈ Pr K , the restriction along the inclusion BEnd(A) ⊆ Mod A of K-linear categories induces an equivalence (i.e. a fully faithful and essentially surjective functor) from the full subcategory of Fun(Mod A , D) that consists of K-linear and co-continuous functors to the full subcategory of Fun(BEnd(A), D) that consists of K-linear functors. Using this result, we can check that the functor (4.18) is fully faithful and essentially surjective as claimed.
Full: Let ζ, κ : ι(A) → B be 1-morphisms in 2AQFT(C) such that ζ = κ : A → π(B) in AQFT(C). (Recall that AQFT(C) only has identity 2-morphisms.) For each c ∈ C, consider the morphism κ * c • (ζ * c ) −1 : ζ c (A(c)) → κ c (A(c)) in B(c). Using ζ = κ, one shows that this defines a natural transformation between the restrictions along the inclusion functor BEnd(A(c)) ⊆ Mod A(c) of the co-continuous K-linear functors ζ c , κ c : Mod A(c) → B(c). Recalling that the restriction functor (4.21) is full, there exists a natural transformation Γ c : ζ c ⇒ κ c whose A(c)component is κ * c • (ζ * c ) −1 . We still have to prove that the collection Γ c , for all c ∈ C, defines a 2-morphism Γ : ζ ⇒ κ between the 1-morphisms ζ, κ : ι(A) → B in 2AQFT(C). This amounts to showing that the diagram ). This functor is by construction K-linear, i.e. ζ c ∈ Lin K (BEnd(A(c)), B(c)). Since the functor (4.21) is essentially surjective, there exists a K-linear and co-continuous functor κ c ∈ Lin K,c (Mod A(c) , B(c)) and a natural isomorphism κ * c from the functor ζ c to the restriction along the inclusion BEnd(A(c)) ⊆ Mod A(c) of the functor κ c . Because A(c) ∈ BEnd(A(c)) is the only object, the natural isomorphism κ * c consists of a single B(c)-isomorphism κ * c : b c → κ c (A(c)), with naturality being encoded in the condition κ c (h) • κ * c = κ * c • ζ c (h), for all h ∈ End(A(c)). Note that we have just constructed part of the data defining a 1-morphism κ : ι(A) → B in 2AQFT(C) (cf. Remark 3.4). To complete the data, we have to construct, for each f ∈ P C t c , a natural isomorphism to B(t) that are K-linear and co-continuous in each variable. Using again the equivalences in (3.4) and (4.21), this problem is equivalent to constructing a B(t)-isomorphism, denoted with a slight abuse of notation also by κ f : The counit allows us to detect whether an object B ∈ 2AQFT(C) lies in the essential image of the inclusion pseudo-functor ι : AQFT(C) → 2AQFT(C). We say that B ∈ 2AQFT(C) is truncated if the corresponding component ǫ B : ι π(B) → B of the counit is an equivalence in 2AQFT(C). This means that a truncated 2AQFT B is fully determined by its truncation π(B) ∈ AQFT(C) as it can be reconstructed (up to equivalence) by applying the inclusion pseudo-functor ι. Our goal in Section 5 is to construct examples of 2AQFTs that are not truncated in the above sense. △

Gauging construction and orbifold 2AQFTs
We present a construction of 2AQFTs from the data of an ordinary AQFT A ∈ AQFT(C) that is endowed with an action of a finite group G. This construction can be interpreted physically as a local gauging of A with respect to G and the resulting 2AQFT as the corresponding categorified orbifold theory, see Proposition 5.4 and Remark 5.6 below. Let us start with introducing some relevant terminology. Remark 5.2. Our choice of terminology in Definition 5.1 is motivated by the following equivalent perspective on G-equivariant AQFTs. Let us denote by Rep K (G) the category of K-linear representations of G. Recall that this is a (closed) symmetric monoidal category with monoidal product given by the tensor product V ⊗ W of representations, monoidal unit given by the trivial representation K and symmetric braiding given by the flip map. Hence, we can introduce the category G-Alg K := Alg As (Rep K (G)) of associative and unital algebras in Rep K (G), which are also called G-equivariant associative and unital K-algebras. (Note that for the trivial group G = {e}, this is just the category Alg K that we considered in Section 2.) It is then easy to check that a G-equivariant AQFT (A, ρ), as introduced in Definition 5.1, is the same data as a functor C → G-Alg K to the category of G-equivariant associative and unital K-algebras that satisfies the ⊥-commutativity property from Definition 2.4. From this perspective, morphisms of G-equivariant AQFTs are just natural transformations of functors from C to G-Alg K . △ Given any G-equivariant AQFT (A, ρ) ∈ G-AQFT(C), one can construct its associated orbifold theory A G 0 ∈ AQFT(C) by taking the invariants of the action ρ : G → Aut(A), see e.g. [Xu00,Mug05]. We have added a subscript 0 to emphasize that, as we shall show in Proposition 5.4, the traditional orbifold theory A G 0 ∼ = π(A G ) is only the truncation of a richer (in general not truncated) categorified orbifold theory A G ∈ 2AQFT(C). The latter will be described by a gauging construction that we develop in this section. We also refer to Remark 5.6 for a physical interpretation of our gauging construction and the resulting categorified orbifold theory.
As a preparation, let us briefly recall some standard facts and constructions from the theory of equivariant algebras and modules. As already mentioned in Remark 5.2, the representation category Rep K (G) of a finite group G is a (closed) symmetric monoidal category, hence we can introduce the category G-Alg K of G-equivariant associative and unital K-algebras. Analogously to the non-equivariant case Alg K from Section 2, this category is symmetric monoidal with monoidal product the tensor product algebra A ⊗ B (endowed with the tensor product G-action), monoidal unit the algebra K (endowed with the trivial G-action) and symmetric braiding given by the flip map. For every object A ∈ G-Alg K , one can introduce the locally presentable K-linear category G-Mod A := Mod A (Rep K (G)) of G-equivariant right A-modules. An object in this category is an object V ∈ Rep K (G) together with a Rep K (G)-morphism V ⊗ A → V that satisfies the usual axioms of a right A-action. Morphisms are Rep K (G)-morphism that preserve the right A-actions. Similarly to the non-equivariant case, given any morphism κ : A → B in G-Alg K , one can define a K-linear induced module functor κ ! = (−) ⊗ A B : G-Mod A → G-Mod B . This functor has a right adjoint given by the restriction functor κ * : G-Mod B → G-Mod A . As a consequence, κ ! is a co-continuous K-linear functor between locally presentable K-linear categories, i.e. a 1-morphism in the 2-category Pr K .
Let now (A, ρ) ∈ G-AQFT(C) be any G-equivariant AQFT on C. We define its gauging A G ∈ 2AQFT(C) by a G-equivariant generalization of the inclusion pseudo-functor ι from Section 4.2. Concretely, A G is described by the following data as listed in Remark 3.4: (1) For each c ∈ C, we set to be the locally presentable K-linear category of G-equivariant right modules over the G-equivariant associative and unital K-algebra A(c) ∈ G-Alg K .
(2) For each non-empty tuple f = (f 1 , . . . , f n ) ∈ P C t c of mutually orthogonal C-morphisms, we define the functor where ⊗ n : (V 1 , . . . , V n ) → V 1 ⊗ · · · ⊗ V n is the functor assigning the n-ary tensor product of representations (equipped with the induced structure of a G-equivariant module over the tensor product of algebras) and A(f ) ! is the induced module functor for G-equivariant modules along the G-Alg K -morphism A(f ) : Note that the functor (5.2) is K-linear and co-continuous in each variable, i.e. it defines a 1-operation in Pr K . For the empty tuple c = ∅, we set the pointing to be A G ( * t ) := A(t) ∈ G-Mod A(t) .
(3-5) The coherence isomorphisms for A G are completely analogous to the ones for the inclusion ι(A) ∈ 2AQFT(C) from Section 4.2.
Let us consider now a morphism ζ : (A, ρ) → (B, σ) in G-AQFT(C). Then the following data defines a 1-morphism ζ G : A G → B G in 2AQFT(C), see also Remark 3.4: (1) For each c ∈ C, we set to be the K-linear and co-continuous induced module functor for G-equivariant modules along the G-Alg K -morphism ζ c : A(c) → B(c).
Proposition 5.3. For every orthogonal category C and finite group G, the construction above defines a pseudo-functor which we call the gauging construction.
The following result relates our novel gauging construction to the traditional concept of orbifold theories from [Xu00,Mug05].
Proposition 5.4. For every G-equivariant AQFT (A, ρ) ∈ G-AQFT(C), there exists a natural isomorphism A G 0 ∼ = π(A G ) in AQFT(C) between the traditional orbifold theory A G 0 (that assigns subalgebras of G-invariants) and the truncation (cf. Section 4.1) of the gauging construction A G ∈ 2AQFT(C) from Proposition 5.3.
Proof. From the description of the truncation 2-functor in Section 4.1, we obtain that π(A G )(c) = End(A(c)) is the endomorphism algebra of the G-equivariant module A(c) ∈ G-Mod A(c) , for each c ∈ C. Since morphisms in G-Mod A(c) preserve by definition the G-action, it follows that End(A(c)) is isomorphic to the subalgebra of G-invariants in A(c), hence π(A G )(c) ∼ = A G 0 (c) is isomorphic to the algebra that is assigned by the traditional orbifold theory A G 0 . Using further the explicit description of the factorization products of π(A G ) from Section 4.1, one checks that this family of Alg K -isomorphisms defines an AQFT(C)-isomorphism π(A G ) ∼ = A G 0 . Naturality of this isomorphism in (A, ρ) ∈ G-AQFT(C) is obvious.
The previous proposition provides a justification for the following terminology.
Remark 5.6. In addition to our result in Proposition 5.4, there is further justification for calling the 2AQFT A G a categorified orbifold theory. The presentation in this remark is intentionally rather informal, which is convenient to convey our main message.
Let us recall that the field configurations of a classical σ-model are given by maps φ : Σ → X from a world-sheet Σ to a target space X. When a finite group G acts on the target space X, one can form the orbifold (i.e. quotient stack) X//G and consider the corresponding orbifold σ-model whose field configurations are now maps φ : Σ → X//G with values in a stack. As a consequence, the "space" of field configurations is a stack too, namely the mapping stack Things get more interesting when we consider the stacky quotient by G in (5.6). From the perspective of [Bra14,Lur04], which we recalled in Section 1, it is better to assign to the stack Fields(U ) in (5.6) its category of quasi-coherent sheaves which is the category of G-equivariant modules over the classical G-equivariant function algebra O (Map(U, X)). The G-equivariant quantization A(U ) of O(Map(U, X)) from the previous paragraph then determines a quantization of this category A G (U ) = G-Mod A(U ) , which we recognize as our gauging construction. Hence, our gauging construction encodes the local aspects of orbifold σ-models. △ We still have to address the important question whether or not it is possible to obtain genuine non-truncated A G ∈ 2AQFT(C) from our gauging construction. This will of course depend on details of the group G and its action ρ : G → Aut(A) on A ∈ AQFT(C). For example, if G = {e} is the trivial group, then the gauging construction from Proposition 5.3 coincides with the inclusion pseudo-functor ι from Proposition 4.2, hence gauging the trivial group G = {e} always leads to truncated 2AQFTs in the sense of Remark 4.4. On the other hand, it is very easy to give simple examples of gaugings that define non-truncated 2AQFTs.
Example 5.7. Let G be a finite group and consider the trivial AQFT A = K ∈ AQFT(C) that assigns A(c) = K ∈ Alg K to every c ∈ C. When endowed with the trivial G-action ρ : G → Aut(K) , g → id K , this defines a G-equivariant AQFT (K, ρ) ∈ G-AQFT(C). Note that the corresponding traditional orbifold theory K G 0 = K is of course the trivial AQFT too. In contrast, the categorified orbifold theory K G ∈ 2AQFT(C) that is obtained from our gauging construction is much more interesting. It assigns to every c ∈ C the representation category K G (c) = G-Mod K = Rep K (G) of the group G and its factorization products K G (f ) = ⊗ n : n i=1 Rep K (G) → Rep K (G) are given by the n-ary tensor products of representations. The pointing K G ( * t ) = K ∈ Rep K (G) is given by the trivial representation. By Proposition 5.4, the truncation π(K G ) ∼ = K G 0 = K of this 2AQFT is the trivial theory Our claim is that the categorified orbifold theory K G ∈ 2AQFT(C) is not truncated, whenever G = {e} is non-trivial. To prove this claim, we consider as explained in Remark 4.4 the corresponding component ǫ K G : ι π(K G ) → K G of the counit of the inclusion-truncation biadjunction. This is a 1-morphism in 2AQFT(C) whose components (ǫ K G ) c : ι π(K G )(c) ≃ Vec K → K G (c) = Rep K (G) are given by co-continuous K-linear functors from the category of vector spaces to the representation category of G. Because 1-morphisms in 2AQFT(C) preserve the pointings (up to coherence isomorphisms), we know that the 1-dimensional vector space K ∈ Vec K is mapped to a trivial representation (ǫ K G ) c (K) ∼ = K ∈ Rep K (G). Using further that every vector space V ∼ = b∈B K ∈ Vec K is isomorphic to a coproduct of the 1-dimensional vector space K (by choosing a basis B) and co-continuity of the functor ǫ K G , we observe that the essential image of (ǫ K G ) c : Vec K → Rep K (G) lies in the full subcategory of trivial G-representations, hence it cannot be an equivalence of categories as every finite group G = {e} admits non-trivial K-linear representations. As a consequence, the component ǫ K G : ι π(K G ) → K G of the counit is not an equivalence in 2AQFT(C) and hence the categorified orbifold theory K G ∈ 2AQFT(C) is not truncated. ▽ Quite remarkably, it is possible to characterize precisely those G-equivariant AQFTs (A, ρ) ∈ G-AQFT(C) whose associated gauging construction A G ∈ 2AQFT(C) is truncated. Our arguments below make use of some standard terminology and results from Hopf-Galois theory, see e.g. [DT89] and also the review article [Mon09]. Let H be a Hopf algebra over K. (In our applications below, H = O(G) = Map(G, K) is the function Hopf algebra of a finite group G.) A right H-comodule algebra is an algebra A ∈ Alg K endowed with a right H-coaction δ : A → A ⊗ H that is an Alg K -morphism. We denote by B := A coH := {a ∈ A : δ(a) = a ⊗ 1 H } ⊆ A the subalgebra of H-coaction invariants.
Definition 5.8. The algebra extension B = A coH ⊆ A is called H-Hopf-Galois if the canonical map is bijective.
Associated to any right H-comodule algebra A are two K-linear categories of interest: First, we have the category Mod H A of right (H, A)-Hopf modules. An object in this category is a right A-module V ∈ Mod A that is endowed with a compatible right H-comodule structure Proof. The left adjoint functor Φ = (−) ⊗ B A is clearly K-linear and co-continuous, i.e. a 1morphism in Pr K . The right adjoint functor Ψ = (−) coH = (−) G 0 assigns the G-invariants (given by a categorical limit), which for actions of finite groups G and char(K) = 0 coincides with the Gcoinvariants (i.e. a categorical colimit). Hence, the right adjoint Ψ is a K-linear and co-continuous functor too and the adjunction (5.10) is in the 2-category Pr K .
The unit η : id ⇒ Ψ Φ of the adjunction (5.10) is given by the components η W : Using again that forming G-invariants coincides with forming G-coinvariants, we find that η : id ⇒ Ψ Φ is a natural isomorphism. Our claim then follows from Theorem 5.9.

Theorem 5.11. Let G be a finite group and (A, ρ) ∈ G-AQFT(C) a G-equivariant AQFT. Then the categorified orbifold theory A G ∈ 2AQFT(C) is truncated if and only if the algebra extension
is O(G)-Hopf-Galois, for all c ∈ C. Proof. Recalling Remark 4.4, the 2AQFT A G ∈ 2AQFT(C) is by definition truncated if the corresponding component ǫ A G : ι π(A G ) → A G of the counit of the inclusion-truncation biadjunction from Theorem 4.3 is an equivalence in 2AQFT(C). The component ǫ A G of the counit is determined uniquely (up to invertible 2-morphisms in 2AQFT(C)) by the condition ǫ A G = id π(A G ) : π(A G ) → π(A G ) on its adjunct under (4.18). Using the explicit description of the inclusion and truncation pseudo-functors from Section 4 and the one of the gauging construction from the present section, one observes that the induced module functors By a straightforward but slightly lengthy calculation, one proves that a 1-morphism in 2AQFT(C) is an equivalence if and only if all its components are equivalences in the 2-category Pr K . (In this proof one uses that every equivalence in any 2-category (here Pr K ) can be upgraded to an adjoint equivalence in order to define the quasi-inverse 1-morphism in 2AQFT(C).) Thus, to prove that A G ∈ 2AQFT(C) is truncated we can equivalently study the components , for all c ∈ C. By Corollary 5.10, these components are equivalences in Pr K if and only if the algebra extension This completes the proof.
Remark 5.12. We would like to emphasize that our result in Theorem 5.11 matches perfectly our physical interpretation of the gauging construction A G ∈ 2AQFT(C) in terms of orbifold σ-models from Remark 5.6. The H = O(G)-Hopf-Galois condition from Definition 5.8 should be interpreted as a non-commutative algebraic generalization of a free G-action on a space, see e.g. [Mon09, Examples 2.11 and 2.12]. Because the quotient stack X//G ≃ X/G corresponding to a free G-action is equivalent to the ordinary quotient space, the resulting "orbifold" σ-model in this case is just an ordinary σ-model with target space X/G. In particular, for free G-actions one does not expect higher categorical features in the corresponding "orbifold" σ-model. This is precisely what we have proven in Theorem 5.11 for orbifold quantum field theories. △ We conclude this section by presenting more examples of non-truncated and also truncated categorified orbifold theories A G ∈ 2AQFT(C).
Example 5.13. Let us denote by Disk(S 1 ) ⊂ Open(S 1 ) the full subcategory of all non-empty open intervals I ⊂ S 1 in the circle S 1 . Restricting the orthogonality relation ⊥ S 1 from Example 2.2, we obtain a full orthogonal subcategory Disk(S 1 ) ⊂ Open(S 1 ). Objects A ∈ AQFT(Disk(S 1 )) are interpreted as chiral conformal AQFTs [Kaw15]. In this example we set K = C to be the field of complex numbers. Let us consider the following specific theory, which is called the chiral free boson. To each interval I ⊂ S 1 , we assign the canonical commutation relations (CCR) algebra where C ∞ c (I) denotes the vector space of compactly supported real-valued functions on I ⊂ S 1 and T ⊗ C C ∞ c (I) := ∞ n=0 (C ∞ c (I) ⊗ R C) ⊗n ∈ Alg C is the complexified free algebra. (For later use, we have made the deformation parameter ∈ R explicit in the commutation relations for ϕ 1 , ϕ 2 ∈ C ∞ c (I).) To each interval inclusion ι J I : I → J, we assign the Alg K -morphism A(ι J I ) : A(I) → A(J) that is defined on the generators by pushforward (i.e. extension by zero) of compactly supported functions. This defines an AQFT A ∈ AQFT(Disk(S 1 )) in the sense of Definition 2.4. Let us consider the representation ρ : G = Z 2 → Aut(A) of the cyclic group of order 2 that is defined on the generators of A(I) by multiplication with ±1, i.e. ρ(±1)(ϕ) = ±ϕ, for all ϕ ∈ A(I). This defines a Z 2 -equivariant AQFT (A, ρ) and we can form the corresponding categorified orbifold theory A Z 2 ∈ 2AQFT(Disk(S 1 )) from Definition 5.5. To find out whether this theory is truncated or not, we use our results from Theorem 5.11. Let us consider an arbitrary interval I ⊂ S 1 and set A := A(I). Observe that the subalgebra B := A Z 2 0 ⊂ A of Z 2 -invariants is the even part of the algebra (5.11). Regarding A = A(I) as a B-bimodule, we obtain a direct sum decomposition A = B ⊕ V , where V is the odd part of (5.11). Hence, the source of the canonical map (5.8) is isomorphic to Using further that A ⊗ O(G) ∼ = g∈G A, the canonical map (5.8) explicitly reads as Note that the canonical map β is bijective if and only if the map µ : Let us consider first the case where the deformation parameter = 0 is zero, which describes a classical (i.e. not quantized) field theory. In this case (5.11) is a complexified symmetric algebra over C ∞ c (I) and the map µ : V ⊗ B V → B is not surjective because its image is at least quadratic in the generators. This implies that the canonical map β in (5.12) is not bijective, hence by Theorem 5.11 the categorified orbifold theory A Z 2 ∈ 2AQFT(Disk(S 1 )) for = 0 is non-truncated.
The situation changes drastically in the quantum case = 0. From the canonical commutation relations in (5.11), we deduce that one can always find two generators ϕ 1 , ϕ 2 ∈ C ∞ c (I) ⊆ V ⊆ A that satisfy [ϕ 1 , ϕ 2 ] = i 1. Dividing by , which is possible because we assumed that = 0, we can now prove that the map µ : V ⊗ B V → B is bijective. For surjectivity, consider an arbitrary b ∈ B and observe that where we also used that ϕ 1 v j ∈ B and ϕ 2 v j ∈ B. Theorem 5.11 then implies that the categorified orbifold theory A Z 2 ∈ 2AQFT(Disk(S 1 )) for = 0 is truncated. Summing up, we have seen an example of a non-truncated classical orbifold field theory that is quantized to a truncated orbifold quantum field theory. We would like to emphasize that this result crucially relies on inverting the deformation parameter 0 = ∈ R and hence it does not arise in formal deformation quantization. (In fact, treating in (5.11) as a formal parameter, the categorified orbifold theory A Z 2 ∈ 2AQFT(Disk(S 1 )) is non-truncated as in the classical case = 0.) A similar interplay between quantization and orbifold singularities was observed before within a different framework [Brz14,BS17]. ▽ Remark 5.14. We would like to emphasize that the results of Example 5.13 hold true in much greater generality. Let C be any orthogonal category and A ∈ AQFT(C) any AQFT that assigns, to every c ∈ C, a CCR-algebra A(c) = CCR(L(c), σ c ) of a symplectic vector space (L(c), σ c ), i.e. σ c is non-degenerate. Using similar arguments as in Example 5.13, one shows that the categorified orbifold theory corresponding to the Z 2 -action ρ(±1) : (L(c), σ c ) → (L(c), σ c ) , ϕ → ±ϕ is truncated, provided that R ∋ = 0. The same holds true for AQFTs assigning canonical anticommutation relation (CAR) algebras of non-degenerate inner product spaces. △

Fredenhagen's universal category
The goal of this section is to present a categorified version of Fredenhagen's universal algebra. Let us briefly recall the original 1-categorical construction for ordinary AQFTs from [Fre90,Fre93,FRS92], see also [Lan14,BSW17] for more details. Given a full orthogonal subcategory embedding J : C → D and any ordinary AQFT A ∈ AQFT(C) on C, operadic left Kan extension along the induced operad morphism J : P C → P D determines a canonical extension J ! (A) ∈ AQFT(D) of A to the larger orthogonal category D. The algebra J ! (A)(d) ∈ Alg K that is assigned by the extended AQFT J ! (A) to an object d ∈ D is usually referred to as Fredenhagen's universal algebra. This extension prescription is canonical in the sense that it is part of an adjunction We will study a generalization of this extension construction to 2AQFTs, which is based on the biadjunction where the right adjoint 2-functor J * is given by restriction of 2AQFTs along J. Hence, the left adjoint pseudo-functor J ! is a 2-categorical generalization of operadic left Kan extension. Given any A ∈ 2AQFT(C) on C, this determines a canonical extension J ! (A) ∈ 2AQFT(D) to the larger orthogonal category D. Following AQFT terminology, we shall refer to the locally presentable K-linear category J ! (A)(d) ∈ Pr K that is assigned by the extended 2AQFT J ! (A) to an object d ∈ D as Fredenhagen's universal category. In the context of Example 6.1, we will provide examples of such categories for 2AQFTs on the circle M = S 1 .

Morphisms
Applying the same construction to P D defines a symmetric monoidal category P ⊗ D . Furthermore, the orthogonal functor J : C → D induces an operad morphism J : P C → P D and hence a symmetric monoidal functor J ⊗ : P ⊗ C → P ⊗ D between the monoidal envelopes. The latter reads explicitly as follows: Recall from Definition 3.3 that 2AQFTs on C are by definition P C -algebras. Hence, by the universal property of monoidal envelopes, we can associate to every A ∈ 2AQFT(C) a symmetric monoidal pseudo-functor from the monoidal envelope of P C . This pseudo-functor acts on objects c = (c 1 , . . . , c n ) ∈ P ⊗ C as the n-ary Kelly-Deligne tensor product of the locally presentable K-linear categories A(c i ) ∈ Pr K , cf. Remark 3.2. (By convention, we set A(∅) := Vec K to be the monoidal unit of Pr K .) On morphisms (α, f ) : c → t in P ⊗ C , this pseudo-functor acts as where ≃ α is the equivalence in the symmetric monoidal 2-category Pr K that is associated to the displayed permutation determined by α. The coherence data for the symmetric monoidal pseudo-functor A : P ⊗ C → Pr K are canonically given by the coherence data for A ∈ 2AQFT(C) and the symmetric monoidal structure on Pr K .

Extension
The extension pseudo-functor J ! : 2AQFT(C) −→ 2AQFT(D) in the biadjunction (6.1) is obtained canonically via operadic left pseudo-Kan extension along J : P C → P D . Passing from colored operads to their monoidal envelopes, J ! can be obtained via (categorical) left pseudo-Kan extension along J ⊗ : P ⊗ C → P ⊗ D , cf. [Hor17]. Furthermore, the latter left pseudo-Kan extension can be computed in terms of suitable bicolimits [Lac10,LN16]. Using this approach, we can now describe the extension J ! (A) ∈ 2AQFT(D) of a 2AQFT A ∈ 2AQFT(C). For each d ∈ D, Fredenhagen's universal category is the locally presentable K-linear category obtained as a bicolimit in Pr K , where J ⊗ /(d) denotes the slice category for the functor J ⊗ : Recall also (6.2) for the construction of the pseudo-functor A : P ⊗ C → Pr K . (To avoid confusion, let us stress that the symbol (d) stands for the tuple of length one that is defined by the object d ∈ D, i.e. (d) ∈ P ⊗ D is an object in the monoidal envelope.) This bicolimit always exists because Pr K is bicategorically cocomplete, see e.g. [BCJF15, Lemma 2.5]. For each tuple g = (g 1 , . . . , g n ) ∈ P D s d of mutually orthogonal D-morphisms, we set the factorization product to be the functor that is defined below, which is co-continuous and K-linear in each entry: Consider the diagram where g * : n i=1 J ⊗ /(d i ) → J ⊗ /(s) is the functor induced by post-composition with g in the colored operad P D . By direct inspection, the left square commutes. In the right square, instead, the clockwise and counter-clockwise paths give functors that are related by the natural isomorphism (⋆) determined by the symmetric monoidal structure on the pseudo-functor A. Passing to bicolimits and recalling that the Kelly-Deligne tensor product ⊠ commutes with bicolim (in each entry) provides a co-continuous K-linear , which is co-continuous and Klinear in each entry, completes the construction of (6.4a). For the empty tuple d = ∅, the pointing J ! (A)( * s ) ∈ J ! (A)(s) of Fredenhagen's universal category J ! (A)(s) is obtained in the same fashion from (6.4b). (Notice that empty products are initial categories, while ⊗ 0 and ⊠ 0 assign the respective monoidal units.) The coherence data, cf. Remark 3.4, for the extended 2AQFT J ! (A) ∈ 2AQFT(D) are obtained canonically from the construction above and the symmetric monoidal pseudo-functor A : P ⊗ C → Pr K . For an arbitrary d ∈ D, we shall now describe Fredenhagen's universal category J ! (A)(d) in fully explicit terms, using the prescription in [BCJF15, Lemma 2.5] to compute the relevant bicolimit (6.3). This is a two-step procedure: 1.) Every co-continuous K-linear functor between two locally presentable K-linear categories admits a right adjoint by the special adjoint functor theorem, cf. [AR94,BCJF15]. Hence, from the pseudo-functor A : P ⊗ C → Pr K , we obtain a new pseudo-functor A R : (P ⊗ C ) op → Cat that acts on objects as A, i.e. A R (c) := A(c) for all c ∈ P ⊗ C , and that assigns to a morphism (α, f ) : c → t in P ⊗ C the right adjoint of the co-continuous K-linear functor assigned by A, i.e. A(α, f ) ⊣ A R (α, f ) : A R (t) → A R (c). (Note that A R is just a pseudofunctor to Cat and not necessarily to Pr K because the right adjoint functors A R (α, f ) may fail to be co-continuous.) 2.) The category underlying the bicolimit (6.3) of A • forget : J ⊗ /(d) → Pr K can be computed as a bilimit of the pseudo-functor A R • forget : (J ⊗ /(d)) op → Cat. The outcome is a locally presentable K-linear category in a canonical way, cf. [BCJF15].
Using the explicit model [Str80,LN16] for computing bilimits of pseudo-functors to Cat, we obtain the following description of Fredenhagen's universal category J ! (A)(d) in terms of explicit data and conditions: consists of the following data: ( in the category A(c).
These data have to satisfy the following cocycle conditions: denotes the coherence isomorphisms for identities that are associated with the pseudo-functor A R .
(ii) For all composable pairs of morphisms (α, in A(c) commutes, where A R 2 ((β,g),(α,f )) denotes the coherence isomorphisms for compositions that are associated with the pseudo-functor A R .

consists of a family of A(c)-morphisms
Identities and composition: Identities and composition are defined component-wise. Example 6.3. Let us consider first the simplest case where A is truncated, i.e. A = ι(A) with A ∈ AQFT(Disk(S 1 )) an ordinary AQFT. We have the following square of biadjunctions The horizontal biadjunctions are the inclusion-truncation biadjunctions from Theorem 4.3, the left vertical (bi)adjunction is an ordinary operadic left Kan extension and the right vertical biadjunction is the operadic left pseudo-Kan extension (6.1). By direct inspection, one confirms that the square formed by the right adjoint 2-functors commutes, i.e. π J * = J * π, hence the square formed by the left adjoint pseudo-functors commutes up to an equivalence, i.e. ι J ! ≃ J ! ι. As a consequence, Fredenhagen's universal category for a truncated 2AQFT is equivalent to the category of right modules over Fredenhagen's universal algebra J ! (A)(S 1 ) ∈ Alg K . The latter is given by the ordinary colimit → Alg K is the symmetric monoidal functor from the monoidal envelope that is determined by A ∈ AQFT(Disk(S 1 )).
To obtain a better understanding of the objects and morphisms in our general presentation of Fredenhagen's universal category J ! (ι(A))(S 1 ), we construct explicitly a functor Mod J ! (A)(S 1 ) → J ! (ι(A))(S 1 ) that implements the equivalence (6.12). Let us first describe this functor on objects. Given any right module V ∈ Mod J ! (A)(S 1 ) over Fredenhagen's universal algebra, we use the canonical Alg K -morphisms χ I : and therefore we can set to be the identity morphism. One easily checks that the coherence conditions (6.7) are satisfied, hence we have defined an object (V, ξ V ) ∈ J ! (ι(A))(S 1 ). Let us now define the functor for all I ∈ J ⊗ /(S 1 ). One easily checks that the coherence conditions (6.9) are satisfied, hence we have defined a morphism L : (V, ξ V ) → (W, ξ W ) in J ! (ι(A))(S 1 ). Using the universal property of the colimit (6.13), one checks that the resulting functor Mod J ! (A)(S 1 ) → J ! (ι(A))(S 1 ) implements the equivalence (6.12). Summing up, we have found that, in the case of a truncated 2AQFT A = ι(A), the objects of Fredenhagen's universal category can be described as families of right modules (6.14) over the local algebras A(I) = n i=1 A(I i ) on disjoint unions of intervals, whose restrictions along inclusions α : I → J coincide (6.15). Morphisms in Fredenhagen's universal category can be described by locally defined module morphisms (6.16), whose restrictions along inclusions α : I → J coincide. ▽ Example 6.4. Let us consider the gauging K G ∈ 2AQFT(Disk(S 1 )) of the trivial AQFT K ∈ AQFT(Disk(S 1 )) with respect to the trivial action of a finite group G, which is a non-truncated 2AQFT for every non-trivial group G = {e}, see Example 5.7. Because 2AQFTs are by definition prefactorization algebras with values in Pr K (cf. Definition 3.3) and K G ∈ 2AQFT(Disk(S 1 )) is also locally constant, we can compute Fredenhagen's universal category J ! (K G )(S 1 ) for this particular example by factorization homology [AF15]. Using in particular [AF15, Theorem 3.19], we obtain that is equivalent to the Hochschild homology of the associative and unital algebra (Rep K (G), ⊗, K) ∈ Alg As (Pr K ) in Pr K . (The latter is just the usual monoidal category of K-linear representations of G, regarded internally in the symmetric monoidal 2-category Pr K .) Hochschild homology can be computed as a bicolimit (in Pr K ) of the simplicial diagram associated with (Rep K (G), ⊗, K) ∈ Alg As (Pr K ), see e.g. [BZFN10, Section 5.1]. (As usual, we suppress the degeneracy maps in (6.18).) Since we are working in a 2-categorical context, this simplicial diagram may be truncated after Rep K (G) ⊠3 .
We will now compute the bicolimit (6.18) explicitly by using the techniques of [BCJF15], see also the end of Section 6.2 for a short summary. A more conceptual explanation of the obtained result is given in Remark 6.5 below. By [BCJF15, Lemma 2.5], we can compute this bicolimit in terms of the bilimit (in the 2-category Cat of categories) of the truncated cosimplicial diagram obtained by taking right adjoints of the face and degeneracy maps in (6.18). In this expression we have also used that Rep K (G) ⊠n ≃ Rep K (G n ) is equivalent to the representation category of the product group G n . The coface and codegeneracy maps in (6.19) are given by coinduced representation functors φ * : Concretely, we have that for the diagonal map ∆ : G → G 2 , g → (g, g), and that (g 1 , g 1 , g 2 ) , for i = 0 , (g 1 , g 2 , g 2 ) , for i = 1 , (g 1 , g 2 , g 1 ) , for i = 2 . (6.21b) The codegeneracy map ǫ 0 : Rep K (G 2 ) → Rep K (G) is given by the coinduced representation functor for G 2 → G , (g 1 , g 2 ) → g 1 .
We are now ready to describe the bilimit (6.19) and hence the category HH • Rep K (G) in more explicit terms: in Rep K (G 2 ) commutes.
We can simplify this description further by using explicit models for the coinduced representation functors φ * : Rep K (G ′ ) → Rep K (G ′′ ) for group maps φ : G ′ → G ′′ . Since we consider only finite groups and a base field K of characteristic 0, there exists a natural isomorphism between the coinduced and the induced representation functors φ * ∼ = φ ! : Rep K (G ′ ) → Rep K (G ′′ ). The latter is easy to describe: Mod is the category of left K[G ′ ]-modules, and similar for G ′′ .) Given any object (V, θ V ) ∈ HH • Rep K (G) , we use this explicit description to deduce that θ V : , which is G-equivariant with respect to the adjoint action on K[G] and satisfies the axioms of a left K[G]-coaction. Moreover, we deduce that a morphism in HH • Rep K (G) is a G-equivariant map that preserves these K[G]-coactions. In summary, we have obtained the following chain of equivalences Let us briefly explain the physical interpretation of this result. By Remark 5.6, we can interpret K G ∈ 2AQFT(Disk(S 1 )) as an orbifold σ-model that is defined on intervals and whose target is the classifying stack BG = { * }//G of G. Indeed, the stack of fields on an interval I ⊂ S 1 is Fields(I) = Map(I, BG) ≃ { * }//G and its category of quasi-coherent sheaves is QCoh(Fields(I)) ≃ Rep K (G), which coincides with the category that the 2AQFT K G assigns to intervals. On the whole circle S 1 , the stack of fields of this orbifold σ-model is given by the loop stack Fields(S 1 ) = Map(S 1 , BG) ≃ Bun G (S 1 ), which is equivalent to the stack of principal Gbundles on S 1 . The non-trivial bundles can be interpreted physically as "twisted sectors" of this orbifold σ-model, see e.g. [DVVV89]. The category of quasi-coherent sheaves on this stack is given by QCoh(Fields(S 1 )) ≃ G-Mod O(G) , which coincides with our result for Fredenhagen's universal category (6.23). Hence, Fredenhagen's universal category successfully detects all "twisted sectors" for this simple example of an orbifold σ-model. ▽ Remark 6.5. The category (6.23) that we obtain for the circle is the representation category of the groupoid of principal G-bundles over the circle, i.e. the representation category of the loop groupoid G//G of G (the action groupoid of the action of G on itself by conjugation). This category is also the Drinfeld center of Rep K (G), i.e. the Hochschild cohomology. As a consequence, the Hochschild homology and Hochschild cohomology for Rep K (G) are equivalent. More general conditions under which one finds such an equivalence are given in [DSPS13, Corollary 3.1.5] within the framework of finite tensor categories and in [BZFN10, Theorem 1.7] within the framework of derived algebraic geometry. △ Example 6.6. As a last example, we discuss briefly the gauging A G ∈ 2AQFT(Disk(S 1 )) of an arbitrary G-equivariant AQFT (A, ρ) ∈ G-AQFT(Disk(S 1 )), which includes Examples 6.3 and 6.4 as very special cases. Unfortunately, it seems to be very hard to simplify our explicit description of Fredenhagen's universal category J ! (A G )(S 1 ) in this general case. (Note that computing this category as in Example 6.4 by importing techniques from factorization homology is in general not possible, because we are also interested in 2AQFTs that are not locally constant.) In order to develop a better understanding of the category J ! (A G )(S 1 ), we shall specialize our general description of Fredenhagen's universal category from the end of Section 6.2 to our example at hand. Concretely, an object (V, ξ V ) ∈ J ! (A G )(S 1 ) consists of the following data: (1) For each tuple I = (I 1 , . . . , I n ) ∈ J ⊗ /(S 1 ) of mutually disjoint intervals, a G n -equivariant module V I ∈ G n -Mod A(I) (6.24) over the tensor product algebra A(I) = n i=1 A(I i ). (The G n -action on the tensor product algebra is given by the component-wise G-actions.) (2) For each morphism α : I = (I 1 , . . . , I n ) → J = (J 1 , . . . , J m ) in J ⊗ /(S 1 ), a G n -Mod A(I)isomorphism for each t ∈ O, a ∈ O n and b i ∈ O k i , for i = 1, . . . , n.
These data are required to satisfy the following axioms: Remark A.6. Cat-enriched colored operads, pseudo-morphisms, pseudo-transformations and modifications assemble into a tricategory. The various compositions are similar to the case of the tricategory of bicategories and hence will not be displayed in full detail here. We refer the reader to [SP09, Appendix A.1] for a brief review of the tricategory of bicategories and to [GPS95] for the details.
Let us nevertheless fix the relevant notations that will appear in the bulk of this paper. Given two Cat-enriched colored operads O and P, the tricategory structure implies that there exists a Hom-2-category Let us note that in the case O and P are Set-valued colored operads, i.e. all categories of operations in Definition A.1 are sets, then Alg O (P) = [O, P] ∈ Cat is an ordinary category that coincides with the usual category of O-algebras with values in P, see e.g. [Yau16,BSW17]. △