Group actions on 2-categories
Eugenia Bernaschini, C\'esar Galindo, Mart\'in Mombelli
TL;DR
This paper investigates group actions on 2-categories, introduces the concept of equivariant objects, and establishes coherence theorems and equivalences relating equivariant 2-categories to module categories over G-extensions.
Contribution
It develops a framework for understanding group actions on 2-categories, including the construction of equivariant objects and a coherence theorem, extending the theory of tensor categories and their centers.
Findings
Constructed the 2-category of equivariant objects for a group action.
Proved a coherence theorem for 2-categories with group actions.
Established an equivalence between the 2-category of equivariant objects and module categories over G-extensions.
Abstract
We study actions of discrete groups on 2-categories. The motivating examples are actions on the 2-category of representations of finite tensor categories and their relation with the extension theory of tensor categories by groups. Associated to a group action on a 2-category, we construct the 2-category of equivariant objects. We also introduce the G-equivariant notions of pseudofunctor, pseudonatural transformation and modification. Our first main result is a coherence theorem for 2-categories with an action of a group. For a 2-category B with an action of a group G, we construct a braided G-crossed monoidal category Z_G(B) with trivial component the Drinfeld center of B. We prove that, in the case of a G-action on the 2-category of representation of a tensor category C, the 2-category of equivariant objects is biequivalent to the module categories over an associated G-extension of C.…
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models · Intracranial Aneurysms: Treatment and Complications
Group actions on 2-categories
Eugenia Bernaschini, César Galindo and Martín Mombelli
Facultad de Matemática, Astronomía y Física
Universidad Nacional de Córdoba
CIEM – CONICET
Medina Allende s/n
(5000) Ciudad Universitaria, Córdoba, Argentina
URL: http://www.famaf.unc.edu.ar/$\sim$mombelli](mailto:[email protected]%0A)
Departamento de Matemáticas, Universidad de los Andes,
Carrera 1 N. 18A - 10, Bogotá, Colombia
Abstract.
We study actions of discrete groups on 2-categories. The motivating examples are actions on the 2-category of representations of finite tensor categories and their relation with the extension theory of tensor categories by groups. Associated to a group action on a 2-category, we construct the 2-category of equivariant objects. We also introduce the -equivariant notions of pseudofunctor, pseudonatural transformation and modification. Our first main result is a coherence theorem for 2-categories with an action of a group. For a 2-category with an action of a group , we construct a braided -crossed monoidal category with trivial component the Drinfeld center of . We prove that, in the case of a -action on the 2-category of representation of a tensor category , the 2-category of equivariant objects is biequivalent to the module categories over an associated -extension of . Finally, we prove that the center of the equivariant 2-category is monoidally equivalent to the equivariantization of a relative center, generalizing results obtained in [8].
Key words and phrases:
tensor category; module category; bicategory
2000 Mathematics Subject Classification:
18D05, 18D10
Introduction
The theory of 2-categories appears in a natural way in diverse contexts. For example, it was used by Rouquier to “categorify” certain algebraic objects [22] and appears in topological field theories [6], [19]. The theory of representations of 2-categories has been initiated in a series of papers [14, 15, 16].
Our motivation for the study of 2-categories comes from the theory of tensor categories. For a tensor category , a representation of , or -module category, is a category equipped with an associative action satisfying certain conditions. Given two -module categories , the category is the category whose objects are -module functor between and , and morphisms are -module natural transformations. The 2-category of (left) -modules has as 0-cells -module categories, 1-cells -module functors between them and 2-cells are -module natural transformations. This 2-category is a strong invariant of the tensor category .
Given a 2-category and a 2-monad on , in [17], the notion of the equivariantization 2-category was presented. The equivariantization of a 2-category by a group was studied later in [12].
One of the purposes of the paper is to explicitly describe an action of a group on a 2-category , and describe all ingredients of the resulting equivariantization 2-category . An action of a group on a 2-category consists of
- •
a family of pseudofunctors , ,
- •
pseudonatural equivalences ,
- •
invertible modifications
[TABLE]
for any , satisfying certain axioms. We also prove a coherence theorem for group action, stating that there exists another equivalent action of on , such that all pseudofunctors involved in the group action are 2-functors, , and , are all the identity. As an application of the coherent theorem we prove that associated to every action of group on a 2-category there is a braided -crossed monoidal category such that the trivial component is , the Drinfeld center of .
An important example comes from the theory of tensor categories. We show that, if is a -graded tensor category, and , there is an action of the group acts on , the 2-category of representations of , and there is a biequivalence
[TABLE]
The coherence theorem for group actions allows us to construct an associated strict braided crossed monoidal category and to prove that there is a monoidal equivalence between the center of the equivariantization and the monoidal category of pseudonatural transformations of the forgetful pseudofunctor . When applied this result to the 2-category , we recover the results from [8], on the center of graded tensor categories.
The contents of the paper are organized as follows. In Section 1 we recall the basics of 2-categories. For any pseudofunctor we define the monoidal category of pseudonatural transformations . When is the identity pseudofunctor, is a braided monoidal category called the center of the 2-category.
In Section 2 we explicitly describe the notion of a group action on a 2-category. Given two 2-categories equipped with an action of a group , we define the notion of -pseudofunctor between them. When a -pseudofunctor is a biequivalence, we say that are -biequivalent. Also, we define the notions of -pseudonatural transformation and -modifications. All these data, turns out to be a 2-category, denoted by . The equivariant 2-category is , where is the unit 2-category, where acts trivially.
In Section 3 we prove that any 2-category with a group action is -biequivalent to another one where the action is strict. Section 4 is devoted to explicitly describe all ingredients in the equivariant 2-category .
In Section 5 we show an example coming from graded tensor categories. If is a -graded tensor category, then the group acts on the 2-category of left -modules. The resulting equivariant 2-category is biequivalent to . In Section 6 we define the -braided center of a 2-category with an action of a group . In Section 7, we show that there is a monoidal equivalence where is the forgetful pseudofunctor. When applied to the example , we recover results from [8].
Acknowledgments
The work of E.B and M.M. was partially supported by CONICET, Secyt (UNC), Argentina. M.M. is grateful to the department of mathematics at Universidad de los Andes, Bogotá, where part of this work was done, for the kind hospitality.
1. 2-categories
Let us briefly recall the notion of a 2-category. For more details, the reader is referred to [13, 20]. For any 2-category , the set of objects, also called 0-cells, will be denoted by . The composition in each hom-category , that is, the vertical composition of 2-cells, is denoted by juxtaposition , while the symbol is used to denote the horizontal composition functors
[TABLE]
The identity of a 0-cell is written as . For any 1-cell the identity will be denoted or sometimes simply as , when space saving is needed. For any 2-category , we shall denote by the 2-category that is obtained from by reversing 1-cells.
Example 1.1**.**
The unit 2-category has a single 0-cell, named . The monoidal category is the unit monoidal category.
A pseudofunctor , consists of a function , a family of functors for each , a collection of isomorphisms and a family of natural isomorphisms
[TABLE]
for 0-cells , subject to the usual axioms. A pseudofunctor is called unital if , for any 0-cell , and the isomorphisms are the identities. A pseudofunctor is called a 2-functor if the associativity isomorphisms are the identities.
If , are pseudofunctors, a pseudonatural transformation
\textstyle{{\mathcal{B}}{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\downarrow\chi}$$\scriptstyle{F}$$\scriptstyle{G}$$\textstyle{{\mathcal{B}}^{\prime}} consists of a family of 1-cells , and isomorphisms
[TABLE]
natural in , subject to the usual axioms. If are pseudonatural transformations, a modification from
\textstyle{{\mathcal{B}}{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\downarrow\chi}$$\scriptstyle{F}$$\scriptstyle{G}$$\textstyle{{\mathcal{B}}^{\prime}} to
\textstyle{{\mathcal{B}}{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\downarrow\theta}$$\scriptstyle{F}$$\scriptstyle{G}$$\textstyle{{\mathcal{B}}^{\prime}} , consists of a family of 2-cells , such that the diagrams
[TABLE]
commute for all . This modification will be denoted as . Given pseudofunctors , we shall denote the category where objects are pseudonatural transformations from to and arrows are modifications.
A 1-cell is called an equivalence if there exists a 1-cell such that and . We will say that an invertible 1-cell is an isomorphism if there is such that and . The next result will be useful later to simplify some proofs.
Proposition 1.2**.**
Every 2-category (or bicategory) is biequivalent to a 2-category where every equivalence 1-cell is an isomorphism.
Proof.
The proof goes along the lines of [9, Theorem 1.4]. Since every category is equivalent to a skeletal one. Every bicategory is biequivalent to a locally skeletal one , that is, each of its hom-category is skeletal. Then in , every 1-cell equivalence is an isomorphism. By Street’s Yoneda lemma for bicategories [21, p.117 ], the Yoneda embedding
[TABLE]
is locally an equivalence. Therefore, is biequivalent to ; the full sub-2category of determined by the contravariant representables. Since every equivalence in is an isomorphism, every equivalence in is an isomorphism and is biequivalent to . ∎
1.1. The tricategory of 2-categories
Given a pair of 2-categories and , we can define the functor 2-category, 2Cat, whose 0-cells are pseudofunctors , whose 1-cells are pseudonatural transformations, and whose 2-cells are modifications. Given 2-categories and , we define a pseudo-functor
[TABLE]
called the tensor product. The tensor product at the level of pseudofunctors is the composition. The tensor product of pseudonatural transformations is
[TABLE]
where
[TABLE]
Here, the isomorphisms constraints of the pseudofunctors have been omitted as a space-saving measure. If and are another pseudonatural transformations and and are modifications, their tensor product is defined as , , for any 0-cell .
If and are pseudonatural transformations between pseudofunctors , then there is a modification
[TABLE]
given by
[TABLE]
This modification is called the comparison constraint.
The tensor product is associative only at the level of pseudofunctors, but not for pseudonatural transformations. There exists an associativity constraint
[TABLE]
for pseudonatural transformations , and . The modification
[TABLE]
is defined by . It is easy to see that satisfies the pentagonal identity.
1.2. Finite tensor categories
A (strict) monoidal category is a 2-category with one single 0-cell. A finite tensor category over is a finite -linear abelian rigid monoidal category such that the tensor product functor is -linear in each variable. The reader is referred to [5].
Suppose and are strict tensor categories. A monoidal functor is a pseudofunctor between the corresponding 2-categories. Explicitly, it consists of a functor , natural isomorphisms , and isomorphism , satisfying certain axioms. If are monoidal functors , a natural monoidal transformation is a natural transformation , such that for any pair of objects
[TABLE]
1.3. The endomorphism category of a pseudofunctor
If is a 2-category, the monoidal category
[TABLE]
is exactly the center of , i.e., the obvious generalization of the center construction of a monoidal category. See [18].
Let be two 2-categories and be a unital pseudofunctor. Denote ; the category of pseudonatural transformations of the pseudofunctor . This is a monoidal category with tensor product described in the previous section. Explicitly, objects in are pairs , where
[TABLE]
[TABLE]
where, for any , is a natural isomorphism 2-cell such that
[TABLE]
for any 0-cells , and any pair of 1-cells , .
If , are two objects in , a morphism in is a collection of 2-cells , such that
[TABLE]
for any 1-cell . The category has a monoidal product defined as follows. Let be two objects. Then , where for any 0-cells , and
[TABLE]
If are objects, and are morphisms in , then is defined by
[TABLE]
for any 0-cell . The unit is the object
[TABLE]
for any 0-cells and any 1-cell The center of the identity pseudofunctor is denoted as , and it coincides with the definition presented in [18].
2. Group actions on 2-categories
Assume is a group and is a 2-category. We shall denote by the 2-category that has 0-cells the elements of the group . For any pair
[TABLE]
Moreover, is a monoidal 2-category, see [9]. Since is also a monoidal 2-category, we define an action of on as a weak monoidal homomorphism . See for example [9].
Explicitly, an action of on a 2-category consists of the following data:
- •
A family of pseudofunctors , ,
- •
pseudonatural equivalences , ,
- •
a pseudonatural equivalence ,
- •
for any invertible modifications
[TABLE]
[TABLE]
such that for any 0-cell
[TABLE]
[TABLE]
for any . Where,
[TABLE]
[TABLE]
In equation (2.2), we are omitting the associativity isomorphisms of the pseudofunctors . In the following diagrams we shall denote by the pseudofunctor , the composition of functors as juxtaposition and the tensor product of pseudonatural transformations also by juxtaposition. Diagrammatically, we have modifications
[TABLE]
such that the next diagrams are equal for all ,
[TABLE]
[TABLE]
We say that a group acts trivially on if the weak monoidal homomorphism is the trivial one. This means that for any , the pseudofunctors are the identity, are the identity pseudonatural transformations and all the modifications are identities.
Definition 2.1**.**
An action is called unital if is a unital pseudofunctor, , and , for any . A unital -action will be denoted simply by .
Definition 2.2**.**
An action is called strict if each pseudofunctor is a 2-functor, and , and the pseudonatural transformations and the modifications are the identities for any .
A similar argument as in [7, Proposition 3.1] applied in this case, allows us to consider only unital actions. Assume that are 2-categories equipped with unital actions of a group via
[TABLE]
Definition 2.3**.**
A -pseudofunctor between and is a triple , where
- •
is a unital pseudofunctor,
- •
for any , pseudonatural equivalences
- •
invertible modifications
[TABLE]
such that such that for all
[TABLE]
[TABLE]
[TABLE]
holds in 2Cat. In the above diagrams, we are using the comparison constraints defined in (1.2).
Definition 2.4**.**
Assume that are -pseudofunctors. A -pseudonatural transformation is a pair , where is a pseudonatural transformation, and are invertible modifications
[TABLE]
such that for all , the equation
[TABLE]
[TABLE]
holds in 2Cat.
Definition 2.5**.**
Assume that are -pseudonatural transformations. A -modification is a modification such that
[TABLE]
[TABLE]
Assume that are -pseudofunctors, and , are -pseudonatural transformations. The composition
[TABLE]
is defined as follows. The pseudonatural transformation . For any 0-cell and any
[TABLE]
Here, we are also ommiting the associativity constraints of the pseudofunctor . The composition of modifications of -categories is the usual composition of modifications.
Definition 2.6**.**
is the 2-category in which 0-cells are pseudofunctors of -categories, 1-cells are pseudonatural transformations of -categories and 2-cells are modifications of -categories.
The next result is a consequence of [9, Corollary 8.3].
Proposition 2.7**.**
* is a 2-category. ∎*
Definition 2.8**.**
We say that the 2-categories and are -biequivalent if there exists a -pseudofunctor that is also a biequivalence.
Lemma 2.9** (Transport of structure).**
Let be a 2-category with an action of given by . Let be a biequivalence,
[TABLE]
a -indexed family of pseudofunctors and pseudonatural equivalences, respectively. Then, there is a way to endowed with a -action such that is a -biequivalence .
Proof.
Since and are psedonatural equivalences, we can simultaneously provide the datum and the pseudonatural equivalences , . Now, axiom 2.5 uniquely determines the modifications . Axiom 2.3 follows from the corresponding axioms of -action via . The pseudofunctor is a -biequivalence by construction. ∎
Corollary 2.10**.**
Every 2-category with a -action is -biequivalent to a 2-category where acts by 2-functors, that is, all are 2-functors.
Proof.
By the coherence of theorem for pseudofunctor, see [11, Section 2.3], every bicategory is biequivalent to a 2-category such that every pseudo-functor is pseudo-natural equivalent to a 2-functor. Then applying Lemma 2.9 we can transport the action of to a -biequivalent action on where acts by 2-functors. ∎
3. Coherence for group actions on 2-categories
The main result of this section is to prove the following coherence theorem for a group action on a 2-category.
Theorem 3.1** (Coherence for group actions on 2-categories).**
Let be a group. Every 2-category with an action of is -biequivalent to a 2-category with a strict action of .∎
Assume is a 2-category equipped with a unital action of , . By Corollary 2.10 we can assume that is a 2-functor for any . We shall first construct a 2-category with a strict action of .
Objects of are triples , where is a -indexed family of objects, is a -indexed family of 1-cell equivalences and
[TABLE]
a -index family of isomorphism 2-cells, such
[TABLE]
that for all , and equation
[TABLE]
[TABLE]
holds in . If is a 0-cell, the identity 1-cell is defined as follows. , where , for any .
If and are objects in , a 1-cell is a pair , where is a -indexed family of 1-cells and
[TABLE]
is a -indexed family of isomorphism 2-cells, such that for all , and equation
[TABLE]
[TABLE]
holds in . If are 1-cells, a 2-cell is a -indexed family of 2-cells such that for all , equation
[TABLE]
[TABLE]
holds in .
The (vertical) composition in each category is defined pointwise.
Now, let us define the horizontal composition If and are 0-cells, and
[TABLE]
are 1-cells, define
[TABLE]
where , and t_{g,h}=\big{(}1_{X_{gh}}\circ s_{g,h}\big{)}\big{(}l_{g,h}\circ 1_{F_{g}(Y_{h})}\big{)}, for any . The horizontal composition of 2-cells in is just the horizontal composition of 2-cells in .
Lemma 3.2**.**
* is a 2-category endowed with a strict action of .*
Proof.
The proof that is indeed a 2-category follows by a straightforward calculation. Let us define now a canonical strict action of on the 2-category . For any define the 2-functors as follows. If is a 0-cell, , then
[TABLE]
If is a 1-cell,
[TABLE]
If is a 2-cell, then , for any . Since the are 2-functors such that for all and , defines a strict action of on . ∎
There is a pseudofunctor defined as follows. If is a 0-cell in , then
[TABLE]
if is a 1-cell, then and for 2-cells , , where . The fact that are modifications implies that is indeed a 1-cell in . The following proposition implies immediately Theorem 3.1
Proposition 3.3**.**
* is a -biequivalence.*
Proof.
If is an object in , then the 1-equivalences and the 2-cells
[TABLE]
defines a 1-equivalence from to , that is, is bi-essentially surjective.
Let and be objects in , and be a 1-cell in . The invertible 2-cells define an invertible 2-cell from to . Then is locally essentially surjective.
If and such that . Thus, , but since we are considering a unital action, , that is, is locally faithful. Suppose is a 2-cell in , condition (3.3) implies that , then . Since, is bi-essentially surjective and locally fully faithful, is a biequivalence.
To see that has a canonical structure of -pseudofunctor, we note that
[TABLE]
for any . Then, using the pseudonatural transformations , we define a pseusonatural transformation
[TABLE]
as follows. For any 0-cell we have to define an equivalence 1-cell in . Set , where, for any
[TABLE]
Axiom (2.3) implies that morphisms fulfill condition (3.2). Thus, is indeed a 1-cell in . To complete the definition of of the pseudonatural equivalence , we have to define, 2-cells in
[TABLE]
for any 1-cell . Set \big{(}(\gamma_{g})_{X}\big{)}_{x}=(\chi_{x,g})_{X}, for any . The fact that are modifications, imply that 2-cells \big{(}(\gamma_{g})_{X}\big{)}_{x} satisfy (3.3). To define the modifications
[TABLE]
we note that
[TABLE]
and
[TABLE]
Then we define for all .
Since are modifications, turns out to be modifications for any . Condition described in diagram (2.5) is exactly diagram (2.3). ∎
4. The equivariant 2-category
Let be a group. Denote by the unit 2-category endowed with the trivial action of , and assume that is a 2-category with an action of .
Definition 4.1**.**
The *equivariant 2-category * is . 0-cells, 1-cells and 2-cells in will be called equivariant 0-cells, 1-cells and 2-cells, respectively.
Proposition 4.2**.**
Assume and are -biequivalent. Then the 2-categories , are biequivalent.
Proof.
Straightforward.∎
Lemma 4.3**.**
There exists a forgetfull 2-functor . ∎
Proof.
If is an equivariant 0-cell in , then . If is an equivariant 1-cell, then . On 2-cells the functor is the identity. ∎
4.1. Unpacking definition of equivariantization
We shall explicitly describe the 2-category . This would allows us to show concrete examples and obtain some results in Section 7.
We shall assume that there is a unital action of on the 2-category such that all pseudofunctors are 2-functors. This is possible using Corollary 2.10. The 2-category has 0-cells triples , where
is a 0-cell in ;
are invertible 1-cells in ;
are isomorphisms 2-cells in the category such that
[TABLE]
[TABLE]
for all . For short, the collection will be denoted simply as .
Given two equivariant 0-cells , , an *equivariant 1-cell * is a pair where
- •
is a 1-cell,
- •
and for any , , are invertible 2-cells such that and such that for any
[TABLE]
If are equivariant 1-cells, an equivariant 2-cell is a 2-cell such that for all
[TABLE]
Suppose that , , are equivariant 0-cells, and
[TABLE]
are equivariant 1-cells, then the composition is defined as , where for any
[TABLE]
5. Group actions from graded tensor categories
Starting with a -graded tensor category , we shall construct a -action on the 2-category of -representations.
5.1. Group actions on tensor categories
Let be a finite group and be a finite tensor category. An action of on consists of the following data:
- •
tensor autoequivalences for any ,
- •
a natural isomorphism ,
- •
and monoidal natural isomorphisms ,
such that for all ,
[TABLE]
[TABLE]
For simplicity, we shall assumed that , and for all .
If a finite group acts on a finite tensor category , there is associated a new finite tensor category called the equivariantization of by . An object in is a pair , where is an object together with isomorphisms satisfying
[TABLE]
for all . A -equivariant morphism between -equivariant objects and , is a morphism in such that for all . The category has a monoidal product as follows. If , then , where for any
[TABLE]
For more details we refer the reader to [1], [2], [3].
There is also associated the graded tensor category , with underlying abelian category , where for any . If is an object, the object in is denoted by . The tensor product is
[TABLE]
The reader is refered to [23] for the complete monoidal structure of this tensor category.
5.2. Representations of tensor categories
A left -module category over a tensor category is a finite -linear abelian category equipped with
a -bilinear bi-exact bifunctor ;
natural associativity and unit isomorphisms , , such that
[TABLE]
[TABLE]
A module functor between module categories and over a tensor category is a pair , where
is a left exact functor; 2.
natural isomorphism: , , , such that for any , :
[TABLE]
Let and be -module categories. We denote by the category whose objects are module functors from to . A morphism between and is a natural transformation such that for any , :
[TABLE]
We shall also say that is a -module transformation.
Let be a tensor functor and let be a -module category. We shall denote by the -module category with the same underlying abelian category and action, associativity and unit morphisms defined, respectively, by
[TABLE]
for all , . Right -module and -bimodule categories are defined in a similar way. For the complete definition see [10].
A -module category is exact [5] if, for any projective object , the object is projective in for all . If is a left -module then is the right -module over the opposite Abelian category with action
[TABLE]
associativity isomorphisms for all . Analogously, if is a right -module category, then is a left -module category. If is a -bimodule category, we denote the opposite Abelian category, with left and right -module structure given as in (5.10).
5.3. 2-categories of representations of tensor categories
Suppouse that is a tensor category. The 2-category has as 0-cells, left -module categories, if are -module categories, then the category Analogously we define the 2-category of right -module categories.
If is a finite tensor category, the 2-category of exact left -module categories is defined in a similar way as , with 0-cells being exact left -module categories. It is known that is 2-equivalent to if and only if is Morita equivalent to . See for example [FMM, Thm. 3.4].
5.4. -Graded tensor categories
Let be a finite group. A (faithful) -grading on a finite tensor category is a decomposition , where are full abelian subcategories of such that
;
for all
In this case is a tensor subcategory of and each is an exact -bimodule category. We shall assume that for any . The tensor category is called a -graded extension of . This class of extensions of tensor categories were studied and classified in [4].
If is a left -module category, , , the functor defined by
[TABLE]
for any , is a -module functor. Moreover, the functor
[TABLE]
is an equivalence of -module categories.This is a particular case of [10, Thm. 3.20].
5.5. The relative center of a bimodule category
The next definition appeared in [8].
Definition 5.1**.**
Let be a tensor category and a -bimodule category. The * relative center* of is the category of -bimodule functors from to .
Explicitly, objects of are pairs , where is an objects of and
[TABLE]
is a natural family of isomorphisms such that
[TABLE]
where are the associativity constraints in .
Let be a -graded tensor category, with The inclusion functor induces the forgetful pseudofunctor .
Proposition 5.2**.**
There is a monoidal equivalence
Proof.
Let us define the functor as follows. For any set Here, for each , is the -module functor given by
[TABLE]
The isomorphisms endowing the functor structure of -module functor are
[TABLE]
given by the following composition:
[TABLE]
[TABLE]
for any . It follows that is a -module functor.
Now, we shall explain the definition of . Take , and a -module functor. Define
[TABLE]
[TABLE]
for any . Then, is a -module natural isomorphism.
Now, we shall define the functor on morphisms. Let , be objects in and be an arrow in . Define as follows. For any -module , define the -module natural transformation
[TABLE]
for any .
Now, we shall define a functor that will be the inverse of . Any object induces a -module functor , .
Let be an object in . For any -module category , is a -module functor. We shall denote it by . In particular, . We have natural -module isomorphisms In particular, we have isomorphisms
[TABLE]
Using that has a -module structure, there is a natural isomorphism
[TABLE]
Let be the natural isomorphism defined as
[TABLE]
The natural transformation satisfies 5.11 since is a -module natural transformation. Then Whence, we define .
Let be a morphism in , then is a morphism in since is a -module natural transformation. Set It follows straightforward that is well-defined and that and are inverse of each other. ∎
The center of the 2-category of representations of a tensor category coincides with the Drinfeld center of .
Corollary 5.3**.**
**
Proof.
Take and the identity pseudofunctor. ∎
5.6. Group actions coming from graded tensor categories
Throughout this section will denote a finite group. Assume that is a finite tensor category and is a -graded extension of . Set . We shall further assume that is a strict monoidal category.
In this section we aim to prove the following result.
Theorem 5.4**.**
There is an action of on the 2-category . Moreover, there are 2-equivalences
[TABLE]
Proof.
First, let us define an action of on the 2-category . For any define the 2-functors as follows. For any left -module category , set . If are left -module categories, and is a -module functor, then
[TABLE]
Now, we shall define the pseudonatural equivalences , for any . For any left -module category
[TABLE]
[TABLE]
for any , . It follows that is a well-defined -module functor. For any -module functor we have that , whence, we can define
[TABLE]
to be the identities. Since , for any , then we can choose to be the identities.
Now, we shall define a biequivalence . Assume is an equivariant 0-cell. This means that we have -module functors
[TABLE]
together with -module natural isomorphisms
[TABLE]
satisfying the required axioms. Recall the definition of the functors given in Section 5.4.
Claim 5.1**.**
Let be . If , then, there exists a family of -module natural isomorphisms
[TABLE]
Proof of Claim.
If , then
[TABLE]
[TABLE]
Note that there are module natural isomorphisms
[TABLE]
Combining these two isomorphisms we get that
[TABLE]
Using this isomorphism and the fact that is a -module functor, we get that
[TABLE]
obtaining the desired isomorphisms. ∎
We define as Abelian categories. We must endowed the category with a structure of -module category. If , set
[TABLE]
We have to define associativity isomorphisms
[TABLE]
Suppouse that , . Then
[TABLE]
Hence, we define
[TABLE]
Axiom (5.4) is equivalent, in this case, to axiom (4.1). It is clear that is a biequivalence and restricted to the category of exact modules gives the second biequivalence. ∎
6. Braided -crossed tensor categories from actions on 2-categories
In this section actions of groups on 2-categories are assumed to be strict. This does not lead to any loss of generality, since, in view of Theorem 3.1, all definitions and statements remain valid for non-strict actions after insertion of the suitable isomorphisms.
6.1. Strict braided -crossed tensor categories
Braided -crossed fusion categories play the same role in homotopy quantum field theory that braided fusion categories in the topological quantum field theory, see [24, 25, 26].
Definition 6.1**.**
Let be a groups and a strict monoidal category. A strict braided -crossed structure on consist of the following data:
- (1)
a decomposition (coproduct of categories) such that
- •
,
- •
for all , 2. (2)
a -indexed family of strict monoidal functor , such that
- •
, , , 3. (3)
a family of natural isomorphisms
[TABLE]
such that
- •
- •
- •
for all , .
Even when the definition of strict braided -crossed monoidal category is too restrictive, every weak braided -crossed category is equivalent to a strict braided -crossed category, see [7].
6.2. Center of a -action
Let be a group acting strictly on a 2-category , where , denotes the associated 2-functors. We shall introduce a -graded monoidal category equipped with an action of .
6.2.1. The -graded monoidal category
Define the strict monoidal category where and the product induced by the tensor product of pseudonatural transformation defined in (1.1). In other words, if and , we define as folows: for any object , and for any 1-cell
[TABLE]
The unit object is .
6.2.2. The action of on
Given and , we define as follows: for objects , and for any 1-arrow
[TABLE]
Analogously, the functor is defined for morphism in .
6.2.3. The -braiding of
Let and . By the definition of pseudo-natural transformation we have
[TABLE]
but and , then the define natural isomorphism .
Theorem 6.2**.**
Let be a groups with a strcit action on a 2-categoy . Then the monoidal category defined in 6.2.1 is a strict braided -crossed monoidal category with action defined in 6.2.2 and -braiding defined in 6.2.3. Moreover, the braided category is exactly the Drinfeld center of .
Proof.
Since the action of on is strict, it follows by definition the equations
- •
- •
- •
.
∎
6.3. Example
Let be a faithfully -graded fusion category.
Since every is a -bimodule category, they define 2-functors
[TABLE]
the tensor products induce pseudo-natural equivalences and the associator of induce invertible modifications that defines an action of on . See [4] for details.
In this case the category is just , the category of -bimodule functors and natural transformations from to . The category is canonically equivalent to the category defined in [8, Definition 2.1] (use that is a category equivalence). Then the -graded category is equivalent to the monoidal category . The braided -crossed category is equivalent to the -crossed category defined in [8].
7. The center of the equivariant 2-category
This section is devoted to prove the following result. Let be a finite group acting on a 2-category . Recall the forgetful 2-functor described in Lemma 4.3.
Theorem 7.1**.**
The group acts on by monoidal autoequivalences, and there is a monoidal equivalence
[TABLE]
As a consequence, we have the following result.
Corollary 7.2**.**
[8, Thm. 3.5]** Let be a faithfully graded tensor category, with . There is an action of the group on the relative center and a monoidal equivalence
[TABLE]
Proof.
Let be the forgetful pseudofunctor. Then
[TABLE]
The first equivalence follow from Corollary 5.3, the second one is Theorem 5.4, and the last one is Proposition 5.2. ∎
For the rest of this section we shall use the notation introduced in Section 4.1. There is no harm in assuming that the action is unital and strict, see definitions 2.1, 2.2. By Proposition 1.2, we can assume that any invertible 1-cell is an isomorphism. In particular, if is an equivariant 0-cell, for any , the 1-cell is invertible. Thus, we can choose a 1-cell such that
[TABLE]
If are 1-cells, we shall sometimes denote , as a space saving measure.
7.1. A group action on
For any , we shall define tensor autoequivalences such that they define an action of on First, let us explicitly describe objects in . An object consists of
[TABLE]
[TABLE]
where is an equivariant 1-cell. The isomorphisms satisfy (1.4). If , a morphism is a collection of 2-cells in
[TABLE]
such that for any equivariant 1-cell
[TABLE]
Lemma 7.3**.**
Suppose and is an equivariant 0-cell. There are isomorphisms 2-cells
[TABLE]
such that
[TABLE]
[TABLE]
for any .
Proof.
Take Equation (7.2) follow from (4.1). ∎
For any , let us define the functors , . Where, for any equivariant 0-cell
[TABLE]
Remark 7.4*.*
As a saving space measure, if are equivariant 0-cells, we are going to denote , . Also, we shall denote and when no confusion arises.
If is an equivariant 1-cell, then
[TABLE]
If is a morphism in , then
[TABLE]
The proof of the next result follows straightforwardly.
Proposition 7.5**.**
The functors are well-defined monoidal functors. ∎
Now, for any , we shall define monoidal natural isomorphisms satisfying (5.1) and (5.2). Take , so we must define an arrow
[TABLE]
For each equivariant 0-cell we define the map
[TABLE]
[TABLE]
Proposition 7.6**.**
For any , the following assertions holds.
- (i)
* are well-defined natural isomorphisms in .*
- (ii)
* are monoidal natural transformations.*
- (iii)
For any and any , the following equation holds
[TABLE]
Proof.
(i). We must verify that are morphisms in the category , that is, equation
[TABLE]
is fulfilled for any equivariant 1-cell . The left hand side of (7.4) equals to
[TABLE]
The second equation follows from the definition of , the fourth equality follows from (4.2). The right hand side of (7.4) equals to
[TABLE]
It follows from Equation (7.1) that both sides are equal.
(ii). Let be objects in . Since the functors are strict, this means that , we must prove that
[TABLE]
Let be an equivariant 0-cell. The left hand side of (7.5) evaluated in equals to
[TABLE]
The right hand side of (7.5) evaluated in equals to
[TABLE]
It follows from (7.1) that both sides are equal.
(iii). Let be an equivariant 0-cell. The left hand side of (7.3) evaluated in is equal to
[TABLE]
The right hand side of (7.3) evaluated in is equal to
[TABLE]
Now, that both expressions are equal follow by (7.2) and (4.1). ∎
7.1.1. Proof of Theorem 7.1
Let us first describe an object in the equivariantization of the category An object in is a collection where , and is a morphism in the category, for any . This means, that is a 1-cell, for any equivariant 0-cell , and for any equivariant 1-cell there is an isomorphism such that equation (1.4) is fulfilled. Also, for any and any equivariant 0-cell there are morphisms
[TABLE]
such that
[TABLE]
[TABLE]
for any equivariant 0-cells , any equivariant 1-cell , and any . Equation (7.6) follows from the fact that is a morphism in the category , and equation (7.7) follows from (5.3).
Define the functor as follows. Let , then . For any equivariant 0-cell , must be an equivariant 1-cell in the category . Define , where
[TABLE]
If is an equivariant 1-cell, then
[TABLE]
[TABLE]
Claim 7.1**.**
The following statements hold.
- (i)
, for any equivariant 0-cell .
- (ii)
The object belongs to the category In particular, the functor is well-defined.
- (iii)
The functor is an equivalence of categories, and it has a monoidal structure.
Proof of Claim.
(i). We must check that the maps satisfy (4.2). In this case, we must prove that for any
[TABLE]
is equal to
[TABLE]
Using the definition of , we get that the first expression is equal to
[TABLE]
The second equality follows from (7.7), and the last one follows from (7.1).
(ii). Since for any equivariant 1-cell , then satisfy (1.4). We must verify only that is an equivariant 2-cell, that is (4.3) is satisfied. To simplify the notation, let us denote In this particular case, using the composition of equivariant 1-cells given by (4.4), we have to prove that
[TABLE]
The left hand side of equation (7.9) is equal to
[TABLE]
The first equality follows by using the definition of given in (7.8), the second equality follows from (7.6), and the third one follows from the definition of .
(iii). The fact that is an equivalence follows easily. A direct computation shows that
[TABLE]
for any pair of objects . ∎
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