nLab equivariantization


Monoidal categories

Fusion categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products



Internal monoids



In higher category theory


On this whole page, assume that GG is a finite group. Denote by GΜ²\underline{G} the corresponding discrete monoidal category, and by Rep G\operatorname{Rep}_G its finite dimensional kk-representations. Further, for a monoidal category π’ž\mathcal{C}, denote by π’žβˆ’β„³π’ͺπ’Ÿ\mathcal{C}-\mathcal{MOD} the bicategory of kk-linear π’ž \mathcal{C} -module categories, module functors and natural transformations.

Main idea

There are 2-functors D:Rep Gβˆ’β„³π’ͺπ’Ÿβ‡„GΜ²βˆ’β„³π’ͺπ’Ÿ:ED : \operatorname{Rep}_G-\mathcal{MOD} \rightleftarrows \underline{G}-\mathcal{MOD} : E, called equivariantisation and deequivariantisation, respectively. They become weak inverses once we restrict to semisimple module categories. In a sense, (de)equivariantisation generalises the Morita equivalence between Rep G\operatorname{Rep}_G and Vec G\operatorname{Vec}_G for finite groups.

Detailed definitions

A GΜ²\underline{G}-action (a GΜ²\underline{G}-module category structure) on a kk-linear semisimple category π’ž\mathcal{C} amounts to the following data:

  • for each group element g∈Gg \in G a (linear) equivalence ρ(g):π’žβ†’π’ž\rho(g)\colon \mathcal{C} \to \mathcal{C}
  • natural isomorphisms ρ 2(g,h) X:ρ(g)(ρ(h)X)→ρ(gh)X\rho^{2}(g,h)_X\colon \rho(g)(\rho(h) X) \to \rho(gh)X
  • a natural isomorphism ρ X 0:X→ρ(1)X\rho^0_X\colon X \to \rho(1)X
  • satisfying certain obvious coherence axioms


Definition A GG-equivariant object in π’ž\mathcal{C} is then an object XX in π’ž\mathcal{C} and a family of isomorphisms u g:ρ(g)Xβ†’Xu_g\colon \rho(g) X \to X compatible with the action and Ξ³\gamma.

The GG-equivariant objects form a category (where morphisms need to commute with the u gu_g), which is denoted by π’ž G\mathcal{C}^G.

The primordial example for this construction is the category of finite dimensional vector spaces kβˆ’Vectk-\operatorname{Vect}, which has a trivial GG-action. Its GG-equivariant objects kβˆ’Vect Gk-\operatorname{Vect}^G are simply Rep G\operatorname{Rep}_G.

Since kβˆ’Vectk-\operatorname{Vect} has a natural action on any kk-linear category like π’ž\mathcal{C}, kβˆ’Vect G≃Rep Gk-\operatorname{Vect}^G \simeq \operatorname{Rep}_G acquires an action on π’ž G\mathcal{C}^G. Explicitly, let (V,Ο•:Gβ†’End(V))(V, \phi\colon G \to \operatorname{End}(V)) a representation, and (X,u g)(X, u_g) an equivariant object, then the module structure βˆ’β–Ήβˆ’:Rep(G)βŠ π’ž Gβ†’π’ž G-\triangleright-\colon \operatorname{Rep}(G) \boxtimes \mathcal{C}^G \to \mathcal{C}^G is defined as (V,Ο•)β–Ή(X,u g)=(Vβ–ΉX,Ο•(g)βŠ—u g)(V, \phi) \triangleright (X, u_g) = (V \triangleright X, \phi(g) \otimes u_g). Thus we define the 2-functor EE as Eπ’ž=π’ž GE\mathcal{C} = \mathcal{C}^G.


First note that the function algebra k(G)k(G) is an object in Rep(G)\operatorname{Rep}(G) by means of left multiplication with GG. It has an additional GG-representation by right inverse multiplication. Furthermore, it is an internal algebra.

Definition Let π’ž\mathcal{C} be a β„³\mathcal{M}-module category, and AA an algebra internal to β„³\mathcal{M}. An AA-module in π’ž\mathcal{C} is an object XX in π’ž\mathcal{C} and a morphism Aβ–ΉXβ†’XA \triangleright X \to X satisfying the obvious action axioms.

We can therefore form the category of k(G)k(G)-modules in π’ž\mathcal{C}, since we have a Rep(G)\operatorname{Rep}(G) module structure, and denote it by π’ž G\mathcal{C}_G. Since k(G)k(G) has the additional right GG-representation, π’ž G\mathcal{C}_G inherits the GG-action. Thus the 2-functor Dπ’ž=π’ž GD\mathcal{C} = \mathcal{C}_G is defined.

Mutual inverses

Theorem The previously defined 2-functors DD and EE are mutually (weakly) inverse 2-equivalences between the bicategory of semisimple kk-linear categories with GG-action, and the bicategory of semisimple kk-linear categories with Rep g\operatorname{Rep}_g-action. In particular, π’žβ‰ƒ(π’ž G) G\mathcal{C} \simeq (\mathcal{C}^G)_G.

Forgetful and induction functors

  • There is an obvious faithful forgetful functor U:π’ž Gβ†’π’žU \colon \mathcal{C}^G \to \mathcal{C}.
  • There is a left and right adjoint to UU, the induction functor IX=βŠ• g∈Gρ(g)XIX = \oplus_{g \in G} \rho(g) X.
  • There is an obvious forgetful functor Uβ€²:π’ž Gβ†’π’žU'\colon \mathcal{C}_G \to \mathcal{C}.
  • There is a left adjoint to UU, the canonical functor FX=k(G)βŠ—XFX = k(G) \otimes X, with the trivial action on XX.

Monoidal and braided categories, and central functors

The previous constructions generalise easily when our categories acquire monoidal or braided structures.

Actions of groups on monoidal categories are simply a monoidal equivalence for every group element with the same extra structure as mentioned before. Actions of groups on braided categories are additionally required to preserve the braiding. (Following the idea of stuff, structure, and properties, an action on a symmetric category are simply actions on the underlying braided category.)

The action of kβˆ’Vectk-\operatorname{Vect} on a monoidal linear category π’ž\mathcal{C} can also be understood as the canonical monoidal functor kβˆ’Vectβ†’π’žk-\operatorname{Vect} \to \mathcal{C} sending kk to the monoidal unit, followed by the tensor product. In this situation, the Rep G\operatorname{Rep}_G module structure on π’ž\mathcal{C} can then also be understood as the canonical functor Rep Gβ†’π’ž G\operatorname{Rep}_G \to \mathcal{C}^G.

By abstract nonsense, an action on an algebraic object should be a morphism into its center. Consequently, the action of a braided category ℬ\mathcal{B} on a monoidal category π’ž\mathcal{C} should be given by a functor ℬ→𝒡(π’ž)\mathcal{B} \to \mathcal{Z}(\mathcal{C}), where 𝒡\mathcal{Z} is the Drinfeld center. This is called a central functor. Similarly, the action of a symmetric category π’œ\mathcal{A} on a braided category ℬ\mathcal{B} should be a functor π’œβ†’β„¬β€²\mathcal{A} \to \mathcal{B}', where ℬ′\mathcal{B}' is the β€œsymmetric center”, or β€œMΓΌger center”.

The previously mentioned functor Rep Gβ†’π’ž G\operatorname{Rep}_G \to \mathcal{C}^G factors through the forgetful functor 𝒡(π’ž)β†’π’ž\mathcal{Z}(\mathcal{C}) \to \mathcal{C} (or, in the case of a braided category, through the inclusion π’žβ€²β†ͺπ’ž\mathcal{C}' \hookrightarrow \mathcal{C}), so we have an action in this sense.

Theorem By equivariantisation, the bicategory of fusion categories with GG-actions is 2-equivalent to the bicategory of fusion categories with central functors from Rep G\operatorname{Rep}_G. Similarly, the bicategory of braided fusion categories with braided GG-actions is 2-equivalent to the bicategory of fusion categories (symmetric central functors from Rep G\operatorname{Rep}_G.


Each braided fusion category ℬ\mathcal{B} has a canonical symmetric subcategory, its symmetric centre ℬ′\mathcal{B}'. By choosing a trivial twist, ℬ′\mathcal{B}' has a canonical spherical structure. If all the quantum dimensions are positive, that is, if ℬ′\mathcal{B}' is tannakian, it is equivalent to Rep G\operatorname{Rep}_G for a unique group GG (since there is a canonical fibre functor). The obvious construction is then to deequivariantise over ℬ′≃Rep G\mathcal{B}' \simeq \operatorname{Rep}_G, which yields ℬ G\mathcal{B}_G, the modularisation of ℬ\mathcal{B}.

There is an β€œexact sequence” of braided fusion categories:

kβˆ’Vectβ†ͺℬ′≃Rep Gβ†ͺℬ↠ℬ G k-\operatorname{Vect} \hookrightarrow \mathcal{B}' \simeq \operatorname{Rep}_G \hookrightarrow \mathcal{B} \twoheadrightarrow \mathcal{B}_G

It is exact in the sense that ℬ′\mathcal{B}' is the maximal symmetric subcategory of ℬ\mathcal{B}, and that ℬ G\mathcal{B}_G is the maximal modular category such that ℬ′\mathcal{B}' is sent to sums of the monoidal unit.


  • Vladimir Drinfeld, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik, On braided fusion categories I, section 4 arXiv
  • Alain BruguiΓ¨res, CatΓ©gories prΓ©modulaires, modularisations et invariants des variΓ©tΓ©s de dimension 3 (french) pdf
  • Michael MΓΌger, Galois Theory for Braided Tensor Categories and the Modular Closure arXiv

Last revised on September 5, 2017 at 15:16:11. See the history of this page for a list of all contributions to it.