Group actions on 2-categories

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 2category 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 ZG(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. 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].

Equivariant Morita theory for graded tensor categories

“…Recall the notion of group actions on 2-categories, as discussed for example in [HSV17] and [BGM19]. A strict G-action on a 2-category B is a collection of 2-functors {g • :…”

A 3-categorical perspective on G-crossed braided categories

“…Example 1.11 (Generalized relative center construction). The article [BGM19] shows that every (weak) G-action on a 2-category may be strictified to a strict G-action on a strict 2-category, encoded by a group homomorphism π : G → Aut st (B), where Aut st (B) is the group of strict 2-equivalences of B which admit strict inverses. From such a strict G-action, the authors then construct a G-crossed braided monoidal category Z G (B) whose g-graded component is the category of pseudonatural transformations and modifications PseudoNat(id B ⇒ π(g)).…”

“…Our construction of a G-crossed braided monoidal category from a G-pointed 3-category may be understood as a generalization of [BGM19] from G-actions on 2-categories, encoded by 3-functors BG → 2Cat from BG into the 3-category of 2-categories, to arbitrary 3-functors BG → C. In particular, we show in Section 3.2 that we may strictify a 1-surjective weak 3-functor BG → C to a Gray-functor BG → C into a Gray-category C equivalent to C, and construct a G-crossed braided category from this data.…”

“…Our main theorem was inspired by, and is closely related to, the following two results: Passing from a G-pointed 3-category to the associated G-crossed braided category generalizes a result of [BGM19] which constructs G-crossed braided categories from group actions on 2-categories; see Example 1.11 for more details. A version of the construction of a G-pointed 3-category from a G-crossed braided category is discussed in [Cui19], and we use this construction in Section 4.3 to prove essential surjectivity of the 2functor 3Cat st G → GCrsBrd st .…”

The Serre automorphism via homotopy actions and the Cobordism Hypothesis for oriented manifolds

“…In section 5, we use the wire-diagram calculus developed in [Bar14]. It is worth noticing that the study of actions of groups on higher categories and their homotopy fixed points is also of independent interest, see for instance [EGNO15,BGM17] for the case of finite groups. The paper is organized as follows.…”