Finitary birepresentations of finitary bicategories

Abstract

In this paper, we discuss the generalization of finitary 2-representation theory of finitary 2-categories to finitary birepresentation theory of finitary bicategories. In previous papers on the subject, the classification of simple transitive 2-representations of a given 2-category was reduced to that for certain subquotients. These reduction results were all formulated as bijections between equivalence classes of 2-representations. In this paper, we generalize them to biequivalences between certain 2-categories of birepresentations.

Weighted colimits of 2-representations and star algebras

“…The resulting setting of bicategories, pseudofunctors, strong transformations and modifications is what we will call the bicategorical setting, and we will give our results in this setup. In particular, we will study birepresentations of finitary bicategories, following [MMMTZ1]. Most of our results also hold in what we call the 2 -categorical setting, where we require the structure 2-morphisms to be the identities, thus working with 2-categories, 2functors, 2-transformations and modifications.…”

“…Notation. Our notational conventions largely follow those of [MMMTZ1], with one difference and a few additions. The identity 1-morphisms of objects will be denoted by 1 i and the like, and the identity 2-morphisms of 1-morphisms will be denoted by id F , and the like.…”

“…The whiskering of a 1-morphism G with a 2-morphism α, given by id G • h α, will be denoted by G Similarly to, [MMMTZ1], we only indicate the 1-morphisms indexing the structure 2morphisms of a bicategorical structure, while omitting the indexing objects from the notation, thus writing a H,G,F rather than a i,j,k,l H,G,F for the associator…”

“…The 2-category B str n can be viewed as a strictification of a monoidal category B n of certain projective bimodules over a quotient of a zig-zag algebra A n , where the monoidal structure is given by − ⊗ An −. We follow [MMMTZ1] in relaxing the 2categorical setup to the bicategorical setup, which allows us to consider simple transitive birepresentations of B n rather than simple transitive 2-representations of B str n . As observed in [MMMTZ1], the resulting two classification problems are equivalent.…”

“…We follow [MMMTZ1] in relaxing the 2categorical setup to the bicategorical setup, which allows us to consider simple transitive birepresentations of B n rather than simple transitive 2-representations of B str n . As observed in [MMMTZ1], the resulting two classification problems are equivalent. The bicategorical setup allows for greater flexibility in the computation of colimits.…”