Weighted Colimits of 2-Representations and Star Algebras

Abstract

We apply the theory of weighted bicategorical colimits to study the problem of existence and computation of such colimits of birepresentations of finitary bicategories. The main application of our results is the complete classification of simple transitive birepresentations of a bicategory studied previously by Zimmermann. The classification confirms a conjecture he has made.

Module categories, internal bimodules, and Tambara modules

“…This adjunction is an adjoint equivalence. An analogous statement holds in the bicategory $$\mathcal {C}\!\operatorname{-Tamb}$$: using the universal property of Cauchy completion, given a $$\mathcal {C}\!$$‐module category $$\mathbf {M}$$, there is a canonical way to extend the $$\mathcal {C}\!$$‐module category structure to $$\mathbf {M}^{\mathsf {c}}$$; see, for example, [34, Section 3.2]. The embedding $$\iota _{\mathcal {C}}^{\mathsf {c}}$$ then becomes a $$\mathcal {C}\!$$‐module functor.…”