**Data**

- Title: Finitary birepresentations of finitary bicategories
- Authors: Marco Mackaay, Volodymyr Mazorchuk, Vanessa Miemietz, Daniel Tubbenhauer and Xiaoting Zhang
- Status: Preprint. Last update: Fri, 8 Jan 2021 10:36:39 UTC
- ArXiv link: https://arxiv.org/abs/2008.01658
- LaTex Beamer presentation: Slides1, Slides2, Slides3, Slides4

**Abstract**

**A few extra words**

The main purpose of this paper is therefore to discuss the generalization of some important foundational results on finitary $2$-representation theory to finitary birepresentation theory. By discussing, we mean formulating those results carefully in the greatest possible generality (or at least as generally as we currently can) and proving them in detail whenever the proof is not straightforward and cannot be found in the literature. A lot of the results in this paper will not surprise the experts, but we think that it is important to have the statements and their proofs, which sometimes involve quite complicated diagrams, in written form somewhere in the literature.

An example of what changes in this bicategorical setup compared to the $2$-categorical one is

which is an adjunction (a.k.a. zigzag) relation, coming now with associators and unitors. As a bonus, we prove a strong version of the $\mathcal{H}$-reduction theorem as well as a version of the double centralizer theorem in our context, both of which are new.