Thomason's colimit theorem for the double category of elements

TL;DR

This paper extends Thomason's colimit theorem to the double category of elements for any 2-category and 2-functor to Cat, establishing a weak homotopy equivalence between the homotopy colimit and the classifying space of the double category.

Contribution

It demonstrates that the double category of elements satisfies a version of Thomason's colimit theorem, generalizing previous results to a broader categorical context.

Findings

Establishes a weak homotopy equivalence between B hocolim F and B(∫_C F)

Extends Thomason's colimit theorem to double categories of elements

Provides a new perspective on the homotopy theory of 2-categories

Abstract

We show that, for any 2-category $C$ and 2-functor $F\colon C \to Cat$, the double category of elements $\iint_C F$ introduced by Grandis and Par\'e satisfies a version of Thomason's colimit theorem; that is, there is a weak homotopy equivalence $B hocolim F\simeq B(\iint_C F)$.