A homotopy colimit theorem for diagrams of braided monoidal categories

TL;DR

This paper extends the homotopy colimit theorem to diagrams of braided monoidal categories, showing their homotopy type can be represented by a simplicial set nerve and is equivalent to the homotopy colimit of the associated simplicial sets.

Contribution

It introduces a homotopy colimit theorem for diagrams of braided monoidal categories using the simplicial nerve, bridging categorical structures with homotopy theory.

Findings

Homotopy type of diagrams represented by simplicial set nerve.

Weak homotopy equivalence between nerve-based and homotopy colimit constructions.

Extension of Thomason's theorem to braided monoidal categories.

Abstract

Thomason's Homotopy Colimit Theorem has been extended to bicategories and this extension can be adapted, through the delooping principle, to a corresponding theorem for diagrams of monoidal categories. In this version, we show that the homotopy type of the diagram can be also represented by a genuine simplicial set nerve associated with it. This suggests the study of a homotopy colimit theorem, for diagrams $\b$ of braided monoidal categories, by means of a simplicial set {\em nerve of the diagram}. We prove that it is weak homotopy equivalent to the homotopy colimit of the diagram, of simplicial sets, obtained from composing $\b$ with the geometric nerve functor of braided monoidal categories.