On the homotopy type of a cofibred category

TL;DR

This paper introduces two new simplicial constructions, the fibred and cleaved nerves, to analyze the homotopy type of cofibred categories, extending classical theorems and providing new insights into their homotopy and homology.

Contribution

It presents the fibred and cleaved nerves as novel tools for studying the homotopy type of cofibred categories, generalizing classical nerve constructions.

Findings

Fibred nerve extends Segal's nerve to cofibred categories.

Cleaved nerve offers a smaller simplicial model for splitting fibrations.

New results on homotopy and homology of small categories.

Abstract

In this paper we describe two ways on which cofibred categories give rise to bisimplicial sets. The "fibred nerve" is a natural extension of Segal's classical nerve of a category, and it constitutes an alternative simplicial description of the homotopy type of the total category. If the fibration is splitting, then one can construct the "cleaved nerve", a smaller variant which emerges from a distinguished closed cleavage. We interpret some classical theorems by Thomason and Quillen in terms of our constructions, and use the fibred and cleaved nerve to establish new results on homotopy and homology of small categories.