A Twisted Version of the Classifying Space Functor

TL;DR

This paper demonstrates a homeomorphism between geometric realizations of small categories and their cofibrant replacements, introducing a twisted classifying space functor that nearly forms a natural transformation.

Contribution

It establishes a homeomorphism specific to small categories and explores the naturality of these maps within the context of classifying space functors.

Findings

Homeomorphism between geometric realizations and cofibrant replacements for small categories

Introduction of a twisted version of the classifying space functor

Analysis of naturality of the constructed homeomorphisms

Abstract

It is known that there is a weak-equivalence between the geometric realization of a simplicially enriched small category and its cofibrant replacement [12]. In this paper, we show that when only small categories are considered there exists a homeomorphism between these geometric realizations. We also discuss the naturality of these homoemorphisms. The inclusion of the category of small categories to the category of simplicially enriched categories, the cofibrant replacement of simplicially enriched categories, and the geometric realization of simplicially enriched categories are three composable functors. Hence one can ask if the collection of all these homeomorphisms gives a natural transformation from the composition of these three functors to the classifying space functor. We show that this is almost the case and that this composition can be considered as some twisted version of the classifying space functor. \end{abstract}