On the subdivision of small categories

TL;DR

This paper develops an intrinsic approach to subdividing small categories, providing new insights into their properties and a novel proof of the equivalence between homotopy categories of spaces and small categories using posets.

Contribution

It introduces an intrinsic subdivision method for small categories and offers a new conceptual proof of the equivalence between homotopy categories of spaces and small categories.

Findings

Provides an intrinsic subdivision framework for small categories

Derives fundamental properties of the subdivision process

Offers a new proof of the equivalence between homotopy categories

Abstract

We present an intrinsic and concrete development of the subdivision of small categories, give some simple examples and derive its fundamental properties. As an application, we deduce an alternative way to compare the homotopy categories of spaces and small categories, by using partially ordered sets. This yields a new conceptual proof to the well-known fact that these two homotopy categories are equivalent.