The Euler characteristic of an enriched category

TL;DR

This paper extends the concept of Euler characteristic to categories enriched over monoidal model categories, especially topological categories, and applies it to compute the Euler characteristic of cellular stratified spaces.

Contribution

It introduces a definition of Euler characteristic for enriched categories compatible with model structures and demonstrates its application to topological categories and stratified spaces.

Findings

Euler characteristic defined for enriched categories compatible with model structures

Application to topological categories and cellular stratified spaces

Computed Euler characteristic of a stratified space via its face category

Abstract

We define Euler characteristic of a category enriched by a monoidal model category. If a monoidal model category V is equipped with Euler characteristic that is compatible with weak equivalences and fibrations in V, then our Euler characteristic is also compatible with weak equivalences and fibrations in the model structure induced by that of V. In particular, we focus on the case of topological categories; that is, categories enriched by the category of topological spaces. As its application, we obtain the ordinary Euler characteristic of a cellular stratified space X by computing the Euler characteristic of the face category C(X) induced from X.