The Euler characteristic of a category as the sum of a divergent series

TL;DR

This paper explores a method to assign a finite Euler characteristic to infinite cell complexes, especially the nerve of a finite category, by evaluating divergent series in a meaningful way.

Contribution

It introduces an alternative approach to define the Euler characteristic of categories using divergent series, extending classical concepts to infinite complexes.

Findings

Provides a new method to evaluate Euler characteristics of infinite complexes

Shows equivalence of the new definition with classical one in many cases

Applies to the nerve of finite categories for meaningful Euler characteristic calculation

Abstract

The Euler characteristic of a cell complex is often thought of as the alternating sum of the number of cells of each dimension. When the complex is infinite, the sum diverges. Nevertheless, it can sometimes be evaluated; in particular, this is possible when the complex is the nerve of a finite category. This provides an alternative definition of the Euler characteristic of a category, which is in many cases equivalent to the original one (math.CT/0610260).