\v{C}ech complexes for covers of small categories

TL;DR

This paper introduces a combinatorial analogue of the nerve theorem for small categories using the Grothendieck construction, and applies it to establish an inclusion-exclusion principle for the Euler characteristic of finite categories.

Contribution

It develops a new combinatorial approach to the nerve theorem for small categories and proves an inclusion-exclusion principle for their Euler characteristic.

Findings

Established a combinatorial nerve theorem analogue for small categories

Proved the inclusion-exclusion principle for Euler characteristics of finite categories

Applied Grothendieck construction to categorical covers

Abstract

We present a combinatorial analogue of the nerve theorem for covers of small categories, using the Grothendieck construction. We apply our result to prove the inclusion-exclusion principle for the Euler characteristic of a finite category.