Coverings of small categories and nerves

TL;DR

This paper explores the relationship between coverings of small categories and their nerves, establishing formulas for zeta functions and Euler characteristics in the context of finite categories and coverings.

Contribution

It introduces new formulas linking coverings of small categories with their nerves, including zeta functions and Euler characteristics.

Findings

Zeta function of E equals that of B to the number of sheets in the covering.

Euler characteristic of E equals the product of the Euler characteristics of the fiber and base categories.

Abstract

We prove a certain proposition which states a relationship between coverings of small categories and nerves. As its application, we prove that for a covering $\map{P}{E}{B}$ of finite categories, the zeta function of $E$ is the zeta function of $B$ to the number of sheet of $P$. Moreover, we prove the formula $\chi(E)=\chi(F)\chi(B)$ for Euler characteristic of categories and coverings.