Ramified coverings of small categories

TL;DR

This paper introduces the concept of ramified coverings for small categories, establishing key properties such as the Riemann-Hurwitz formula, zeta function divisibility, and classifying space behavior, extending classical ideas to categorical structures.

Contribution

It defines ramified coverings of small categories and proves fundamental properties analogous to classical covering theory, including Riemann-Hurwitz, zeta function relations, and classifying space correspondence.

Findings

Riemann-Hurwitz formula holds for finite categories

Zeta function of base divides that of the covering

Classifying space of a ramified covering is also a ramified covering

Abstract

We introduce a ramified covering of small categories, and we show three properties of the notion: the Riemann-Hurwitz formula holds for a ramified covering of finite categories, the zeta function of $C$ divides that of $\widetilde{C}$ for a ramified covering $\map{P}{\widetilde{C}}{C}$ of finite categories, and the classifying space of a $d$-fold ramified covering of small categories is also a $d$-fold ramified covering in the sense of Dold \cite{Dol86}.