Ramified coverings of small categories (Q2263813)

Given a (small) category \(C\) and an object \(x \in C\) let \(S(x)\) denote the set of all morphisms in~\(C\) with source~\(x\). Similarly, denote by \(T(x)\) the set of morphisms with target~\(x\). A functor \(P : E \rightarrow B\) is called an \textit{unramified covering} if \(B\) is connected, and if \(P\) induces bijections \(S(x) = S\big( P(x) \big)\) and \(T(x) = T\big( P(x) \big)\) for each object \(x \in E\). An equivalence relation ``\(\sim\)'' on the set of objects of~\(E\) is called an \textit{equivalence relation for ramification} if \(x \sim y\) implies \(P(x) = P(y)\), and if for distinct objects \(x \sim y\) implies that either both \(S(x)\) and~\(S(y)\), or both \(T(x)\) and~\(T(y)\) are trivial (contain the relevant identity morphism only). Given an unramified covering \(P : E \rightarrow B\) and an equivalence relation for ramification on the set of objects of~\(E\), the author defines a new category \(\tilde E\); its objects are equivalence classes \([x]\) of objects of~\(E\), and its morphism sets are given by the formula \[ \hom_{\tilde E} \big( [x], [y] \big) = \coprod_{a \in [x] \atop b \in [y]} \hom_{E} (a,b) \;, \] with composition induced by that of~\(E\). Setting \(\tilde P\big([x] \big) = P(x)\) defines a functor \(\tilde P : \tilde E \rightarrow B\) which is called the \textit{ramified covering of~\(P\) by~''\(\sim\)''}. Based on these definitions, the author proves three results. First, an analogue of the Riemann-Hurwitz formula holds for ramified coverings of finite categories (Theorem~2.5), relating the series Euler characteristic of~\(\tilde E\) as defined by \textit{C. Berger} and \textit{T. Leinster} [Homology Homotopy Appl. 10, No. 1, 41--51 (2008; Zbl 1132.18007)] to the series Euler characteristic of~\(B\): \[ \chi_{\Sigma} (\tilde E) = d \cdot \chi_{\Sigma} (B) - V \;, \] where \(d = \# P^{-1} (b)\) for any object \(b \in B\) is the degree of~\(\tilde P\), and where \(V\) is given by \(V = \sum_{[x] \in \tilde E} \big( \#[x]-1 \big)\). Second, again for ramified coverings of finite categories, the zeta function of~\(\tilde E\) is computed in terms of the zeta function of~\(B\) (Theorem~2.6): \[ \zeta_{\tilde E}(z) = \zeta_{B} (z)^{d} \cdot (1-z)^{V} \;, \] with \(d\) and~\(V\) as before. The zeta function ``enumerates'' simplices on the nerve according to their dimension, \[ \zeta_{C}(z) = \exp \Big( \sum_{n=1}^{\infty} {\# n\text{-simplices in }NC \over n} z^{n}\Big) \;, \] and has been analysed by the author [Doc. Math., J. DMV 18, 1243--1274 (2013; Zbl 1291.18023)]. Finally, it is shown that the geometric realisation of a ramified covering of small (not necessarily finite) categories yields a ramified covering (Theorem~2.7 and Corollary~2.8).