Coverings of small categories and nerves

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 $χ(E)=χ(F)χ(B)$ for Euler characteristic of categories and coverings.

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