Subsection 3.1. (0MT1) · ScalingStacks

3.1. The nerve of a category[0MT1]

Given a category $C$ , let $\nerve C$ denote the nerve of $C$ ; that is, $\nerve C$ is a simplicial set whose $n$ -simplices consist of the set of functors $[n]\rightarrow C$ . (The classifying space $BC$ of a category is a topological space which is the geometric realization of the nerve.) The following is well-known.

[05TX]

Proposition 3.2. The nerve of $[n]$ is $\Delta[n]$ . For categories $C$ and $D$ there are natural isomorphisms

\nerve(C\times D)\approx\nerve C\times\nerve D\qquad\text{and}\qquad\nerve(D^{C})\approx\nerve(D)^{\nerve(C)}.

The functor $\nerve\colon{\operatorname{\mathcal{C}at}}\rightarrow{\operatorname{\mathcal{S}}}$ is a full embedding of categories. Furthermore, if $C$ is a groupoid then $\nerve(C)$ is a Kan complex.

Although the nerve functor is a full embedding, it is awkward from our point of view, since non-equivalent categories may give rise to weakly equivalent nerves.

Charles Rezk

Original source: arXiv:math/9811037v3

Original source · math/9811037v3