Nerve of a category

Given a category $ C$, its nerve $ N(C)$ is the simplicial set constructed as follows. The set $ N(C)_{k}$ of $ k$-simplices is the set of diagrams

$\displaystyle A_{0} \rightarrow A_{1} \rightarrow \dots \rightarrow A_{k}

of objects and morphisms from $ C$. The face maps $ \partial_i:N(C)_{k}
\rightarrow N(C)_{k-1}$ are given by composition of morphisms atthe $ i^{th}$ node in the diagram (or dropping the first or last arrow if $ i=0$ or $ k$ respectively), and the degeneracy maps are given by inserting identity morphisms. The intuition here is that a $ k$-simplex in $ N(C)$is precisely a commutative diagram in $ C$ with the shape of a $ k$-simplex. If $ C$ is a topological category, we can enrich $ N(C)$ to be a simplicial space.

Jeffrey Herschel Giansiracusa 2005-06-27