Nerve of a category

Given a category , its nerve is the simplicial set constructed as follows. The set of -simplices is the set of diagrams

of objects and morphisms from . The face maps are given by composition of morphisms atthe node in the diagram (or dropping the first or last arrow if or respectively), and the degeneracy maps are given by inserting identity morphisms. The intuition here is that a -simplex in is precisely a commutative diagram in with the shape of a -simplex. If is a topological category, we can enrich to be a simplicial space.

Jeffrey Herschel Giansiracusa 2005-06-27