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