Motivation
The nerve of a category is often used to construct topological versions of moduli spaces. If X is an object of C, its moduli space should somehow encode all objects isomorphic to X and keep track of the various isomorphisms between all of these objects in that category. This can become rather complicated, especially if the objects have many non-identity automorphisms. The nerve provides a combinatorial way of organizing this data. Since simplicial sets have a good homotopy theory, one can ask questions about the meaning of the various homotopy groups πn(N(C)). One hopes that the answers to such questions provide interesting information about the original category C, or about related categories.
The notion of nerve is a direct generalization of the classical notion of classifying space of a discrete group; see below for details.
Read more about this topic: Nerve (category Theory)
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