Nerve (category theory)
In category theory, a discipline within mathematics, the nerve N(C) of a small category C is a simplicial set constructed from the objects and morphisms of C. The geometric realization of this simplicial set is a topological space, called the classifying space of the category C. These closely related objects can provide information about some familiar and useful categories using algebraic topology, most often homotopy theory.
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.
Let C be a small category. There is a 0-simplex of N(C) for each object of C. There is a 1-simplex for each morphism f : x → y in C. Now suppose that f: x → y and g : y → z are morphisms in C. Then we also have their composition gf : x → z.
The diagram suggests our course of action: add a 2-simplex for this commutative triangle. Every 2-simplex of N(C) comes from a pair of composable morphisms in this way. The addition of these 2-simplices does not erase or otherwise disregard morphisms obtained by composition, it merely remembers that this is how they arise.
In general, N(C)k consists of the k-tuples of composable morphisms
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