Nerve (category theory)

The nerve of a category is often used to construct topological versions of moduli spaces.

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 of C. To complete the definition of N(C) as a simplicial set, we must also specify the face and degeneracy maps.

Similarly, the degeneracy maps are given by inserting an identity morphism at the object Ai.

We can now describe the nerve of the category C as the functor Δop → Set This description of the nerve makes functoriality transparent; for example, a functor between small categories C and D induces a map of simplicial sets N(C) → N(D).

After passing to the geometric realization, this k-tuple can be identified with the unique k-cell in the usual CW structure on infinite-dimensional real projective space.

Here, "reasonable" means that the space in question is the geometric realization of a simplicial set.

Note that the realization of this nerve is not generally homeomorphic to X (or even homotopy equivalent): homotopy equivalence will usually hold only for a good cover by contractible sets having contractible intersections.

Theorem — A simplicial set is the nerve of a category if and only if it satisfies the Segal conditions.