Nerve (category Theory) - Construction

Construction

Let C be a small category. It is easy to define the sets N(C)_k for small k, which leads to the general definition. In particular, 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. Note that the addition of these 2-simplices does not erase or otherwise disregard morphisms obtained by composition, it merely remembers that that 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. These are also provided to us by the structure of C as a category. The face maps

are given by composition of morphisms at the ith object (or dropping an object on the end, when i is 0 or k). This means that d_i sends the k-tuple

to the (k-1)-tuple

.

That is, the map d_i composes the morphisms A_i-1 → A_i and A_i → A_i+1 into the morphism A_i-1 → A_i+1, yielding a (k−1)-tuple for every k-tuple.

Similarly, the degeneracy maps

are given by inserting an identity morphism at the object A_i.

Recall that simplicial sets may also be regarded as functors Δop → Set, where Δ is the category of totally ordered finite sets and order-preserving morphisms. Every partially ordered set P yields a (small) category i(P) with objects the elements of P and with a unique morphism from p to q whenever p ≤ q in P. We thus obtain a functor i from the category Δ to the category of small categories. We can now describe the nerve of the category C as the functor Δop → Set

.

This description of the nerve makes functoriality quite transparent; for example, a functor between small categories C and D induces a map of simplicial sets N(C) → N(D). Moreover a natural transformation between two such functors induces a homotopy between the induced maps. This observation can be regarded as the beginning of one of the principles of higher category theory. It follows that adjoint functors induce homotopy equivalences. In particular, if C has an initial or final object, its nerve is contractible.