the map di composes the morphisms Ai−1 → Ai and Ai → Ai+1 into the morphism Ai−1 → Ai+1, yielding a (k − 1)-tuple for every k-tuple. Similarly, the degeneracy maps are given by inserting an identity morphism at the object Ai. 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 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. Examples The primordial example is the classifying space of a discrete group G. We regard G as a category with one object whose endomorphisms are the elements of G. Then the k-simplices of N(G) are just k-tuples of elements of G. The face maps act by multiplication, and the degeneracy maps act by insertion of the identity element. If G is the group with two elements, then there is exactly one nondegenerate k-simplex for each nonnegative integer k, corresponding to the unique k-tuple of elements of G containing no identities. 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. The latter is the most popular model for the classifying space of the group with two elements. See (Segal 1968) for further details and the relationship of the above to Milnor's join construction of BG. Most spaces are classifying spaces Every "reasonable" topological space is homeomorphic to the classifying space of a small category. Here, "reasonable" means that the space in question is the geometric realization of a simplicial set. This is obviously a necessary