Here, NC([n]) is the set of all functors from [n] to C, where we consider [n] as a category with objects 0,1,...,n and a single morphism from i to j whenever i ≤ j. Concretely, the n-simplices of the nerve NC can be thought of as sequences of n composable morphisms in C: a0 → a1 → ... → an. (In particular, the 0-simplices are the objects of C and the 1-simplices are the morphisms of C.) The face map d0 drops the first morphism from such a list, the face map dn drops the last, and the face map di for 0 < i < n drops ai and composes the ith and (i + 1)th morphisms. The degeneracy maps si lengthen the sequence by inserting an identity morphism at position i. We can recover the poset S from the nerve NS and the category C from the nerve NC; in this sense simplicial sets generalize posets and categories. Another important class of examples of simplicial sets is given by the singular set SY of a topological space Y. Here SYn consists of all the continuous maps from the standard topological n-simplex to Y. The singular set is further explained below. The standard n-simplex and the category of simplices The standard n-simplex, denoted Δn, is a simplicial set defined as the functor homΔ(-, [n]) where [n] denotes the ordered set {0, 1, ... ,n} of the first (n + 1) nonnegative integers. (In many texts, it is written instead as hom([n],-) where the homset is understood to be in the opposite category Δop.) By the Yoneda lemma, the n-simplices of a simplicial set X stand in 1–1 correspondence with the natural transformations from Δn to X, i.e. . Furthermore, X gives rise to a category of simplices, denoted by , whose objects are maps (i.e. natural transformations) Δn → X and whose morphisms are natural transformations Δn → Δm over X arising from maps [n] → [m] in Δ. That is, is a slice category of Δ over X. The following isomorphism shows that a simplicial set X is a colimit of its simplices: where the colimit is taken over the category of simplices of X. Geometric realization There is a functor |•|: sSet → CGHaus called the geometric realization taking a simplicial set X to its corresponding realization in the category of compactly-generated Hausdorff topological spaces. Intuitively, the