a list of three vertices A, B, C and three arrows B → C, A → C and A → B. In general, an n-simplex is an object made up from a list of n + 1 vertices (which are 0-simplices) and n + 1 faces (which are (n − 1)-simplices). The vertices of the i-th face are the vertices of the n-simplex minus the i-th vertex. The vertices of a simplex need not be distinct and a simplex is not determined by its vertices and faces: two different simplices may share the same list of faces (and therefore the same list of vertices), just like two different arrows in a multigraph may connect the same two vertices. Simplicial sets should not be confused with abstract simplicial complexes, which generalize simple undirected graphs rather than directed multigraphs. Formally, a simplicial set X is a collection of sets Xn, n = 0, 1, 2, ..., together with certain maps between these sets: the face maps dn,i : Xn → Xn−1 (n = 1, 2, 3, ... and 0 ≤ i ≤ n) and degeneracy maps sn,i : Xn→Xn+1 (n = 0, 1, 2, ... and 0 ≤ i ≤ n). We think of the elements of Xn as the n-simplices of X. The map dn,i assigns to each such n-simplex its i-th face, the face "opposite to" (i.e. not containing) the i-th vertex. The map sn,i assigns to each n-simplex the degenerate (n+1)-simplex which arises from the given one by duplicating the i-th vertex. This description implicitly requires certain consistency relations among the maps dn,i and sn,i. Rather than requiring these simplicial identities explicitly as part of the definition, the short and elegant modern definition uses the language of category theory. Formal definition Let Δ denote the simplex category. The objects of Δ are nonempty linearly ordered sets of the form [n] = {0, 1, ..., n} with n≥0. The morphisms in Δ are (non-strictly) order-preserving functions between these sets. A simplicial set X is a contravariant functor X : Δ → Set where Set is the category of sets. (Alternatively and equivalently, one may define simplicial sets as covariant functors from the opposite category Δop to Set.) Given a simplicial set X, we often write Xn instead of X([n]). Simplicial sets form a category, usually denoted sSet, whose objects are simplicial sets and whose morphisms are natural transformations between them.