and every face is contained in a facet of dimension . One-dimensional abstract simplicial complexes are mathematically equivalent to simple undirected graphs: the vertex set of the complex can be viewed as the vertex set of a graph, and the two-element facets of the complex correspond to undirected edges of a graph. In this view, one-element facets of a complex correspond to isolated vertices that do not have any incident edges. A subcomplex of is an abstract simplicial complex L such that every face of L belongs to ; that is, and L is an abstract simplicial complex. A subcomplex that consists of all of the subsets of a single face of is often called a simplex of . (However, some authors use the term "simplex" for a face or, rather ambiguously, for both a face and the subcomplex associated with a face, by analogy with the non-abstract (geometric) simplicial complex terminology. To avoid ambiguity, we do not use in this article the term "simplex" for a face in the context of abstract complexes). The d-skeleton of is the subcomplex of consisting of all of the faces of that have dimension at most d. In particular, the 1-skeleton is called the underlying graph of . The 0-skeleton of can be identified with its vertex set, although formally it is not quite the same thing (the vertex set is a single set of all of the vertices, while the 0-skeleton is a family of single-element sets). The link of a face in , often denoted or , is the subcomplex of defined by Note that the link of the empty set is itself. Simplicial maps Given two abstract simplicial complexes, and , a simplicial map is a function that maps the vertices of to the vertices of and that has the property that for any face of , the image is a face of . There is a category SCpx with abstract simplicial complexes as objects and simplicial maps as morphisms. This is equivalent to a suitable category defined using non-abstract simplicial complexes. Moreover, the categorical point of view allows us to tighten the relation between the underlying set S of an abstract simplicial complex and the vertex set of : for the purposes of defining a category of abstract simplicial complexes, the elements of S not lying in are irrelevant. More precisely, SCpx is equivalent to the category where: an