object is a set S equipped with a collection of non-empty finite subsets that contains all singletons and such that if is in and is non-empty, then also belongs to . a morphism from to is a function such that the image of any element of is an element of . Geometric realization We can associate to any abstract simplicial complex (ASC) K a topological space , called its geometric realization. There are several ways to define . Geometric definition Every geometric simplicial complex (GSC) determines an ASC: the vertices of the ASC are the vertices of the GSC, and the faces of the ASC are the vertex-sets of the faces of the GSC. For example, consider a GSC with 4 vertices {1,2,3,4}, where the maximal faces are the triangle between {1,2,3} and the lines between {2,4} and {3,4}. Then, the corresponding ASC contains the sets {1,2,3}, {2,4}, {3,4}, and all their subsets. We say that the GSC is the geometric realization of the ASC. Every ASC has a geometric realization. This is easy to see for a finite ASC. Let . Identify the vertices in with the vertices of an (N-1)-dimensional simplex in . Construct the GSC {conv(F): F is a face in K}. Clearly, the ASC associated with this GSC is identical to K, so we have indeed constructed a geometric realization of K. In fact, an ASC can be realized using much fewer dimensions. If an ASC is d-dimensional (that is, the maximum cardinality of a simplex in it is d+1), then it has a geometric realization in , but might not have a geometric realization in The special case d=1 corresponds to the well-known fact, that any graph can be plotted in where the edges are straight lines that do not intersect each other except in common vertices, but not any graph can be plotted in in this way. If K is the standard combinatorial n-simplex, then can be naturally identified with . Every two geometric realizations of the same ASC, even in Euclidean spaces of different dimensions, are homeomorphic. Therefore, given an ASC K, one can speak of the geometric realization of K. Topological definition The construction goes as follows. First, define as a subset of consisting of functions satisfying the two conditions: Now think of the set of elements of with finite support as the direct limit of where A ranges over finite subsets