Topology, combinatorics, and categories: from ∆ to Θ_2

The predominant purpose of this paper is to provide a set of generating maps for the category Θ_2 and the relations amongst them. To understand Θ_2 we examine the category ∆, of which Θ_2 is a generalization. We describe a natural way that ∆ arises as one tries to model topological spaces combinatorially by means of simplices, leading to the concept of a simplicial set. We show that the generating maps of ∆ can be utilized to define generating maps for Θ_2, with many of the relations amongst these generating maps for Θ_2 coming from the relations of the generating maps of ∆.