An introduction to simplicial sets

This is an elementary introduction to simplicial sets, which are generalizations of ∆-complexes from algebraic topology. The theory of simplicial sets provides a way to express homotopy and homology without requiring topology. This paper is meant to be accessible to anyone who has had experience with algebraic topology and has at least basic knowledge of category theory. An important part of simplicial homology is the idea of using ∆-complexes instead of simplicial complexes (see [6, Ch. 2]). They allow one to deal with the combinatorial data associated with a simplicial complex (which is important for homology) instead of the actual topological structure (which is not). Another simplex-based homology theory is singular homology, whose singular maps (see [6, Ch. 2]) represent simplices in a given topological space. While singular maps have properties analogous to simplices, such as a sensible definition for the faces of a singular map, singular maps are not in general injective, which means the data for gluing the faces of a singular map together might not be able to be described as a ∆-complex. The theory of simplicial sets generalizes the idea of ∆-complexes to encompass other objects with simplex structure, such as singular maps. The theory provides a realization functor |−| from simplicial sets to topological spaces which preserves homotopy. This functor is left adjoint to the functor S which takes topological spaces and gives a simplicial set consisting of the singular maps. Perhaps the most difficult part for a newcomer to the subject of simplicial sets is getting used to the category theory involved. Because of this, this paper limits discussion to simplicial sets and algebraic topology. The author found [1] very useful when trying to understand the idea of simplicial sets and [4] illuminating for the derivation of the relations (8), (9), and (10). Much of the material comes from [3], but it was corroborated with [2] to determine what is modern notation.