The classification of topological spaces up to homotopy equivalence is the central problem of algebraic topology. The method of attack on this problem is the application of various functors from the category of topological spaces (or a suitable subcategory) to certain algebraic categories. These functors do not distinguish between homotopy equivalent spaces, and the algebraic data they provide may be enough to distinguish nonequivalent spaces. In order to make the problem of classification more reasonable, it is necessary to identify a tractable category of spaces. A candidate for such a category should be large enough to contain all ‘important’ spaces (such as finite-dimensional manifolds) as well as contain the results of various constructions applied to these spaces (for example, it should be closed under suspension, loops, etc.). Furthermore, the classical homotopy functors, singular homology and cohomology and the homotopy groups, should be effective in distinguishing spaces in this category.

In this chapter, we present two categories of spaces that satisfy the desiderata of homotopy theory. In §4.1, CW-complexes are defined. The homotopy groups of CW-complexes are effective enough for a classification scheme, and the combinatorial structure allows the computation of their singular homology groups. In §4.2, the equivalent category of simplicial sets and mappings is presented. This category features a more rigid combinatorial structure in which the classical homotopy invariants can be defined and studied.