Homotopy Category - Homotopy Theory

Homotopy Theory

Many of the elementary results in homotopy theory can be formulated for arbitrary topological spaces, but as one goes deeper into the theory it is often necessary to work with a more restrictive category of spaces. For most purposes, the homotopy category of CW complexes is the appropriate choice. In the opinion of some experts the homotopy category of CW complexes is the best, if not the only, candidate for the homotopy category. One basic result is that the representable functors on the homotopy category of CW complexes have a simple characterization (the Brown representability theorem).

The category of CW complexes is deficient in the sense that the space of maps between two CW complexes is not always a CW complex. A more well-behaved category commonly used in homotopy theory is the category of compactly generated Hausdorff spaces (also called k-spaces). This category includes all CW complexes, locally compact spaces, and first-countable spaces (such as metric spaces).

One important later development was that of spectra in homotopy theory, essentially the derived category idea in a form useful for topologists. Spectra have also been defined in various cases using the model category approach, generalizing the topological case. Many theorists interested in the classical topological theory consider this more axiomatic approach less useful for their purposes. Finding good replacements for CW complexes in the purely algebraic case is a subject of current research.