Homotopy category

In mathematics, a homotopy category is a category whose objects are topological spaces and whose morphisms are homotopy classes of continuous functions. The homotopy category of all topological spaces is often denoted hTop or Toph.

Homotopy categories are well-defined since the homotopy relation is compatible with function composition. That is, if "f"₁, "g"₁ : "X" → "Y" are homotopic and "f"₂, "g"₂ : "Y" → "Z" are homotopic then their compositions "f"₂ o "f"₁, "g"₂ o "g"₁ "X" → "Z" are homotopic as well.

While the objects of a homotopy category are sets (with additional structure), the morphisms are not actual functions between them, but rather classes of functions. Indeed, hTop is an example of a category that is not concretizable, meaning there does not exist a faithful forgetful functor:"U" : hTop → Setto the category of sets. Homotopy categories are examples of quotient categories. The category hTop is a quotient of Top, the ordinary category of topological spaces.

The set of morphisms between spaces "X" and "Y" in a homotopy category is commonly denoted ["X","Y"] rather than Hom("X","Y"). To a large extent the business of homotopy theory is to describe the homotopy category hTop; in fact it turns out that calculating ["X","Y"] is hard as a general problem, and much effort has been put into the most interesting cases, for example where "X" and "Y" are spheres (c.f. the homotopy groups of spheres).

Isomorphisms in a homotopy category are just homotopy equivalences of spaces. That is, two topological spaces are isomorphic in hTop if and only if they are homotopy equivalent (i.e. have the same homotopy type).

Basepoints

For the purposes of homotopy theory it is usually necessary to keep track of basepoints in each space: for example the fundamental group of topological space is, properly speaking, dependent on the basepoint chosen. A topological space with a distinguished basepoint is called a pointed space.

A pointed homotopy category is then a category of pointed spaces whose morphisms are homotopy classes of pointed maps (a homotopy of pointed maps is a homotopy relative to the basepoint). The pointed homotopy category of all pointed spaces is denoted hTop_•. The set of maps between pointed spaces "X" and "Y" in hTop_• is commonly denoted ["X","Y"] _•.

The need to use basepoints has a significant effect on the products (and other limits) appropriate to use. For example, in homotopy theory, the "smash product" "X" ∧ "Y" of spaces "X" and "Y" is used.

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.

See also

Homotopy category of chain complexes