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.

Definition and examples

The homotopy category hTop of topological spaces is the category whose objects are topological spaces. Instead of taking continuous functions as morphisms between two such spaces, the morphisms in hTop between two spaces X and Y are given by the equivalence classes of all continuous functions X → Y with respect to the relation of homotopy. That is to say, two continuous functions are considered the same morphism in hTop if they can be deformed into one another via a (continuous) homotopy. The set of morphisms between spaces X and Y in a homotopy category is commonly denoted [X,Y] rather than Hom(X,Y).

The composition

[X, Y] × [Y, Z] → [X, Z]

is defined by

[f] o [g] = [f o g].

This is well-defined since the homotopy relation is compatible with function composition. That is, if f₁, g₁ : X → Y are homotopic and f₂, g₂ : Y → Z are homotopic then their compositions f₂ o f₁, g₂ o g₁ 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 → Set

to 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.

Pointed version

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. The pointed homotopy category hTop_• is defined to be the category whose objects are pointed topological spaces and whose morphisms are equivalence classes of pointed maps (i.e., sending the distinguished base point to the base point) modulo pointed homotopy (i.e., the homotopy fixes the base points, as well). 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 equivalences

A continuous map f : X → Y is called a homotopy equivalence, if there is another continuous map g : Y → X such that the two compositions f o g and g o f are homotopic to the respective identity maps. Equivalently, the classes of [f o g] and [g o f] agree with the ones of the identity map of Y and X, respectively. Yet in other words, f is a homotopy equivalence if and only if [f] is an isomorphism in the homotopy category. That is, two topological spaces are isomorphic in hTop if and only if they are homotopy equivalent (i.e. have the same homotopy type).

Homotopy groups

Given the n-sphere Sⁿ, the set

[Sⁿ, X]

of homotopy classes of maps from Sⁿ to some topological space X is the same as the n-th homotopy group π_n(X) (for n ≥ 1, the set of connected path-components for n = 0).

Even immediate examples, such as the homotopy groups of spheres,

π_n(S^m) = [Sⁿ, S^m]

are hard to compute.

Eilenberg-MacLane spaces

Given an abelian group G and n ≥ 0, the Eilenberg-MacLane space K(G,n) is a topological space satisfying, for any CW-complex X,

[X, K(G,n)] = Hⁿ(X; G),

where the right hand side denotes the nth singular cohomology group of X with coefficients in G. In this sense, singular cohomology is representable by the representing space K(G,n). The Brown representability theorem is concerned with the representability of more general functors

hTop^op → Set.

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.

Generalizations

The above definition of the homotopy of topological spaces is a special case of the more general construction of the homotopy category of a model category. Roughly speaking, a model category is a category C with three distinguished types of morphisms called fibrations, cofibrations and weak equivalences. Localizing C with respect to the weak equivalences yields the homotopy category.

This construction, applied to the model category of topological spaces, gives back the homotopy category outlined above. Applied to the model category of chain complexes over some commutative ring R, for example, yields the derived category of R-modules. The homotopy category of chain complexes can also be interpreted along these lines.