Homotopy Category - Definition and Examples

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 rather than Hom(X,Y).

The composition

× →

is defined by

o = .

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.