In mathematics, the homotopy category of topological spaces, often denoted hTop or Toph, is the category whose objects are topological spaces and whose morphisms are homotopy equivalence classes of continuous maps. This is a category because the homotopy relation is compatible with function composition in the following sense: if

f₁, g₁ : X → Y

are homotopic, and

f₂, g₂ : Y → Z

are homotopic, then their compositions

f₂ o f₁

and

g₂ o g₁ : X → Z

are homotopic as well.

hTop is not concrete

While the objects of hTop are sets (with additional structure), the morphisms are not actual functions between them, but rather classes of functions. In this way, hTop is a quotient category of Top.

hTop is an example of a category that is not concretizable. This means that there does not exist a faithful forgetful functor

U : hTop → Set

Limits and colimits

Examples of limits and colimits in hTop include:

• The empty set (considered as a topological space) is the initial object of hTop; any singleton topological space is a terminal object. There are thus no zero objects in hTop.

Category-theoretic categories | Topology

Categoría de espacios topológicos