nLab Ho(Top)



Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts




The “classical homotopy category” Ho(Top)Ho(Top) typically refers to the category of topological spaces with morphisms between them the homotopy classes of continuous functions, or (slightly less classically but more commonly these days) to its full subcategory on those topological spaces homeomorphic to a CW-complex. The latter is technically the homotopy category obtained by localizing the category of topological spaces at those continuous functions that are weak homotopy equivalences, hence it is also the homotopy category of a model category of the classical model structure on topological spaces.


By Ho(Top)Ho(Top) one denotes the category which is the homotopy category of Top with respect to weak equivalences given

Depending on context here Top contains all topological spaces or is some subcategory of nice topological spaces.

The study of Ho(Top)Ho(Top) was the motivating example of homotopy theory. Often Ho(Top)Ho(Top) is called the homotopy category.

The simplicial localization of Top at the weak homotopy equivalences yields the (∞,1)-category of ∞-groupoids/homotopy types.

Compactly generated spaces

Let now TopTop denote concretely the category of compactly generated weakly Hausdorff spaces. And Let CWCW be the subcategory on CW-complexes. We have Ho(CW) whe=Ho(CW) he=Ho(CW)Ho(CW)_{whe} = Ho(CW)_{he} = Ho(CW).

There is a functor

TopHo(CW) Top \to Ho(CW)

that sends each topological space to a weakly homotopy equivalent CW-complex.

By the homotopy hypothesis-theorem Ho(CW)Ho(CW) is equivalent for instance to the homotopy category of a model category Ho(sSet Quillen)Ho(sSet_{Quillen}) of the classical model structure on simplicial sets as well as Ho(Top Quillen)Ho(Top_{Quillen})of the classical model structure on topological spaces.

Shape theory

The category Ho(Top) heHo(Top)_{he} can be studied by testing its objects with objects from Ho(CW)Ho(CW). This is the topic of shape theory.

Related concepts


The exact completion of the category Ho(Top)Ho(Top) is shown to be a pretopos in:

category: category

Last revised on October 13, 2023 at 14:08:41. See the history of this page for a list of all contributions to it.