Ho(Top) (changes) in nLab

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Idea

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.

Definition

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

• either by homotopy equivalences – Ho(Top) heHo(Top)_{he}.

• or by weak homotopy equivalences – Ho(Top) wheHo(Top)_{whe}.

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

Top→Ho(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.

• hypothèse inspiratrice

• Top, Ho(Top)

• sSet, Ho(sSet)

• ∞Grpd, Ho(∞Grpd)

• stable homotopy category

References

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

• Marino Gran , andEnrico Vitale, On the exact completion of the homotopy category , Cahiers de Topologie et Géométrie Différentielle Catégoriques, Catégoriques Volume 39 (1998) no. 4, pp. 287-297. [[numdam](http://www.numdam.org/item/CTGDC_1998__39_4_287_0/)]39 4 (1998) 287-297 [[numdam:CTGDC_1998__39_4_287_0](http://www.numdam.org/item/CTGDC_1998__39_4_287_0)]