nLab
Ho(Top)

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Definition

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 Ho(sSet Quillen)Ho(sSet_{Quillen}) of the standard model structure on simplicial sets.

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.

category: category

Revised on November 26, 2012 00:36:45 by Urs Schreiber (89.204.153.181)