Contents

Definition

By $\mathrm{Ho}\left(\mathrm{Top}\right)$ one denotes the category which is the homotopy category of Top with respect to weak equivalences given

• either by homotopy equivalences – $\mathrm{Ho}\left(\mathrm{Top}{\right)}_{\mathrm{he}}$.

• or by weak homotopy equivalences – $\mathrm{Ho}\left(\mathrm{Top}{\right)}_{\mathrm{whe}}$.

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

The study of $\mathrm{Ho}\left(\mathrm{Top}\right)$ was the motivating example of homotopy theory. Often $\mathrm{Ho}\left(\mathrm{Top}\right)$ 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 $\mathrm{Top}$ denote concretely the category of compactly generated weakly Hausdorff spaces. And Let $\mathrm{CW}$ be the subcategory on CW-complexes. We have $\mathrm{Ho}\left(\mathrm{CW}{\right)}_{\mathrm{whe}}=\mathrm{Ho}\left(\mathrm{CW}{\right)}_{\mathrm{he}}=\mathrm{Ho}\left(\mathrm{CW}\right)$.

There is a functor

$\mathrm{Top}\to \mathrm{Ho}\left(\mathrm{CW}\right)$Top \to Ho(CW)

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

By the homotopy hypothesis-theorem $\mathrm{Ho}\left(\mathrm{CW}\right)$ is equivalent for instance to the homotopy category $\mathrm{Ho}\left({\mathrm{sSet}}_{\mathrm{Quillen}}\right)$ of the standard model structure on simplicial sets.

Shape theory

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

category: category

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