# Contents

## Definition

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

There is a functor

$Top \to Ho(CW)$

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

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

## Shape theory

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

category: category

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