# nLab Top

### Context

#### Topology

topology

algebraic topology

# Contents

## Definition

Top is the category of topological spaces and continuous maps between them.

How exactly this is understood depends a bit on context: of course $Top$ forms an ordinary category. But it is also naturally an (∞,1)-category. This, in turn, may be presented by regarding $Top$ as a model category equipped with the Quillen model structure.

Moreover, what exactly counts as an object in $Top$ often varies in different contexts. For many applications it is useful to restrict to a subcategory of nice topological spaces such as compactly generated spaces or CW-complexes. There other other convenient categories of topological spaces.

The homotopy category of $Top$ with respect to weak homotopy equivalences is Ho(Top). This is the central object of study in homotopy theory. Regarded as an (∞,1)-category $Top$ is the archetypical homotopy theory, equivalent to ∞Grpd.

## References

An axiomatic desciption of $Top$ building along the lines of ETCS for Set is discussed in

• Dana Schlomiuk, An elementary theory of the category of topological space , Transactions of the AMS, volume 149 (1970)

category: category

Revised on June 13, 2013 17:27:21 by Urs Schreiber (82.169.65.155)