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

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

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


An axiomatic desciption of TopTop 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 (