nLab
Top

Regarded as a plain category, $\mathrm{Top}$ is the category whose objects are topological spaces and whose morphisms are continuous maps.

The homotopy category of Top with respect to weak homotopy equivalences is Ho(Top). This is the central object of study in homotopy theory. Notice that often one may want to use instead a category of nice topological spaces such as CW-complexes or a nice category of spaces.

More generally, $\mathrm{Top}$ may denote the archetypical (∞,1)-category that is the archetypical (∞,1)-topos. As such, $\mathrm{Top}$ is also the archetypical homotopy theory.

In this incarnation ${\mathrm{Top}}_{\mathrm{nice}}$ is equivalent to Simp Set (in terms of the standard model structure on simplicial sets) and to ∞Grpd.

category: category