How exactly this is understood depends a bit on context: of course forms an ordinary category. But it is also naturally an (∞,1)-category. This, in turn, may be presented by regarding as a model category equipped with the Quillen model structure.
Moreover, what exactly counts as an object in 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 with respect to weak homotopy equivalences is Ho(Top). This is the central object of study in homotopy theory. Regarded as an (∞,1)-category is the archetypical homotopy theory, equivalent to ∞Grpd.