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By Top is usually denoted the collection 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.

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.

category: category

Revised on March 17, 2010 20:00:28 by Urs Schreiber (131.211.234.61)
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