Editing Category of topological spaces (section)

==Relationships to other categories==

*The category of [[pointed topological space]]s '''Top'''<sub>•</sub> is a [[coslice category]] over '''Top'''. 
* The [[homotopy category of topological spaces|homotopy category]] '''hTop''' has topological spaces for objects and [[homotopy equivalent|homotopy equivalence classes]] of continuous maps for morphisms. This is a [[quotient category]] of '''Top'''. One can likewise form the pointed homotopy category '''hTop'''<sub>•</sub>.
*'''Top''' contains the important category '''Haus''' of [[Hausdorff space|Hausdorff spaces]] as a [[full subcategory]]. The added structure of this subcategory allows for more epimorphisms: in fact, the epimorphisms in this subcategory are precisely those morphisms with [[dense set|dense]] [[image (mathematics)|images]] in their [[codomain]]s, so that epimorphisms need not be [[surjective]].
*'''Top''' contains the full subcategory '''CGHaus''' of [[compactly generated Hausdorff space]]s, which has the important property of being a [[Cartesian closed category]] while still containing all of the typical spaces of interest. This makes '''CGHaus''' a particularly ''convenient category of topological spaces'' that is often used in place of '''Top'''.
* The forgetful functor to '''Set''' has both a left and a right adjoint, as described above in the concrete category section.
* There is a functor to the category of [[Locale (mathematics)|locales]] '''Loc''' sending a topological space to its locale of open sets. This functor has a right adjoint that sends each locale to its topological space of points. This adjunction restricts to an equivalence between the category of [[sober space]]s and spatial locales.
*The [[homotopy hypothesis]] relates '''Top''' with '''∞Grpd''', the category of [[∞-groupoid|∞-groupoids]]. The conjecture states that ∞-groupoids are equivalent to topological spaces modulo [[Weak equivalence (homotopy theory)|weak homotopy equivalence]].