Editing Adjoint functors (section)

====Topology====
* '''A functor with a left and a right adjoint.''' Let ''G'' be the functor from [[topological space]]s to [[Set (mathematics)|sets]] that associates to every topological space its underlying set (forgetting the topology, that is). ''G'' has a left adjoint ''F'', creating the [[discrete space]] on a set ''Y'', and a right adjoint ''H'' creating the [[trivial topology]] on ''Y''.
* '''Suspensions and loop spaces.'''  Given [[topological spaces]] ''X'' and ''Y'', the space [''SX'', ''Y''] of [[homotopy classes]] of maps from the [[suspension (topology)|suspension]] ''SX'' of ''X'' to ''Y'' is naturally isomorphic to the space [''X'', Ω''Y''] of homotopy classes of maps from ''X'' to the [[loop space]] Ω''Y'' of ''Y''.  The suspension functor is therefore left adjoint to the loop space functor in the [[homotopy category]], an important fact in [[homotopy theory]].
* '''Stone–Čech compactification.''' Let '''KHaus''' be the category of [[compact space|compact]] [[Hausdorff space]]s and ''G'' : '''KHaus''' → '''Top''' be the inclusion functor to the category of [[topological spaces]]. Then ''G'' has a left adjoint ''F'' : '''Top''' → '''KHaus''', the [[Stone–Čech compactification]]. The unit of this adjoint pair yields a [[continuous function (topology)|continuous]] map from every topological space ''X'' into its Stone–Čech compactification. 
* '''Direct and inverse images of sheaves.''' Every [[continuous map]] ''f'' : ''X'' → ''Y'' between [[topological space]]s induces a functor ''f''<sub>&thinsp;∗</sub> from the category of [[sheaf (mathematics)|sheaves]] (of sets, or abelian groups, or rings...) on ''X'' to the corresponding category of sheaves on ''Y'', the ''[[direct image functor]]''. It also induces a functor ''f''{{i sup|−1}} from the category of sheaves of abelian groups on ''Y'' to the category of sheaves of abelian groups on ''X'', the ''[[inverse image functor]]''. ''f''{{i sup|−1}} is left adjoint to ''f''<sub>&thinsp;∗</sub>. Here a more subtle point is that the left adjoint for [[coherent sheaf|coherent sheaves]] will differ from that for sheaves (of sets).
* '''Soberification.''' The article on [[Stone duality]] describes an adjunction between the category of topological spaces and the category of [[sober space]]s that is known as soberification. Notably, the article also contains a detailed description of another adjunction that prepares the way for the famous [[duality (category theory)|duality]] of sober spaces and spatial locales, exploited in [[pointless topology]].