Representable functor

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, particularly <a href="/facts/Category_theory/6zS3DXPa">category theory</a>, a representable functor is a certain <a href="/facts/Functor/l2u3rIBE">functor</a> from an arbitrary <a href="/facts/Category_(mathematics)/7WzxUThf">category</a> into the <a href="/facts/Category_of_sets/LF5ykYl0">category of sets</a>. Such functors give representations of an abstract category in terms of known structures (i.e. <a href="/facts/Set_(mathematics)/BfucfHMq">sets</a> and <a href="/facts/Function_(mathematics)/CdReEKCh">functions</a>) allowing one to utilize, as much as possible, knowledge about the category of sets in other settings.
From another point of view, representable functors for a category C are the functors given with C. Their theory is a vast generalisation of <a href="/facts/Upper_set/hSyGJ2WB">upper sets</a> in <a href="/facts/Poset/qb4a3FAX">posets</a>, and <a href="/facts/Yoneda_lemma/j04vXd6x">Yoneda's representability</a> theorem generalizes <a href="/facts/Cayley%27s_theorem/a0M460K8">Cayley's theorem</a> in <a href="/facts/Group_theory/NfowcI5H">group theory</a>.

Representable functor open-in-new

Representable functor