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...subfunctors, then, under certain conditions, it can be shown that ''F'' is representable. This is a useful technique for the construction of ringed spaces. It was [[Category:Functors]] ...

...s implicit in the use of [[Ext functor]]s, the [[derived functor]]s of Hom-functors; if there is a bicommutant aspect, taken over the Steenrod algebra acting, is a [[natural transformation]] of functors ...

...ry [[category (mathematics)|category]] into the [[category of sets]]. Such functors give representations of an abstract category in terms of known structures ( ...another point of view, representable functors for a category ''C'' are the functors ''given'' with ''C''. Their theory is a vast generalisation of [[upper set] ...

...neralized cohomology theories]]) are examples of contravariant half-exact functors. If ''B'' is any fibrant topological space, the (representable) functor ''F(X)=[X,B]'' is half-exact. ...

...[[functor]]s to the [[category of sets]]. These functors are called '''hom-functors''' and have numerous applications in category theory and other branches of For all objects ''A'' and ''B'' in ''C'' we define two functors to the [[category of sets]] as follows: ...

...]]s are [[natural transformation]]s <math>\eta: F \to G</math> between the functors (here, <math>G: C \to D</math> is another object in the category). Functor ...>), <math>[C,D]</math>, or <math>D ^C</math>, has as objects the covariant functors from <math>C</math> to <math>D</math>, ...

...d insight is that over a certain [[category theory|category]], these are [[representable functor]]s. Furthermore, their representatives are related to the algebras ...

==Functors== *[[Representable functor]] ...

...nnected [[CW complex]]es, to the [[category of sets]] '''Set''', to be a [[representable functor]]. ...e then sufficient. For technical reasons, the theorem is often stated for functors to the category of [[pointed set]]s; in other words the sets are also given ...

...resentable functor]] question, then apply a criterion that singles out the representable [[functor]]s for schemes. When this programmatic approach works, the result [[Category:Representable functors]] ...

{{otheruses4|exact functors in homological algebra|exact functors between regular categories|regular category}} ...objects. Much of the work in homological algebra is designed to cope with functors that ''fail'' to be exact, but in ways that can still be controlled. ...

...defined on that category. It also clarifies how the embedded category of [[representable functor]]s and their [[natural transformation]]s relates to the other objec ...category <math> \mathcal{C} </math>, one should study the category of all functors of <math> \mathcal{C} </math> into <math> \mathbf{Set} </math> (the [[categ ...

A [[morphism]] of presheaves is defined to be a [[natural transformation]] of functors. This makes the collection of all presheaves on <math>C</math> into a cate ...or some [[object (category theory)|object]] ''A'' of '''C''' is called a [[representable presheaf]]. ...

...e [[presheaf (category theory)|presheaves]] over '''C''') of retracts of [[representable functor]]s. The category of presheaves on '''C''' is equivalent to the cate *The Karoubi envelope construction takes semi-adjunctions to [[adjoint functors|adjunction]]s.<ref>{{cite journal ...

...are used throughout modern mathematics to relate various categories. Thus, functors are important in every area of mathematics where [[category theory]] is app That is, functors must preserve [[Morphism#Definition|identity morphisms]] and [[Function com ...

...th>C</math> form a full subcategory <math>Ind(C)</math> in the category of functors (presheaves) <math>C^{op}\to Set</math>. The category <math>Pro(C)=Ind(C^{o ...

...he notion of a natural transformation states that a particular map between functors can be done consistently over an entire category. ...ry|functor categories]]. Natural transformations are, after categories and functors, one of the most fundamental notions of [[category theory]] and consequentl ...

...lated notions of [[universal property|universal properties]] and [[adjoint functors]], exist at a high level of abstraction. In order to understand them, it is ...the limit and the colimit functors are [[covariant functor|''covariant'']] functors. ...

=== Representable functor === ...in it, take <math>F = \operatorname{Hom}(-, *)</math>, the contravariant [[representable functor|functor represented by]] ''*''. Then the category <math>C_F</math> ...

...,t,m)</math> forms a groupoid (where <math>R,U</math> are their associated functors). Moreover, this construction is functorial on <math>(\mathrm{Sch}/S)</math ...hcal{X} \to \mathcal{X}\times_S\mathcal{X}</math> of fibered categories is representable as algebraic spaces ...