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...hat every [[presheaf of sets]] is a [[colimit]] of [[representable functor|representable presheaves]] in a canonical way.<ref>{{harvnb|Mac Lane|1998|loc=Ch III, § 7 ...nition, a [[simplicial set]] is a presheaf on the simplex category Δ and a representable simplicial set is exactly of the form <math>\Delta^n = \operatorname{Hom}(- ...

...ries are '''presheaves with transfers'''. These are contravariant additive functors<blockquote><math>F:\text{Cor}_k \to \text{Ab}</math></blockquote>and their === Representable functors === ...

...llowed, desuspension would result in all [[cohomology]] [[functor]]s being representable. ...

[[Category:Representable functors]] ...

...=Anderson |first1=S.O. |last2=Power |first2=A.J. |date=April 1997 |title=A representable approach to finite nondeterminism |url=http://dx.doi.org/10.1016/s0304-3975 ...category|discrete categories]] of objects of <math>C, D</math> and the two functors are inclusions. In the case of groups seen as one-object categories, this i ...

...tion of the associated homology theory, this means that all cycles must be representable by small cycles. For instance, for [[singular homology]], the excision prop In this spirit, for certain homotopy-invariant functors which are not excisive, the corresponding excisive theory may be constructe ...

...|Rutten|1998}}</ref> That is a duality between covariant and contravariant representable [[Presheaf (category theory)|presheave]]s associated with an objects of cat ...al{A}</math>, taking <math>X \in \mathcal{A}</math> to the contravariant [[representable functor]]: {{R|Baez2022}}<ref name=nlab2>{{harv|Yoneda embedding in nlab}} ...

..._C</math> refers to the relative [[functor category]]; the objects are the functors from ''U'' to ''F'' over ''C'' and the morphisms are the base-preserving na It comes with the forgetful functors ''p'', ''q'' from <math>F \times_B G</math> to ''F'' and ''G''. ...

...h>\mathcal{C} \to \mathcal{D}</math> of groupoid fibrations is called a '''representable submersion''' if ...he [[fibred product]] <math>\mathcal{C} \times_{\mathcal{D}} F_U</math> is representable, i.e. it is isomorphic to <math>F_V</math> (for some manifold <math>V</math ...

...d more generally, derived stacks. Heuristically, derived schemes should be functors from some category of derived rings to the category of sets ...ected to be modelled by homotopy types). These derived stacks are suitable functors of the form ...

...system in the [[opposite category]] <math>C^{op}</math> or, equivalently, functors ...tics)|shape theory]]. Pro-objects also arise via their connection to [[pro-representable functor]]s, for example in [[Grothendieck's Galois theory]], and also in [[ ...

...ns. We let <math>\Delta^{op}Shv(Sm_S)_{Nis}</math> denote the category of functors <math>\Delta^{op} \to Shv(Sm_S)_{Nis}</math>. That is, <math>\Delta^{op}Sh ...th>. The collection <math>\{x^*\}</math> is a conservative family of fibre functors for <math>Shv(Sm_S)_{Nis}</math>. ...

...hail | author2-link=Mikhail Kapranov | title=Representable functors, Serre functors, and reconstructions | journal=Mathematics of the USSR-Izvestiya | volume=3 ...

On the category of pointed spaces, there are two important functors: the [[loop functor]] <math>\Omega</math> and the (reduced) [[suspension fu These functors are used to construct [[fiber sequence]]s and [[cofiber sequence]]s. Namely ...

...s <math>\mathcal{Quot}_{\mathcal{E}/X/S}^{\Phi,\mathcal{L}}</math> are all representable by projective schemes <math>\text{Quot}_{\mathcal{E}/X/S}^{\Phi}</math> ove ...

...ted towards the study of the families of varieties, that is towards moduli functors and moduli stacks. The main task is to understand what kind of objects form ...ia [[Yoneda's lemma]], as a way of identifying <math> S </math> with the [[representable functor]] <math> h_{S} </math> it sets up. Historically there was a process ...