View source for Representable functor

A covariant (or contravariant) functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081340/r0813401.png" /> from some [[Category|category]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081340/r0813402.png" /> into the category of sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081340/r0813403.png" /> (cf. [[Sets, category of|Sets, category of]]) that is isomorphic to one of the functors

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or

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A functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081340/r0813406.png" /> is representable if and only if there is an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081340/r0813407.png" /> and an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081340/r0813408.png" /> such that for every element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081340/r0813409.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081340/r08134010.png" />, there is a unique morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081340/r08134011.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081340/r08134012.png" />. The object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081340/r08134013.png" /> is called a representing object for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081340/r08134014.png" />; it is unique up to isomorphism.

In the category of sets the identity functor is representable: a representing object is a singleton. The functor of taking a Cartesian power is also representable: a representing object is a set whose cardinality equals the given power. In an arbitrary category a product of representable functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081340/r08134015.png" /> with representing objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081340/r08134016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081340/r08134017.png" />, is representable if and only if the [[Coproduct|coproduct]] of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081340/r08134018.png" /> exists in the category. Every covariant representable functor commutes with limits, i.e. is continuous (cf. [[Continuous functor|Continuous functor]]).

A representable functor is an analogue of the concept of a  "free universal algebra with one generator" . For any functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081340/r08134019.png" /> and a representable functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081340/r08134020.png" /> the set of natural transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081340/r08134021.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081340/r08134022.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081340/r08134023.png" /> is a representing object for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081340/r08134024.png" />. This shows that representable functors are free objects in the category of functors.

In the case of additive categories one considers additive functors with values in the category of Abelian groups instead of functors with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081340/r08134025.png" />. Therefore, in this case one understands a representable functor to be an additive functor isomorphic to one of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081340/r08134026.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081340/r08134027.png" />.

The concept of a representable functor arose first in algebraic geometry (cf. [[#References|[2]]]). The most important examples of representable functors in this branch are Picard functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081340/r08134028.png" /> and Hilbert functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081340/r08134029.png" />, which are representable in the category of algebraic spaces (cf. [[#References|[1]]] and [[Algebraic space|Algebraic space]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081340/r08134030.png" /> be the field of fractions of a regular discretely-normed ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081340/r08134031.png" /> with perfect field of residues. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081340/r08134032.png" /> is a smooth geometrically non-degenerate singular curve of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081340/r08134033.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081340/r08134034.png" />, then its [[Minimal model|minimal model]] represents the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081340/r08134035.png" /> from the category of regular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081340/r08134036.png" />-schemes. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081340/r08134037.png" /> is an Abelian variety over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081340/r08134038.png" />, then its minimal [[Néron model|Néron model]] is a smooth group scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081340/r08134039.png" />, representing the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081340/r08134040.png" /> from the category of smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081340/r08134041.png" />-schemes.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Artin,   "Algebraic spaces" , Yale Univ. Press  (1971)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Grothendieck,   J. Dieudonné,   "Eléments de géometrie algébrique" , '''I. Le langage des schémes''' , Springer  (1971)</TD></TR></table>



====Comments====
Representable functors occur in many branches of mathematics besides algebraic geometry. S. MacLane [[#References|[a1]]] traces their first appearance to work of J.-P. Serre in algebraic topology, around 1953. The theorem (above) characterizing natural transformations from a representable functor to an arbitrary functor is commonly called the Yoneda lemma. If a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081340/r08134042.png" /> has arbitrary coproducts, then a functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081340/r08134043.png" /> is representable if and only if it has a left adjoint (cf. [[Adjoint functor|Adjoint functor]]).

Note that all above-mentioned functors in algebraic geometry are contravariant.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. MacLane,   "Categories for the working mathematician" , Springer  (1971)  pp. Chapt. IV, Sect. 6; Chapt. VII, Sect. 7</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Grothendieck,   J. Dieudonné,   "Eléments de géometrie algebriques III"  ''Publ. Math. IHES'' , '''11'''  (1961)  pp. 349–356</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Grothendieck,   "Fondements de la géométrie algébrique"  ''Sém. Bourbaki'' , '''195; 221; 232'''  (1960–1962)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Artin,   "Algebraization of formal moduli, I"  D.C. Spencer (ed.)  S. Iyanaga (ed.) , ''Global analysis (papers in honor of K. Kodaira)'' , Princeton Univ. Press  (1969)  pp. 21–72</TD></TR></table>