Representable functor

A covariant (or contravariant) functor $ F $ from some category $ \mathfrak R $ into the category of sets $ \mathfrak S $ (cf. Sets, category of) that is isomorphic to one of the functors

$$ H ( A, -) : \mathfrak R \rightarrow \mathfrak S ,\ \ X \mapsto H( A, X), $$

or

$$ H(-, A): \mathfrak R \rightarrow \mathfrak S ,\ \ X \mapsto H( X, A) . $$

A functor $ F: \mathfrak R \rightarrow \mathfrak S $ is representable if and only if there is an object $ A \in \mathop{\rm Ob} \mathfrak R $ and an element $ a \in F ( A) $ such that for every element $ x \in F ( X) $, $ X \in \mathop{\rm Ob} \mathfrak R $, there is a unique morphism $ \alpha : A \rightarrow X $ for which $ x = F ( \alpha )( a) $. The object $ A $ is called a representing object for $ F $; 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 $ F _ {i} $ with representing objects $ A _ {i} $, $ i \in I $, is representable if and only if the coproduct of the $ A _ {i} $ exists in the category. Every covariant representable functor commutes with limits, i.e. is continuous (cf. Continuous functor).

A representable functor is an analogue of the concept of a "free universal algebra with one generator" . For any functor $ G: \mathfrak R \rightarrow \mathfrak S $ and a representable functor $ F $ the set of natural transformations $ \mathop{\rm Nat} ( F, G) $ is isomorphic to $ G ( A) $, where $ A $ is a representing object for $ F $. 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 $ \mathfrak S $. Therefore, in this case one understands a representable functor to be an additive functor isomorphic to one of the form $ H( A, -) $ or $ H(-, A) $.

The concept of a representable functor arose first in algebraic geometry (cf. [2]). The most important examples of representable functors in this branch are Picard functors $ \mathop{\rm Pic} X/S $ and Hilbert functors $ \mathop{\rm Hilb} X/S $, which are representable in the category of algebraic spaces (cf. [1] and Algebraic space). Let $ K $ be the field of fractions of a regular discretely-normed ring $ O $ with perfect field of residues. If $ X _ {O} $ is a smooth geometrically non-degenerate singular curve of genus $ g > 0 $ over $ K $, then its minimal model represents the functor $ Y \mapsto \mathop{\rm Isom} _ {K} ( Y \otimes _ {O} K, X _ {O} ) $ from the category of regular $ O $-schemes. If $ A $ is an Abelian variety over $ K $, then its minimal Néron model is a smooth group scheme $ X \rightarrow \mathop{\rm Spec} O $, representing the functor $ Y \mapsto \mathop{\rm Hom} _ {K} ( Y \otimes _ {O} K, A) $ from the category of smooth $ O $-schemes.

References

[1]	M. Artin, "Algebraic spaces" , Yale Univ. Press (1971) MR0427316 MR0407012 Zbl 0232.14003 Zbl 0226.14001 Zbl 0216.05501
[2]	A. Grothendieck, J. Dieudonné, "Eléments de géométrie algébrique" , I. Le langage des schémas , Springer (1971) MR0217085 Zbl 0203.23301

Comments

Representable functors occur in many branches of mathematics besides algebraic geometry. S. MacLane [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 $ \mathfrak R $ has arbitrary coproducts, then a functor $ \mathfrak R \rightarrow \mathfrak S $ is representable if and only if it has a left adjoint (cf. Adjoint functor).

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

References

[a1]	S. MacLane, "Categories for the working mathematician" , Springer (1971) pp. Chapt. IV, Sect. 6; Chapt. VII, Sect. 7 Zbl 0232.18001
[a2]	A. Grothendieck, J. Dieudonné, "Eléments de géométrie algébrique III" Publ. Math. IHES , 11 (1961) pp. 349–356 MR0217085 MR0163910
[a3]	A. Grothendieck, "Fondements de la géométrie algébrique" Sém. Bourbaki , 195; 221; 232 (1960–1962) MR1611235 MR1086880 MR0146040 Zbl 0239.14002 Zbl 0239.14001
[a4]	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 MR0260746 Zbl 0205.50402