In mathematics, particularly category theory, a **representable functor** is a certain functor from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures (i.e. sets and functions) allowing one to utilize, as much as possible, knowledge about the category of sets in other settings.

- Definition
- Universal elements
- Examples
- Properties
- Uniqueness
- Preservation of limits
- Left adjoint
- Relation to universal morphisms and adjoints
- See also
- References

From another point of view, representable functors for a category *C* are the functors *given* with *C*. Their theory is a vast generalisation of upper sets in posets, and of Cayley's theorem in group theory.

Let **C** be a locally small category and let **Set** be the category of sets. For each object *A* of **C** let Hom(*A*,–) be the hom functor that maps object *X* to the set Hom(*A*,*X*).

A functor *F* : **C** → **Set** is said to be **representable** if it is naturally isomorphic to Hom(*A*,–) for some object *A* of **C**. A **representation** of *F* is a pair (*A*, Φ) where

- Φ : Hom(
*A*,–) →*F*

is a natural isomorphism.

A contravariant functor *G* from **C** to **Set** is the same thing as a functor *G* : **C**^{op} → **Set** and is commonly called a presheaf. A presheaf is representable when it is naturally isomorphic to the contravariant hom-functor Hom(–,*A*) for some object *A* of **C**.

According to Yoneda's lemma, natural transformations from Hom(*A*,–) to *F* are in one-to-one correspondence with the elements of *F*(*A*). Given a natural transformation Φ : Hom(*A*,–) → *F* the corresponding element *u* ∈ *F*(*A*) is given by

Conversely, given any element *u* ∈ *F*(*A*) we may define a natural transformation Φ : Hom(*A*,–) → *F* via

where *f* is an element of Hom(*A*,*X*). In order to get a representation of *F* we want to know when the natural transformation induced by *u* is an isomorphism. This leads to the following definition:

- A
**universal element**of a functor*F*:**C**→**Set**is a pair (*A*,*u*) consisting of an object*A*of**C**and an element*u*∈*F*(*A*) such that for every pair (*X*,*v*) with*v*∈*F*(*X*) there exists a unique morphism*f*:*A*→*X*such that (*Ff*)*u*=*v*.

A universal element may be viewed as a universal morphism from the one-point set {•} to the functor *F* or as an initial object in the category of elements of *F*.

The natural transformation induced by an element *u* ∈ *F*(*A*) is an isomorphism if and only if (*A*,*u*) is a universal element of *F*. We therefore conclude that representations of *F* are in one-to-one correspondence with universal elements of *F*. For this reason, it is common to refer to universal elements (*A*,*u*) as representations.

- Consider the contravariant functor
*P*:**Set**→**Set**which maps each set to its power set and each function to its inverse image map. To represent this functor we need a pair (*A*,*u*) where*A*is a set and*u*is a subset of*A*, i.e. an element of*P*(*A*), such that for all sets*X*, the hom-set Hom(*X*,*A*) is isomorphic to*P*(*X*) via Φ_{X}(*f*) = (*Pf*)*u*=*f*^{−1}(*u*). Take*A*= {0,1} and*u*= {1}. Given a subset*S*⊆*X*the corresponding function from*X*to*A*is the characteristic function of*S*. - Forgetful functors to
**Set**are very often representable. In particular, a forgetful functor is represented by (*A*,*u*) whenever*A*is a free object over a singleton set with generator*u*.- The forgetful functor
**Grp**→**Set**on the category of groups is represented by (**Z**, 1). - The forgetful functor
**Ring**→**Set**on the category of rings is represented by (**Z**[*x*],*x*), the polynomial ring in one variable with integer coefficients. - The forgetful functor
**Vect**→**Set**on the category of real vector spaces is represented by (**R**, 1). - The forgetful functor
**Top**→**Set**on the category of topological spaces is represented by any singleton topological space with its unique element.

- The forgetful functor
- A group
*G*can be considered a category (even a groupoid) with one object which we denote by •. A functor from*G*to**Set**then corresponds to a*G*-set. The unique hom-functor Hom(•,–) from*G*to**Set**corresponds to the canonical*G*-set*G*with the action of left multiplication. Standard arguments from group theory show that a functor from*G*to**Set**is representable if and only if the corresponding*G*-set is simply transitive (i.e. a*G*-torsor or heap). Choosing a representation amounts to choosing an identity for the heap. - Let
*C*be the category of CW-complexes with morphisms given by homotopy classes of continuous functions. For each natural number*n*there is a contravariant functor*H*^{n}:*C*→**Ab**which assigns each CW-complex its*n*^{th}cohomology group (with integer coefficients). Composing this with the forgetful functor we have a contravariant functor from*C*to**Set**. Brown's representability theorem in algebraic topology says that this functor is represented by a CW-complex*K*(**Z**,*n*) called an Eilenberg–MacLane space. - Let
*R*be a commutative ring with identity, and let**R**-**Mod**be the category of*R*-modules. If*M*and*N*are unitary modules over*R*, there is a covariant functor*B*:**R**-**Mod**→**Set**which assigns to each*R*-module*P*the set of*R*-bilinear maps*M*×*N*→*P*and to each*R*-module homomorphism*f*:*P*→*Q*the function*B*(*f*) :*B*(*P*) →*B*(*Q*) which sends each bilinear map*g*:*M*×*N*→*P*to the bilinear map*f*∘*g*:*M*×*N*→*Q*. The functor*B*is represented by the*R*-module*M*⊗_{R}*N*.^{ [1] }

Representations of functors are unique up to a unique isomorphism. That is, if (*A*_{1},Φ_{1}) and (*A*_{2},Φ_{2}) represent the same functor, then there exists a unique isomorphism φ : *A*_{1} → *A*_{2} such that

as natural isomorphisms from Hom(*A*_{2},–) to Hom(*A*_{1},–). This fact follows easily from Yoneda's lemma.

Stated in terms of universal elements: if (*A*_{1},*u*_{1}) and (*A*_{2},*u*_{2}) represent the same functor, then there exists a unique isomorphism φ : *A*_{1} → *A*_{2} such that

Representable functors are naturally isomorphic to Hom functors and therefore share their properties. In particular, (covariant) representable functors preserve all limits. It follows that any functor which fails to preserve some limit is not representable.

Contravariant representable functors take colimits to limits.

Any functor *K* : *C* → **Set** with a left adjoint *F* : **Set** → *C* is represented by (*FX*, η_{X}(•)) where *X* = {•} is a singleton set and η is the unit of the adjunction.

Conversely, if *K* is represented by a pair (*A*, *u*) and all small copowers of *A* exist in *C* then *K* has a left adjoint *F* which sends each set *I* to the *I*th copower of *A*.

Therefore, if *C* is a category with all small copowers, a functor *K* : *C* → **Set** is representable if and only if it has a left adjoint.

The categorical notions of universal morphisms and adjoint functors can both be expressed using representable functors.

Let *G* : *D* → *C* be a functor and let *X* be an object of *C*. Then (*A*,φ) is a universal morphism from *X* to *G* if and only if (*A*,φ) is a representation of the functor Hom_{C}(*X*,*G*–) from *D* to **Set**. It follows that *G* has a left-adjoint *F* if and only if Hom_{C}(*X*,*G*–) is representable for all *X* in *C*. The natural isomorphism Φ_{X} : Hom_{D}(*FX*,–) → Hom_{C}(*X*,*G*–) yields the adjointness; that is

is a bijection for all *X* and *Y*.

The dual statements are also true. Let *F* : *C* → *D* be a functor and let *Y* be an object of *D*. Then (*A*,φ) is a universal morphism from *F* to *Y* if and only if (*A*,φ) is a representation of the functor Hom_{D}(*F*–,*Y*) from *C* to **Set**. It follows that *F* has a right-adjoint *G* if and only if Hom_{D}(*F*–,*Y*) is representable for all *Y* in *D*.

In mathematics, specifically category theory, a **functor** is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied.

In mathematics, especially in category theory and homotopy theory, a **groupoid** generalises the notion of group in several equivalent ways. A groupoid can be seen as a:

In category theory, a branch of mathematics, a **Grothendieck topology** is a structure on a category *C* that makes the objects of *C* act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a **site**.

In mathematics, an **isomorphism** is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are **isomorphic** if an isomorphism exists between them. The word isomorphism is derived from the Ancient Greek: ἴσος *isos* "equal", and μορφή *morphe* "form" or "shape".

In category theory, a branch of mathematics, a **universal property** is an important property which is satisfied by a **universal morphism**. Universal morphisms can also be thought of more abstractly as initial or terminal objects of a comma category. Universal properties occur almost everywhere in mathematics, and hence the precise category theoretic concept helps point out similarities between different branches of mathematics, some of which may even seem unrelated.

In mathematics, the **Yoneda lemma** is arguably the most important result in category theory. It is an abstract result on functors of the type *morphisms into a fixed object*. It is a vast generalisation of Cayley's theorem from group theory. It allows the embedding of any locally small category into a category of functors defined on that category. It also clarifies how the embedded category, of representable functors and their natural transformations, relates to the other objects in the larger functor category. It is an important tool that underlies several modern developments in algebraic geometry and representation theory. It is named after Nobuo Yoneda.

In category theory, a branch of mathematics, the abstract notion of a **limit** captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a **colimit** generalizes constructions such as disjoint unions, direct sums, coproducts, pushouts and direct limits.

In category theory, a branch of mathematics, a **natural transformation** provides a way of transforming one functor into another while respecting the internal structure of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed, this intuition can be formalized to define so-called functor categories. Natural transformations are, after categories and functors, one of the most fundamental notions of category theory and consequently appear in the majority of its applications.

In mathematics, specifically category theory, **adjunction** is a relationship that two functors may have. Two functors that stand in this relationship are known as **adjoint functors**, one being the **left adjoint** and the other the **right adjoint**. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems, such as the construction of a free group on a set in algebra, or the construction of the Stone–Čech compactification of a topological space in topology.

In category theory, a branch of mathematics, an **initial object** of a category C is an object I in C such that for every object X in C, there exists precisely one morphism *I* → *X*.

In category theory, a **category is Cartesian closed** if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in mathematical logic and the theory of programming, in that their internal language is the simply typed lambda calculus. They are generalized by closed monoidal categories, whose internal language, linear type systems, are suitable for both quantum and classical computation.

In category theory, a branch of mathematics, a **monad** is an endofunctor, together with two natural transformations required to fulfill certain coherence conditions. Monads are used in the theory of pairs of adjoint functors, and they generalize closure operators on partially ordered sets to arbitrary categories.

In mathematics, the **Gelfand representation** in functional analysis has two related meanings:

In category theory, an abstract branch of mathematics, an **equivalence of categories** is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics. Establishing an equivalence involves demonstrating strong similarities between the mathematical structures concerned. In some cases, these structures may appear to be unrelated at a superficial or intuitive level, making the notion fairly powerful: it creates the opportunity to "translate" theorems between different kinds of mathematical structures, knowing that the essential meaning of those theorems is preserved under the translation.

**Fibred categories** are abstract entities in mathematics used to provide a general framework for descent theory. They formalise the various situations in geometry and algebra in which *inverse images* of objects such as vector bundles can be defined. As an example, for each topological space there is the category of vector bundles on the space, and for every continuous map from a topological space *X* to another topological space *Y* is associated the pullback functor taking bundles on *Y* to bundles on *X*. Fibred categories formalise the system consisting of these categories and inverse image functors. Similar setups appear in various guises in mathematics, in particular in algebraic geometry, which is the context in which fibred categories originally appeared. Fibered categories are used to define stacks, which are fibered categories with "descent". Fibrations also play an important role in categorical semantics of type theory, and in particular that of dependent type theories.

This is a glossary of properties and concepts in category theory in mathematics.

In mathematics, specifically in category theory, hom-sets, i.e. sets of morphisms between objects, give rise to important functors to the category of sets. These functors are called **hom-functors** and have numerous applications in category theory and other branches of mathematics.

In category theory, a branch of mathematics, **profunctors** are a generalization of relations and also of bimodules.

In mathematics, the **category of rings**, denoted by **Ring**, is the category whose objects are rings and whose morphisms are ring homomorphisms. Like many categories in mathematics, the category of rings is large, meaning that the class of all rings is proper.

In algebraic geometry, a **prestack***F* over a category *C* equipped with some Grothendieck topology is a category together with a functor *p*: *F* → *C* satisfying a certain lifting condition and such that locally isomorphic objects are isomorphic. A stack is a prestack with effective descents, meaning local objects may be patched together to become a global object.

- ↑ Hungerford, Thomas.
*Algebra*. Springer-Verlag. p. 470. ISBN 3-540-90518-9.

- Mac Lane, Saunders (1998).
*Categories for the Working Mathematician*. Graduate Texts in Mathematics**5**(2nd ed.). Springer. ISBN 0-387-98403-8.

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