# Functor category

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In category theory, a branch of mathematics, a functor category${\displaystyle D^{C}}$ is a category where the objects are the functors ${\displaystyle F:C\to D}$ and the morphisms are natural transformations ${\displaystyle \eta :F\to G}$ between the functors (here, ${\displaystyle G:C\to D}$ is another object in the category). Functor categories are of interest for two main reasons:

## Contents

• many commonly occurring categories are (disguised) functor categories, so any statement proved for general functor categories is widely applicable;
• every category embeds in a functor category (via the Yoneda embedding); the functor category often has nicer properties than the original category, allowing certain operations that were not available in the original setting.

## Definition

Suppose ${\displaystyle C}$ is a small category (i.e. the objects and morphisms form a set rather than a proper class) and ${\displaystyle D}$ is an arbitrary category. The category of functors from ${\displaystyle C}$ to ${\displaystyle D}$, written as Fun(${\displaystyle C}$, ${\displaystyle D}$), Funct(${\displaystyle C}$,${\displaystyle D}$), ${\displaystyle [C,D]}$, or ${\displaystyle D^{C}}$, has as objects the covariant functors from ${\displaystyle C}$ to ${\displaystyle D}$, and as morphisms the natural transformations between such functors. Note that natural transformations can be composed: if ${\displaystyle \mu (X):F(X)\to G(X)}$ is a natural transformation from the functor ${\displaystyle F:C\to D}$ to the functor ${\displaystyle G:C\to D}$, and ${\displaystyle \eta (X):G(X)\to H(X)}$ is a natural transformation from the functor ${\displaystyle G}$ to the functor ${\displaystyle H}$, then the collection ${\displaystyle \eta (X)\mu (X):F(X)\to H(X)}$ defines a natural transformation from ${\displaystyle F}$ to ${\displaystyle H}$. With this composition of natural transformations (known as vertical composition, see natural transformation), ${\displaystyle D^{C}}$ satisfies the axioms of a category.

In a completely analogous way, one can also consider the category of all contravariant functors from ${\displaystyle C}$ to ${\displaystyle D}$; we write this as Funct(${\displaystyle C^{\text{op}},D}$).

If ${\displaystyle C}$ and ${\displaystyle D}$ are both preadditive categories (i.e. their morphism sets are abelian groups and the composition of morphisms is bilinear), then we can consider the category of all additive functors from ${\displaystyle C}$ to ${\displaystyle D}$, denoted by Add(${\displaystyle C}$,${\displaystyle D}$).

## Examples

• If ${\displaystyle I}$ is a small discrete category (i.e. its only morphisms are the identity morphisms), then a functor from ${\displaystyle I}$ to ${\displaystyle C}$ essentially consists of a family of objects of ${\displaystyle C}$, indexed by ${\displaystyle I}$; the functor category ${\displaystyle C^{I}}$ can be identified with the corresponding product category: its elements are families of objects in ${\displaystyle C}$ and its morphisms are families of morphisms in ${\displaystyle C}$.
• An arrow category ${\displaystyle {\mathcal {C}}^{\rightarrow }}$ (whose objects are the morphisms of ${\displaystyle {\mathcal {C}}}$, and whose morphisms are commuting squares in ${\displaystyle {\mathcal {C}}}$) is just ${\displaystyle {\mathcal {C}}^{\mathbf {2} }}$, where 2 is the category with two objects and their identity morphisms as well as an arrow from one object to the other (but not another arrow back the other way).
• A directed graph consists of a set of arrows and a set of vertices, and two functions from the arrow set to the vertex set, specifying each arrow's start and end vertex. The category of all directed graphs is thus nothing but the functor category ${\displaystyle {\textbf {Set}}^{C}}$, where ${\displaystyle C}$ is the category with two objects connected by two parallel morphisms (source and target), and Set denotes the category of sets.
• Any group ${\displaystyle G}$ can be considered as a one-object category in which every morphism is invertible. The category of all ${\displaystyle G}$-sets is the same as the functor category Set ${\displaystyle G}$.
• Similar to the previous example, the category of ${\displaystyle k}$-linear representations of the group ${\displaystyle G}$ is the same as the functor category k-Vect${\displaystyle G}$ (where k-Vect denotes the category of all vector spaces over the field ${\displaystyle k}$).
• Any ring ${\displaystyle R}$ can be considered as a one-object preadditive category; the category of left modules over ${\displaystyle R}$ is the same as the additive functor category Add(${\displaystyle R}$,${\displaystyle {\textbf {Ab}}}$) (where ${\displaystyle {\textbf {Ab}}}$ denotes the category of abelian groups), and the category of right ${\displaystyle R}$-modules is Add(${\displaystyle R^{\textbf {Ab}}}$). Because of this example, for any preadditive category ${\displaystyle C}$, the category Add(${\displaystyle C}$,${\displaystyle {\textbf {Ab}}}$) is sometimes called the "category of left modules over ${\displaystyle C'}$ and Add(${\displaystyle C^{\text{op}}}$,${\displaystyle {\textbf {Ab}}}$) is the category of right modules over ${\displaystyle C}$.
• The category of presheaves on a topological space ${\displaystyle X}$ is a functor category: we turn the topological space into a category ${\displaystyle C}$ having the open sets in ${\displaystyle X}$as objects and a single morphism from ${\displaystyle U}$ to ${\displaystyle V}$ if and only if ${\displaystyle U}$ is contained in ${\displaystyle V}$. The category of presheaves of sets (abelian groups, rings) on ${\displaystyle X}$ is then the same as the category of contravariant functors from ${\displaystyle C}$ to ${\displaystyle {\textbf {Set}}}$ (or ${\displaystyle {\textbf {Ab}}}$ or ${\displaystyle {\textbf {Ring}}}$). Because of this example, the category Funct(${\displaystyle C^{\text{op}}}$, ${\displaystyle {\textbf {Set}}}$) is sometimes called the "category of presheaves of sets on ${\displaystyle C'}$ even for general categories ${\displaystyle C}$ not arising from a topological space. To define sheaves on a general category ${\displaystyle C}$, one needs more structure: a Grothendieck topology on ${\displaystyle C}$. (Some authors refer to categories that are equivalent to ${\displaystyle {\textbf {Set}}^{C}}$ as presheaf categories. [1] )

## Facts

Most constructions that can be carried out in ${\displaystyle D}$ can also be carried out in ${\displaystyle D^{C}}$ by performing them "componentwise", separately for each object in ${\displaystyle C}$. For instance, if any two objects ${\displaystyle X}$ and ${\displaystyle Y}$ in ${\displaystyle D}$ have a product ${\displaystyle X\times Y}$, then any two functors ${\displaystyle F}$ and ${\displaystyle G}$ in ${\displaystyle D^{C}}$ have a product ${\displaystyle F\times G}$, defined by ${\displaystyle (F\times G)(c)=F(c)\times G(c)}$ for every object ${\displaystyle c}$ in ${\displaystyle C}$. Similarly, if ${\displaystyle \eta _{c}:F(c)\to G(c)}$ is a natural transformation and each ${\displaystyle \eta _{c}}$ has a kernel ${\displaystyle K_{c}}$ in the category ${\displaystyle D}$, then the kernel of ${\displaystyle \eta }$ in the functor category ${\displaystyle D^{C}}$ is the functor ${\displaystyle K}$ with ${\displaystyle K(c)=K_{c}}$ for every object ${\displaystyle c}$ in ${\displaystyle C}$.

As a consequence we have the general rule of thumb that the functor category ${\displaystyle D^{C}}$ shares most of the "nice" properties of ${\displaystyle D}$:

• if ${\displaystyle D}$ is complete (or cocomplete), then so is ${\displaystyle D^{C}}$;
• if ${\displaystyle D}$ is an abelian category, then so is ${\displaystyle D^{C}}$;

We also have:

• if ${\displaystyle C}$ is any small category, then the category ${\displaystyle {\textbf {Set}}^{C}}$ of presheaves is a topos.

So from the above examples, we can conclude right away that the categories of directed graphs, ${\displaystyle G}$-sets and presheaves on a topological space are all complete and cocomplete topoi, and that the categories of representations of ${\displaystyle G}$, modules over the ring ${\displaystyle R}$, and presheaves of abelian groups on a topological space ${\displaystyle X}$ are all abelian, complete and cocomplete.

The embedding of the category ${\displaystyle C}$ in a functor category that was mentioned earlier uses the Yoneda lemma as its main tool. For every object ${\displaystyle X}$ of ${\displaystyle C}$, let ${\displaystyle {\text{Hom}}(-,X)}$ be the contravariant representable functor from ${\displaystyle C}$ to ${\displaystyle {\textbf {Set}}}$. The Yoneda lemma states that the assignment

${\displaystyle X\mapsto \operatorname {Hom} (-,X)}$

is a full embedding of the category ${\displaystyle C}$ into the category Funct(${\displaystyle C^{\text{op}}}$,${\displaystyle {\textbf {Set}}}$). So ${\displaystyle C}$ naturally sits inside a topos.

The same can be carried out for any preadditive category ${\displaystyle C}$: Yoneda then yields a full embedding of ${\displaystyle C}$ into the functor category Add(${\displaystyle C^{\text{op}}}$,${\displaystyle {\textbf {Ab}}}$). So ${\displaystyle C}$ naturally sits inside an abelian category.

The intuition mentioned above (that constructions that can be carried out in ${\displaystyle D}$ can be "lifted" to ${\displaystyle D^{C}}$) can be made precise in several ways; the most succinct formulation uses the language of adjoint functors. Every functor ${\displaystyle F:D\to E}$ induces a functor ${\displaystyle F^{C}:D^{C}\to E^{C}}$ (by composition with ${\displaystyle F}$). If ${\displaystyle F}$ and ${\displaystyle G}$ is a pair of adjoint functors, then ${\displaystyle F^{C}}$ and ${\displaystyle G^{C}}$ is also a pair of adjoint functors.

The functor category ${\displaystyle D^{C}}$ has all the formal properties of an exponential object; in particular the functors from ${\displaystyle E\times C\to D}$ stand in a natural one-to-one correspondence with the functors from ${\displaystyle E}$ to ${\displaystyle D^{C}}$. The category ${\displaystyle {\textbf {Cat}}}$ of all small categories with functors as morphisms is therefore a cartesian closed category.

## Related Research Articles

In mathematics, specifically category theory, a functor is a map 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 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, 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 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, 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, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of abelian groups, Ab. The theory originated in an effort to unify several cohomology theories by Alexander Grothendieck and independently in the slightly earlier work of David Buchsbaum. Abelian categories are very stable categories; for example they are regular and they satisfy the snake lemma. The class of abelian categories is closed under several categorical constructions, for example, the category of chain complexes of an abelian category, or the category of functors from a small category to an abelian category are abelian as well. These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometry, cohomology and pure category theory. Abelian categories are named after Niels Henrik Abel.

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 mathematics, specifically in category theory, a preadditive category is another name for an Ab-category, i.e., a category that is enriched over the category of abelian groups, Ab. That is, an Ab-categoryC is a category such that every hom-set Hom(A,B) in C has the structure of an abelian group, and composition of morphisms is bilinear, in the sense that composition of morphisms distributes over the group operation. In formulas:

In mathematics, specifically in category theory, an additive category is a preadditive category C admitting all finitary biproducts.

In mathematics, specifically in category theory, a pre-abelian category is an additive category that has all kernels and cokernels.

In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original one. For example, such data can consist of the rings of continuous or smooth real-valued functions defined on each open set. Sheaves are by design quite general and abstract objects, and their correct definition is rather technical. They are variously defined, for example, as sheaves of sets or sheaves of rings, depending on the type of data assigned to open sets.

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, certain functors may be derived to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics.

In mathematics, a comma category is a construction in category theory. It provides another way of looking at morphisms: instead of simply relating objects of a category to one another, morphisms become objects in their own right. This notion was introduced in 1963 by F. W. Lawvere, although the technique did not become generally known until many years later. Several mathematical concepts can be treated as comma categories. Comma categories also guarantee the existence of some limits and colimits. The name comes from the notation originally used by Lawvere, which involved the comma punctuation mark. The name persists even though standard notation has changed, since the use of a comma as an operator is potentially confusing, and even Lawvere dislikes the uninformative term "comma category".

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 allowing one to utilize, as much as possible, knowledge about the category of sets in other settings.

In mathematics, a simplicial set is an object made up of "simplices" in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined as a contravariant functor from the simplex category to the category of sets. Simplicial sets were introduced in 1950 by Samuel Eilenberg and J. A. Zilber.

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, monoidal functors are functors between monoidal categories which preserve the monoidal structure. More specifically, a monoidal functor between two monoidal categories consists of a functor between the categories, along with two coherence maps—a natural transformation and a morphism that preserve monoidal multiplication and unit, respectively. Mathematicians require these coherence maps to satisfy additional properties depending on how strictly they want to preserve the monoidal structure; each of these properties gives rise to a slightly different definition of monoidal functors

In category theory, a branch of mathematics, a presheaf on a category is a functor . If is the poset of open sets in a topological space, interpreted as a category, then one recovers the usual notion of presheaf on a topological space.

The theory of ologs is an attempt to provide a rigorous mathematical framework for knowledge representation, construction of scientific models and data storage using category theory, linguistic and graphical tools. Ologs were introduced in 2010 by David Spivak, a research scientist in the Department of Mathematics, MIT.

## References

1. Tom Leinster (2004). Higher Operads, Higher Categories. Cambridge University Press. Bibcode:2004hohc.book.....L. Archived from the original on 2003-10-25.