In category theory, a branch of mathematics, a functor category is a category where the objects are the functors and the morphisms are natural transformations is another object in the category). Functor categories are of interest for two main reasons:

• 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

is a small category (i.e. the objects and morphisms form a set rather than a proper class. With this composition of natural transformations (known as vertical composition, see natural transformation satisfies the axioms of a category.

In a completely analogous way, one can also consider the category of all contravariant).

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).

Examples

• is a small discrete category.

  An arrow category, 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 is the category with two objects connected by two parallel morphisms (source and target), and Set denotes the category of sets. When viewed from this perspective, such directed graphs are often called quivers.

• Any group can be considered as a one-object category in which every morphism is invertible. The category of all G-sets is the same as the functor category Set. Natural transformations are G-maps.

• Similar to the previous example, the category of K-linear representations is the same as the functor category VectK (where VectK denotes the category of all vector spaces over the field K).

• Any ring can be considered as a one-object preadditive category; the category of left modules denotes the category of abelian groups".

• The category of presheaves) is sometimes called the "category of presheaves not arising from a topological space. To define sheaves, one needs more structure: a Grothendieck topology. (Some authors refer to categories that are equivalent as presheaf categories.)

Facts

have a product.

As a consequence we have the general rule of thumb:

• is complete;

• is an abelian category;

We also have:

• of presheaves is a topos.

are all abelian, complete and cocomplete.

in a functor category that was mentioned earlier uses the Yoneda lemma be the contravariant representable functor. The Yoneda lemma states that the assignment

is a full embedding naturally sits inside a topos.

naturally sits inside an abelian category.

) can be made precise in several ways; the most succinct formulation uses the language of adjoint functors is also a pair of adjoint functors.

has all the formal properties of an exponential object of all small categories with functors as morphisms is therefore a cartesian closed category.

See also

• Diagram (category theory)

References