# Functor Category

Functor Category

In category theory, a branch of mathematics, the functors between two given categories form a category, where the objects are the functors and the morphisms are natural transformations between the functors. 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.

### Other articles related to "functor category, functors, category, functor":

Functor Category - Facts
... For instance, if any two objects X and Y in D have a product X×Y, then any two functors F and G in DC have a product F×G, defined by (F×G)(c) = F(c)×G(c ... and each ηc has a kernel Kc in the category D, then the kernel of η in the functor category DC is the functor K with K(c) = Kc for every object c in C ... As a consequence we have the general rule of thumb that the functor category DC shares most of the "nice" properties of D if D is complete (or cocomplete), then so is DC if D is ...
Adjoint Functors - Examples - Further Examples - Topology
... A functor with a left and a right adjoint ... Let G be the functor from topological spaces to sets that associates to every topological space its underlying set (forgetting the topology, that is) ... Let KHaus be the category of compact Hausdorff spaces and G KHaus → Top be the inclusion functor to the category of topological spaces ...
Exact Functor - Examples
... The most important examples of left exact functors are the Hom functors if A is an abelian category and A is an object of A, then FA(X) = HomA(A,X) defines a covariant left-exact functor from A ... The functor FA is exact if and only if A is projective ... The functor GA(X) = HomA(X,A) is a contravariant left-exact functor it is exact if and only if A is injective ...
Cartesian Closed Category - Examples
... Examples of cartesian closed categories include The category Set of all sets, with functions as morphisms, is cartesian closed ... The category of finite sets, with functions as morphisms, is cartesian closed for the same reason ... If G is a group, then the category of all G-sets is cartesian closed ...

### Famous quotes containing the word category:

I see no reason for calling my work poetry except that there is no other category in which to put it.
Marianne Moore (1887–1972)