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Concept
Functor category
Summary
In , a branch of mathematics, a functor category is a category where the objects are the functors and the morphisms are natural transformations between the functors (here, 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.
Suppose is a (i.e. the objects and morphisms form a set rather than a proper class) and is an arbitrary category.
The category of functors from to , written as Fun(, ), Funct(,), , or , has as objects the covariant functors from to ,
and as morphisms the natural transformations between such functors. Note that natural transformations can be composed:
if is a natural transformation from the functor to the functor , and
is a natural transformation from the functor to the functor , then the composition defines a natural transformation
from to . 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 functors from to ; we write this as Funct().
If and are both (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 to , denoted by Add(,).
If is a small (i.e. its only morphisms are the identity morphisms), then a functor from to essentially consists of a family of objects of , indexed by ; the functor category can be identified with the corresponding product category: its elements are families of objects in and its morphisms are families of morphisms in .
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