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. == definition == suppose c is a small category (i.e. the objects and morphisms form a set rather than a proper class) and d is an arbitrary category. the category of functors from c to d, written as fun(c, d), funct(c,d) or d'c, has as objects the covariant functors from c to d, and as morphisms the natural transformations between such functors. note that natural transformations can be composed: if Î¼(x) : f(x) â g(x) is a natural transformation from the functor f : c â d to the functor g : c â d, and Î·(x) : g(x) â h(x) is a natural transformation from the functor g to the functor h, then the collection Î·(x)Î¼(x) : f(x) â h(x) defines a natural transformation from f to h. with this composition of natural transformations (known as vertical composition, see natural transformation), d'c satisfies the axioms of a category. in a completely analogous way, one can also consider the category of all contravariant functors from c to d; we write this as funct(cop,d). if c and 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 c to d, denoted by add(c,d). == examples == * if i is a small discrete category (i.e. its only morphisms are the identity morphisms), then a functor from i to c essentially consists of a family of objects of c, indexed by i; the functor category c'i can be identified with the corresponding product category: its elements are families of objects in c and its morphisms are families of morphisms in c. * an arrow category \mathcal{c}^\rightarrow (whose objects are the morphisms of \mathcal{c}, and whose morphisms are commuting squares in \mathcal{c}) is just \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 verti
Category-theoretic_categories Functors