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 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.
In category theory, a branch of mathematics, a functorcategory D C {\displaystyle D^{C}} is a category where the objects are the functors F : C → D {\displaystyle...
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic...
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of...
mathematics, the Yoneda lemma is a fundamental result in category theory. It is an abstract result on functors of the type morphisms into a fixed object. It is...
contravariant functor acts as a covariant functor from the opposite category Cop to D. A natural transformation is a relation between two functors. Functors often...
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal...
in a category is the terminal object of that category. The third condition is equivalent to the requirement that the functor – ×Y (i.e. the functor from...
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...
preadditive category as the "elements" of the "generalized ring". If C {\displaystyle C} and D {\displaystyle D} are preadditive categories, then a functor F :...
In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies...
mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets (or sometimes to another category, see Relative...
In category theory, a faithful functor is a functor that is injective on hom-sets, and a full functor is surjective on hom-sets. A functor that has both...
concrete category with free objects will be the free object generated by the empty set (since the free functor, being left adjoint to the forgetful functor to...
An equivalence of categories consists of a functor between the involved categories, which is required to have an "inverse" functor. However, in contrast...
designed to cope with functors that fail to be exact, but in ways that can still be controlled. Let P and Q be abelian categories, and let F: P→Q be a...
hom-object category happens to be the category of sets with the usual cartesian product, the definitions of enriched category, enriched functor, etc... reduce...
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...
property. Technically, a universal property is defined in terms of categories and functors by means of a universal morphism (see § Formal definition, below)...
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...
are categories (which are small with respect to some universe) and the morphisms functors. Fct(C, D), the functorcategory: the category of functors from...
particularly category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors give representations...
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