In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic...

relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in...

like the strongly related notions of universal properties and adjoint functors, exist at a high level of abstraction. In order to understand them, it...

a branch of mathematics, a functor category D C {\displaystyle D^{C}} is a category where the objects are the functors F : C → D {\displaystyle F:C\to...

up functor in Wiktionary, the free dictionary. A functor, in mathematics, is a map between categories. Functor may also refer to: Predicate functor in...

is a fundamental result in category theory. It is an abstract result on functors of the type morphisms into a fixed object. It is a vast generalisation...

In functional programming, a functor is a design pattern inspired by the definition from category theory that allows one to apply a function to values...

particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations...

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...

mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition...

In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies...

Presh(D) denotes the category of contravariant functors from D to the category of sets; such a contravariant functor is frequently called a presheaf. Giraud's...

between objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category...

composition are as in C. There is an obvious faithful functor I : S → C, called the inclusion functor which takes objects and morphisms to themselves. Let...

theory, monoidal functors are functors between monoidal categories which preserve the monoidal structure. More specifically, a monoidal functor between two...

mathematics, in the area of category theory, a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure...

calculus of functors or Goodwillie calculus is a technique for studying functors by approximating them by a sequence of simpler functors; it generalizes...

properties. An enriched functor is the appropriate generalization of the notion of a functor to enriched categories. Enriched functors are then maps between...

category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors give representations of an...

In some languages, particularly C++, function objects are often called functors (not related to the functional programming concept). A typical use of a...

geometry, a functor represented by a scheme X is a set-valued contravariant functor on the category of schemes such that the value of the functor at each...

direct image functor, taking sheaves and their morphisms on the domain to sheaves and morphisms on the codomain, and an inverse image functor operating in...

that is equipped with a faithful functor to Set, the category of sets. Let C be a concrete category with a faithful functor U : C → Set. Let X be a set (that...

Technically, a universal property is defined in terms of categories and functors by means of a universal morphism (see § Formal definition, below). Universal...

category-theoretic functor axioms for this functor. Functors can also be objects in categories, with "morphisms" called natural transformations. Given two functors F ...

{\displaystyle C} and D {\displaystyle D} are preadditive categories, then a functor F : C → D {\displaystyle F:C\rightarrow D} is additive if it too is enriched...

a triple ( T , η , μ ) {\displaystyle (T,\eta ,\mu )} consisting of a functor T from a category to itself and two natural transformations η , μ {\displaystyle...

category is a category with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category...

In homological algebra, a δ-functor between two abelian categories A and B is a collection of functors from A to B together with a collection of morphisms...

signalizer functor is a mapping from a potential finite subgroup to the centralizers of the nontrivial elements of an Abelian group. The signalizer functor theorem...