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

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

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

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

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

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

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...

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...

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

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

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

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...

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...

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...

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 properties...

mathematics, the Tor functors are the derived functors of the tensor product of modules over a ring. Along with the Ext functor, Tor is one of the central...

In mathematics, the Ext functors are the derived functors of the Hom functor. Along with the Tor functor, Ext is one of the core concepts of homological...

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...

categorical sum. It follows that any functor which preserves limits will take terminal objects to terminal objects, and any functor which preserves colimits will...

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...

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

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...

{\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...

of final functor (resp. initial functor) is a generalization of the notion of final object (resp. initial object) in a category. A functor F : C → D...

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

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

pre-abelian category, exact functors can be described in particularly simple terms. First, recall that an additive functor is a functor F: C → D between preadditive...

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