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...

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

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...

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

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 mathematics, a group functor is a group-valued functor on the category of commutative rings. Although it is typically viewed as a generalization of...

specifically 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...

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

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...

the category of contravariant functors from D {\displaystyle D} to the category of sets; such a contravariant functor is frequently called a presheaf...

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

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...

In category theory, a branch of mathematics, a conservative functor is a functor F : C → D {\displaystyle F:C\to D} such that for any morphism f in C,...

mathematics, a smooth functor is a type of functor defined on finite-dimensional real vector spaces. Intuitively, a smooth functor is smooth in the sense...

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...

{\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...

category that is equipped with a faithful functor to the category of sets (or sometimes to another category). This functor makes it possible to think of the objects...

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

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

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

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...

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...

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...

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

In mathematical logic, predicate functor logic (PFL) is one of several ways to express first-order logic (also known as predicate logic) by purely algebraic...

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...