relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics...

Look up adjoint in Wiktionary, the free dictionary. In mathematics, the term adjoint applies in several situations. Several of these share a similar formalism:...

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

concept of adjoint functors was introduced independently by Daniel Kan in 1958. Mathematics portal Free object Natural transformation Adjoint functor Monad...

pairs of adjoint functors. Functors sometimes appear in functional programming. For instance, the programming language Haskell has a class Functor where...

to the defining properties of pairs of adjoint functors in category theory, and this is where adjoint functors got their name from. Mathematical concepts...

are functors adjoint to each other, then T = G ∘ F {\displaystyle T=G\circ F} together with η , μ {\displaystyle \eta ,\mu } determined by the adjoint relation...

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...

: X → Y {\displaystyle u:X\to Y} is a pair of adjoint functors (u∗,u∗) (where u∗ : Y → X is left adjoint to u∗ : X → Y) such that u∗ preserves finite limits...

properties and adjoint functors. Let 1 be the discrete category with a single object (denoted by •), and let U : C → 1 be the unique (constant) functor to 1. Then...

other settings. From another point of view, representable functors for a category C are the functors given with C. Their theory is a vast generalisation of...

transpose or algebraic adjoint of a linear map is often used to study the original linear map. This concept is generalised by adjoint functors. Let X # {\displaystyle...

F:{\text{Hom}}(A,B)\rightarrow {\text{Hom}}(F(A),F(B))} is a group homomorphism. Most functors studied between preadditive categories are additive. For a simple example...

in topology and more generally model categories. Two functors F: C → D and G: D → C are adjoint if for all objects c in C and d in D HomD(F(c), d) ≅ HomC(c...

Cat is naturally a 2-category, with functors as its arrows and natural transformations as the arrows between functors. In this setting, commutative diagrams...

to be a "morphism of functors". Informally, the notion of a natural transformation states that a particular map between functors can be done consistently...

must be additive functors (see here). Most of the interesting functors studied in category theory are adjoints. When considering functors between R-linear...

D→D denote the identity functors on C and D, assigning each object and morphism to itself. If F and G are contravariant functors one speaks of a duality...

of a forgetful functor with no adjoint. There is no field satisfying a free universal property for a given set. Adjoint functors Functors Projection (set...

mathematics, the formal criteria for adjoint functors are criteria for the existence of a left or right adjoint of a given functor. One criterion is the following...

Adjoint functors: A functor can be left (or right) adjoint to another functor that maps in the opposite direction. Such a pair of adjoint functors typically...

be conceptually explained using the language of adjoint functors and derived categories: the functor between the derived categories of R- and k-modules...

f^{*}N=N_{R}} , formed by restriction of scalars. They are related as adjoint functors: f ! : Mod R ⇆ Mod S : f ∗ {\displaystyle f_{!}:{\text{Mod}}_{R}\leftrightarrows...

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

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...

and homotopy classes of continuous maps, whence the name. A pair of adjoint functors F : C ⇆ D : G {\displaystyle F:C\leftrightarrows D:G} between two model...

then just a contravariant functor I → C. Let C I o p {\displaystyle C^{I^{\mathrm {op} }}} be the category of these functors (with natural transformations...

functors are adjoint to each other as depicted at the right, where, as usual, F ⇆ G {\displaystyle F\leftrightarrows G} means that F is left adjoint to...

objects exist in C, the functor F, called the free functor is a left adjoint to the faithful functor U; that is, there is a bijection Hom S e t ⁡ ( X ...

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...