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...

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

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

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...

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...

: 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...

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

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...

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...

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...

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...

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...

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

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...

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...

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 ...

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...

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...

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...