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...

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

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

: 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...

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...

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...

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...

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...

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

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

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

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...

(f\colon X\to Z)\mapsto (f^{Y}\colon X^{Y}\to Z^{Y})} , is a right adjoint to the product functor − × Y {\displaystyle -\times Y} . For this reason, the morphisms...

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

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...

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...

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...

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

adjoint functors ( j ! , j ∗ , j ∗ ) {\displaystyle (j_{!},j^{*},j_{*})} and ( i ∗ , i ∗ , i ! ) {\displaystyle (i^{*},i_{*},i^{!})} . The functors i...

requirement that the functor – ×Y (i.e. the functor from C to C that maps objects X to X ×Y and morphisms φ to φ × idY) has a right adjoint, usually denoted...

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

and universal quantifiers can be characterized as special cases of adjoint functors. Back in Zürich for 1968 and 1969 he proposed elementary (first-order)...

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...

relative to ≤. The term "adjoint" refers to the fact that monotone Galois connections are special cases of pairs of adjoint functors in category theory as...

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...

forgetful functors A : Ring → Ab M : Ring → Mon which "forget" multiplication and addition, respectively. Both of these functors have left adjoints. The left...

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...

this context, the duality is often called Eckmann–Hilton duality. Adjoint functor Dual object Duality (mathematics) Opposite category Pulation square...

functor category C1, with objects functors c: 1 → C, selecting an object c∈Ob(C), and arrows natural transformations f: c → d between these functors,...

as the existence of a right adjoint functor f ! {\displaystyle f^{!}} , called twisted or exceptional inverse image functor, to a higher direct image with...