“ The slogan is ‘ Adjoint functors arise everywhere ’.” ( Saunders Mac Lane, Categories for the working mathematician )

* Adjoint functors –

Category: Adjoint functors

Category: Adjoint functors

* Adjoint functors

* Adjoint functors in category theory

Category: Adjoint functors

* Adjoint functor

The same year Roméo premiered, Berlioz was appointed Conservateur Adjoint ( Deputy Librarian ) Paris Conservatoire Library.

Deborah S. Jin ( born 15 November 1968 ) is a physicist with the National Institute of Standards and Technology ( NIST ); Professor Adjoint, Department of Physics at the University of Colorado ; a fellow of the JILA, a NIST joint laboratory with the University of Colorado.

* Adjoint.

Adjoint modelling and Automated Differentiation are methods in this class.

* 2002: Prof. Roger-Maurice Bonnet, Directeur General Adjoint for Science, CNES

* The functor category D < sup > C </ sup > has as objects the functors from C to D and as morphisms the natural transformations of such functors.

The Yoneda lemma is one of the most famous basic results of category theory ; it describes representable functors in functor categories.

The main problem is to prove a signalizer functor theorem for nonsolvable signalizer functors.

Forgetful functors: The functor U: Grp → Set which maps a group to its underlying set and a group homomorphism to its underlying function of sets is a functor.

One can compose functors, i. e. if F is a functor from A to B and G is a functor from B to C then one can form the composite functor G ∘ F from A to C. Composition of functors is associative where defined.

Identity of composition of functors is identity functor.

The collection of all functors C → D form the objects of a category: the functor category.

More generally, if C is an arbitrary abelian category that has enough injectives, then so does C < sup > I </ sup >, and the right derived functors of the inverse limit functor can thus be defined.

In the case where C satisfies Grothendieck's axiom ( AB4 *), Jan-Erik Roos generalized the functor lim < sup > 1 </ sup > on Ab < sup > I </ sup > to series of functors lim < sup > n </ sup > such that

This means that T is left adjoint to the forgetful functor U ( see the section below on relation to adjoint functors ).

It also clarifies how the embedded category, of representable functors and their natural transformations, relates to the other objects in the larger functor category.

Mapping each object A in C to its associated hom-functor h < sup > A </ sup > = Hom ( A ,–) and each morphism f: B → A to the corresponding natural transformation Hom ( f ,–) determines a contravariant functor h < sup >–</ sup > from C to Set < sup > C </ sup >, the functor category of all ( covariant ) functors from C to Set.

The meaning of Yoneda's lemma in this setting is that the functor h < sup >–</ sup > is fully faithful, and therefore gives an embedding of C < sup > op </ sup > in the category of functors to Set.

An important property of adjoint functors is that every right adjoint functor is continuous and every left adjoint functor is cocontinuous.

From about 1955 he started to work on sheaf theory and homological algebra, producing the influential " Tôhoku paper " ( Sur quelques points d ' algèbre homologique, published in 1957 ) where he introduced Abelian categories and applied their theory to show that sheaf cohomology can be defined as certain derived functors in this context.

Note that arrows between categories are called functors, subject to specific defining commutativity conditions ; moreover, categorical diagrams and sequences can be defined as functors ( viz.

Basic constructions, such as the fundamental group or fundamental groupoid of a topological space, can be expressed as fundamental functors to the category of groupoids in this way, and the concept is pervasive in algebra and its applications.

Such a pair of adjoint functors typically arises from a construction defined by a universal property ; this can be seen as a more abstract and powerful view on universal properties.

Representable functors: We can generalize the previous example to any category C. To every pair X, Y of objects in C one can assign the set Hom ( X, Y ) of morphisms from X to Y.

This shows that functors can be considered as morphisms in categories of categories, for example in the category of small categories.

One can use Hom functors to relate limits and colimits in a category C to limits in Set, the category of sets.

Hence, a natural transformation can be considered to be a " morphism of functors ".

The notion of a natural transformation is categorical, and states ( informally ) that a particular map between functors can be done consistently over an entire category.

If and are natural transformations between functors, then we can compose them to get a natural transformation.

If is a natural transformation between functors, and is another functor, then we can form the natural transformation by defining

If C is any category and I is a small category, we can form the functor category C < sup > I </ sup > having as objects all functors from I to C and as morphisms the natural transformations between those functors.

As a result, general theorems about left / right adjoint functors, such as the equivalence of their various definitions or the fact that they respectively preserve colimits / limits ( which are also found in every area of mathematics ), can encode the details of many useful and otherwise non-trivial results.

If C is a complete category, then the functors with left adjoints can be characterized by the adjoint functor theorem of Peter J. Freyd: G has a left adjoint if and only if it is continuous and a certain smallness condition is satisfied: for every object Y of D there exists a family of morphisms

* If is a topological space ( viewed as a category as above ) and is some small category, we can form the category of all contravariant functors from to, using natural transformations as morphisms.

In a pre-abelian category, exact functors can be described in particularly simple terms.