Adjoint functors - Motivation

In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another. Such functors are ubiquitous in mathematics. Adjoint functors are studied in a branch of mathematics known as category theory. Like much of category theory, the general notion of adjoint functors arises at an abstract level beyond the everyday usage of mathematicians. Adjoint functors can be considered from several different points of view. This article starts with a number of introductory sections considering some ...

Including:

• Adjoint functors - Motivation
• Adjoint functors - Ubiquity of adjoint functors
• Adjoint functors - Deep problems formulated with adjoint functors
• Adjoint functors - Adjoint functors as solving optimization problems
• Adjoint functors - The case of partial orders
• Adjoint functors - Formal definitions
• Adjoint functors - Examples
• Adjoint functors - Properties
• Adjoint functors - Uniqueness of adjoints
• Adjoint functors - Relation to universal constructions
• Adjoint functors - Characterization via unit and co-unit
• Adjoint functors - Adjoints preserve certain limits
• Adjoint functors - Additivity
• Adjoint functors - Composition
• Adjoint functors - Adjoint pairs extend equivalences
• Adjoint functors - General existence theorem

Read more here: » Adjoint functors: Encyclopedia - Adjoint functors

Adjoint functors - Ubiquity of adjoint functors. The idea of an adjoint functor was formulated by Daniel Kan in 1958. Like many of the concepts in category theory, it was suggested by the needs of homological algebra, which was at the time devoted to computations. Those faced with giving tidy, systematic presentations of the subject would have noticed relations such as Hom(F(X), Y< ...

See also:

Adjoint functors, Adjoint functors - Motivation, Adjoint functors - Ubiquity of adjoint functors, Adjoint functors - Deep problems formulated with adjoint functors, Adjoint functors - Adjoint functors as solving optimization problems, Adjoint functors - The case of partial orders, Adjoint functors - Formal definitions, Adjoint functors - Examples, Adjoint functors - Properties, Adjoint functors - Uniqueness of adjoints, Adjoint functors - Relation to universal constructions, Adjoint functors - Characterization via unit and co-unit, Adjoint functors - Adjoints preserve certain limits, Adjoint functors - Additivity, Adjoint functors - Composition, Adjoint functors - Adjoint pairs extend equivalences, Adjoint functors - General existence theorem

Read more here: » Adjoint functors: Encyclopedia II - Adjoint functors - Motivation

A pair of adjoint functors between two categories C and D consists of two functors F : C → D and G : D → C and a natural isomorphism φ : MorD(F–, –) → MorC(–, G–) consisting of bijections: φX,Y : MorD(F(X), Y) → Mor< ...

See also:

Adjoint functors, Adjoint functors - Motivation, Adjoint functors - Ubiquity of adjoint functors, Adjoint functors - Deep problems formulated with adjoint functors, Adjoint functors - Adjoint functors as solving optimization problems, Adjoint functors - The case of partial orders, Adjoint functors - Formal definitions, Adjoint functors - Examples, Adjoint functors - Properties, Adjoint functors - Uniqueness of adjoints, Adjoint functors - Relation to universal constructions, Adjoint functors - Characterization via unit and co-unit, Adjoint functors - Adjoints preserve certain limits, Adjoint functors - Additivity, Adjoint functors - Composition, Adjoint functors - Adjoint pairs extend equivalences, Adjoint functors - General existence theorem

Read more here: » Adjoint functors: Encyclopedia II - Adjoint functors - Formal definitions

Free objects and forgetful functors. If F : Set → Grp is the functor assigning to each set X the free group over X, and if G : Grp → Set is the forgetful functor assigning to each group its underlying set, then the universal property of the free group shows that F is left adjoint to G. The unit of this adjoint pair is the embedding of a set X into the free group over X. In general, free constructions in mathematics tend to be left adjoints of forgetful functors. Free rings, free abelian groups, ...

See also:

Adjoint functors, Adjoint functors - Motivation, Adjoint functors - Ubiquity of adjoint functors, Adjoint functors - Deep problems formulated with adjoint functors, Adjoint functors - Adjoint functors as solving optimization problems, Adjoint functors - The case of partial orders, Adjoint functors - Formal definitions, Adjoint functors - Examples, Adjoint functors - Properties, Adjoint functors - Uniqueness of adjoints, Adjoint functors - Relation to universal constructions, Adjoint functors - Characterization via unit and co-unit, Adjoint functors - Adjoints preserve certain limits, Adjoint functors - Additivity, Adjoint functors - Composition, Adjoint functors - Adjoint pairs extend equivalences, Adjoint functors - General existence theorem

Read more here: » Adjoint functors: Encyclopedia II - Adjoint functors - Examples