Search results
Results from the WOW.Com Content Network
A functor : is a right adjoint functor if for each object in , there exists a universal morphism from to . Spelled out, this means that for each ...
In category theory, a branch of 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, which first appeared in Peter J. Freyd's 1964 book Abelian Categories, [1] an Introduction to the Theory of Functors:
A functor from G to Set is then nothing but a group action of G on a particular set, i.e. a G-set. Likewise, a functor from G to the category of vector spaces, Vect K, is a linear representation of G. In general, a functor G → C can be considered as an "action" of G on an object in the category C. If C is a group, then this action is a group ...
An important property of adjoint functors is that every right adjoint functor is continuous and every left adjoint functor is cocontinuous. Since adjoint functors exist in abundance, this gives numerous examples of continuous and cocontinuous functors. For a given diagram F : J → C and functor G : C → D, if both F and GF have specified ...
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 arises from a construction defined by a universal property; this can be seen as a more abstract and powerful view on universal properties.
Universal constructions are functorial in nature: if one can carry out the construction for every object in a category C then one obtains a functor on C. Furthermore, this functor is a right or left adjoint to the functor U used in the definition of the universal property. [2] Universal properties occur everywhere in mathematics.
Specifically, adjoint or adjunction may mean: Adjoint of a linear map, also called its transpose in case of matrices; Hermitian adjoint (adjoint of a linear operator) in functional analysis; Adjoint endomorphism of a Lie algebra; Adjoint representation of a Lie group; Adjoint functors in category theory; Adjunction (field theory)
The earlier example of G-sets can be seen as a special case of functor categories: every group can be considered as a one-object category, and G-sets are nothing but functors from this category to Set; The category of all directed graphs is Cartesian closed; this is a functor category as explained under functor category.