Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
For any 2-category , there are two 2-categories (each with several variants) that could be called “the 2-category of adjunctions in ”.
The 2-category which here we call has the same objects as , its morphisms are the adjunctions in (pointing in the direction of, say, the left adjoint), and its 2-morphisms are mate-pairs of 2-morphisms between adjunctions in .
The 2-category is the functor 2-category from the walking adjunction to . Thus its objects are the adjunctions in — or more precisely, triples where are objects of and is an adjunction between and . Its morphisms are pairs of morphisms and such that certain squares commute (perhaps up to a transformation or isomorphism), and its 2-cells are similarly composed of cylinders.
Note that the 1-morphisms of are the objects of .
The morphisms in are the adjoint triples in .
The inclusion of , the free monad, in induces a 2-functor from to , the 2-category of monads in . The adjoints to this 2-functor are the Kleisli and Eilenberg-Moore constructions on monads in .
Claude Auderset?, Adjonctions et monades au niveau des -catégories, Cahiers de topologie et géométrie différentielle 15.1 (1974): 3-20.
Stephen Schanuel and Ross Street, The free adjunction, Cahiers de topologie et géométrie différentielle catégoriques, tome 27, no 1 (1986), p. 81-83
Last revised on September 16, 2024 at 09:55:27. See the history of this page for a list of all contributions to it.