Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
Given a 2-category , adjoint pairs and , and 1-cells and , there is a bijection
given by pasting with the unit of one adjunction and the counit of the other, i.e.
The 2-cells and in prop. 1 are called mates (or sometimes conjugates) with respect to the adjunctions and (and to the 1-cells and ).
Strict 2-functors preserve adjunctions and pasting diagrams, so that if is a 2-functor and if and are mates wrt and in , then and are mates wrt and in .
If is a 2-natural transformation, then the naturality identities and are mates wrt and .
There are two double categories with objects those of , vertical arrows adjoint pairs in and horizontal arrows 1-cells of . In one the 2-cells are those of the form above, while in the other they are those of the form . It is easily shown, as in Kelly–Street, that the triangle identities and the definition of composition of adjoints make these two double categories isomorphic. So for any there is a double category , defined up to isomorphism as above but with mate-pairs in as 2-cells.
What this means is that, for example, the mate of a square coming from a pasting diagram is given by pasting the mates of the individual 2-cells (whenever this makes sense).
In the double category , every vertical arrow has both a companion (the left adjoint) and a conjoint (the right adjoint). (In fact, in some sense it is the universal double category cosntructed from with this property.) Therefore, it is equivalent to a 2-category equipped with proarrows. More explicitly, there is a forgetful functor from the 2-category of objects, adjunctions and mate-pairs in to that sends an adjunction to . It is locally fully faithful, and moreover every has a right adjoint in by definition; this gives the more traditional definition of a proarrow equipment.
Let be an adjunction in the 2-category Cat, i.e. a pair of adjoint functors, and and be objects of and considered as functors out of the terminal category . Then taking mates with respect to and yields the familiar bijection
and the pasting operations as above yield the usual definition of the isomorphism of adjunction by means of unit and counit. Moreover, the naturality of the mate correspondence yields naturality of the bijection.
Suppose given a commutative square (up to isomorphism) of functors:
in which and have left adjoints and , respectively. (The classical example is a Wirthmüller context.) Then the natural isomorphism that makes the square commute
has a mate
defined as the composite
Ones says that the original square satisfies the Beck-Chevalley condition if this mate is an equivalence.
There is a version of the mate correspondence that applies to two-variable adjunctions and -variable adjunctions; see Cheng-Gurski-Riehl.
Max Kelly, Ross Street, Review of the elements of 2-categories, in Kelly (ed.), Category Seminar, LNM 420.
Tom Leinster, Higher operads, higher categories, math.CT/0305049, Section 6.1
Eugenia Cheng, Nick Gurski, Emily Riehl, “Multivariable adjunctions and mates”, arXiv:1208.4520, (to appear: Journal of K-Theory).