Relative adjoint functors
Relative adjoints with respect to a functor are a generalization of adjoints, where in the relative case plays the role of the identity in the standard setting: adjoints are the same as -relative adjoints.
Fix a functor . Then, a functor
has a left -relative adjoint (or -left adjoint) if there is a functor
and a natural isomorphism
Dually, has a -right adjoint if there’s a natural isomorphism
- stands for being the -left adjoint of
- stands for being the -right adjoint of
absolute lifting definition
Just as with regular adjoints, relative adjoints can be defined in a more conceptual way in terms of absolute liftings. We have
- if , and this left lifting is absolute
- if , and this right lifting is absolute
The most important difference with regular adjunctions is the asymmetry of the concept. First, for it makes no sense to ask for (domains and comodomains do not typecheck). And secondly, and more importantly:
- is -left adjoint to : determines
- is -right adjoint to : determines
(this is obvious from the definition in terms of liftings). Because of this, even if most of the properties of adjunctions have a generalization to the relative setting, they do that in a one-sided way.
Asymmetry manifests itself here:
- yields a -relative unit 2-cell
- while gives a -relative counit
with no naturally available counterpart for them in each case.
These 2-cells are directly provided by the definition in terms of liftings, as the universal 2-cells
- given by
- given by
Alternatively, and just as with regular adjunctions, their components can be obtained from the natural hom-isomorphism: in the unit case, evaluating at we get a bijection
is given by evaluating at the aforementioned bijection. A completely analogous procedure yields a description of the counit for .
relative monads and comonads
Just as adjunctions give rise to monads and comonads, for relative adjoints
- If , then is a relative monad?
- If , then is a relative comonad?
with relative units and counits as above, respectively.
There are also relative analogues of Eilenberg-Moore and Kleisli categories for these.
- partially defined adjoints
as remarked in the local definition of adjoint functor, given a functor
it may happen that is representable only for some , but not for all of them. In that case, taking
be the inclusion of the full subcategory determined by representable, and defining accordingly, we have
This can be specialized to situations such as a category having some but not all limits of some kind, partially defined Kan extensions, etc. See also free object.
- fully faithful functors
A functor is fully faithful iff it is representably fully faithful iff , and this lifting is absolute. Thus, fully faithful can be expressed as
Take a locally small category, and a locally left-small functor (one for which is always small). The -nerve induced by is the functor
given by . It is a fundamental fact that and this lifting is absolute; or, in relative adjoint notation, . The universal 2-cell is given by the action of on morphisms:
Note that when specialized to , this reduces to the Yoneda lemma: first , and then absolute in hom-isomorphism terms reads:
One of the axioms of a Yoneda structure? on a 2-category abstract over this situation, by requiring the existence of -nerves with respect to yoneda embeddings such that the 1-cell is an absolute left lifting as above; see Mark Weber or the original Street-Walters papers cited in the references below.
- F Ulmer - Properties of dense and relative adjoint functors Journal of Algebra :: article at mendeley
- Thorsten Altenkirch, James Chapman and Tarmo Uustalu - Monads need not be endofunctors Foundations of Software Science :: pdf
- Mark Weber - Yoneda structures from 2-toposes Applied Categorical Structures :: pdf
- Ross Street, Bob Walters - Yoneda structures on 2-categories Journal of Algebra :: article at mendeley