The notion of relative adjoint functor (with respect to a functor $J$) is a generalisation of the notion of adjoint functor, in which the domain of the left adjoint is not required to be the same as the codomain on the right adjoint. Dually, the notion of relative coadjoint functor is a generalisation of the notion of adjoint functor, in which the domain of the right adjoint is not required to be the same as the codomain of the left adjoint. In both cases, when $J$ is an identity functor, the notions reduce to the notion of adjoint functor.
In generalization of the relation between adjunctions and monads, relative adjoint functors are related to relative monads, whilst relative coadjoint functors are related to relative comonads.
Fix a functor $J\colon A \to E$. Then, for a functor
to have a left $J$-relative adjoint (or left $J$-adjoint) means that there is a functor
and a natural isomorphism of the form
Such a situation is called a $J$-relative adjunction and is denoted $L \dashv_J R$.
Dually, for a functor $L \colon C \longrightarrow E$ to have a right $J$-relative coadjoint (or right $J$-coadjoint) $R \colon A \longrightarrow E$ means that there is a natural isomorphism of the form
Such a situation is called a $J$-relative coadjunction and is denoted $L {\,\,}_J\!\dashv R$.
Importantly, there is a bifurcation of concepts, which is not visible for ordinary adjunctions: the notion of relative adjunction is not self-dual.
In the literature, relative adjunctions and relative coadjunctions have not been adequately distinguished, with the term “relative adjunction” frequently being used for both. However, they are distinct concepts and behave differently (albeit dually).
Similarly, the convention $L \dashv_J R$ for relative adjunctions and $L {\,\,}_J\!\dashv R$ relative coadjunctions is sometimes reversed in the literature. On this page, we follow the conventions of AM24, who give a more detailed history of the concept.
Just as adjunctions give rise to monads and comonads,
(with the units and counits respectively induced as described above).
Conversely, there are anlogues of the Kleisli category and Eilenberg–Moore category for relative monads and relative comonads, which induce the relative monads and relative comonads.
An $id$-relative adjunction is simply an ordinary adjunction. Dually, an $id$-relative coadjunction is simply an ordinary adjunction.
A functor $F: A \to B$ is fully faithful iff the canonical natural transformation $1 \Rightarrow B(F{-}, F{-})$ is invertible iff there exists any such isomorphism, i.e. iff
As remarked in the local definition of adjoint functor, given a functor
it may happen that $E(L(-),e)$ is representable only for some $e \in E$, but not for all of them. In that case, taking
to be the inclusion of the full subcategory determined by $E(L(-),e)$ representable, and defining $R \colon A \to C$ accordingly, we have
This can be specialized to situations such as a category having some but not all limit of some kind, partially defined extensions, etc. See also free object.
Take $A$ a locally small category, and $F\colon A \to B$ a small-admissible functor (one for which $B(Fa,b)$ is always small). The nerve of $F$ is the functor
given by $N_F(b)(a) = B(Fa,b)$. The nerve forms a right adjoint to $F$ relative to the Yoneda embedding: $F \dashv_{y_A} N_F$. The universal 2-cell $\eta\colon y_A \to N_F F$ is given by the action of $F$ on morphisms:
at $a' \colon A$ is
Note that, when specialized to $F = 1_A$, this reduces to full faithfulness? of the Yoneda embedding: first $N_{1_A} \simeq y_A$, and then:
In fact, one of the axioms of a Yoneda structure on a 2-category axiomatises this situation, by requiring the existence of absolute left lifting with respect to Yoneda embeddings, as above: see Street–Walters.
A comprehensive account of relative adjunctions (covering also adjoint functors in enriched category theory, and more generally formal category theory) may be found in:
The original reference for relative adjunctions is:
Relative adjunctions were rediscovered in the context of relative monads in:
For the role of nerves in Yoneda structures, see:
Ross Street, Bob Walters, Yoneda structures on 2-categories, Journal of Algebra, 50 2 (1978) 350-379 [doi:10.1016/0021-8693(78)90160-6, mendeley]
Mark Weber - Yoneda structures from 2-toposes, Appl Categor Struct. 15 (2007) 259. [doi:10.1007/s10485-007-9079-2, pdf]
On the categorical semantics of dependent product types as relative right adjoints to context extension in comprehension categories:
Last revised on April 6, 2024 at 10:54:16. See the history of this page for a list of all contributions to it.