nLab relative adjoint functor

Idea

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.

Definition

Fix a functor $J\colon A \to E$. Then, for a functor

$$R \colon C \longrightarrow E$$

to have a left $J$-relative adjoint (or left $J$-adjoint) means that there is a functor

$$L \colon A \longrightarrow C$$

and a natural isomorphism of the form

$$C\big(L(-),-\big) \;\simeq\; E\big(J(-),R(-)\big) \,.$$

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) $R \colon A \longrightarrow C$ means that there is a natural isomorphism of the form

$$E\big(L(-), J(-)\big) \;\simeq\; C\big(-, R(-)\big)$$

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.

Terminology and notation

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.

Properties

• A left $J$-relative adjoint is unique up to isomorphism. A right $J$-relative adjoint is unique up to isomorphism only if $J$ is dense. See Lemma 5.7 of AM24.
• A left $J$-relative adjoint preserves those colimits that $J$ preserves (Proposition 5.11 of AM24). A right $J$-relative adjoint preserves limits when $J$ is dense (Proposition 5.12 of AM24).
• A left relative adjoint is an absolute lift. In particular, if $L \dashv_J R$, then $L = \mathop{Lift}_R J$, and this left lift is absolute. Dually, for a right relative coadjoint $L {\,\,}_J\!\dashv R$, we have $R = \mathop{Rift}_L J$, and this right lift is absolute. Note that, for the converse to hold, we must additionally require that the lifts are pointwise. See Proposition 5.8 and Remark 5.9 of AM24.
• A relative adjunction has a unit $\eta \colon 1 \Rightarrow R L$; whereas a relative coadjunction has a counit $\varepsilon \colon L R \Rightarrow 1$ (both may be seen to be induced from the hom-set definition, like for ordinary adjunctions). In fact, these may be used to give an alternative definition of relative adjunctions and relative coadjunctions, akin to the unit–counit formulation of an adjunction. See Lemma 5.5 of AM24.

Relative monads and comonads

Just as adjunctions give rise to monads and comonads,

1. For relative adjunctions, if $L \dashv_J R$, then $R L$ admits the structure of a monad relative to $J$.
2. For relative coadjunctions, if $L {\,\,}_J\!\dashv R$, then $L R$ admits the structure of a comonad relative to $J$.

(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.

Examples

ordinary adjointness

An $id$-relative adjunction is simply an ordinary adjunction. Dually, an $id$-relative coadjunction is simply an ordinary adjunction.

fully faithful functors

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

$$1 {\,\,}_F\!\dashv F$$

partially defined adjoints

As remarked in the local definition of adjoint functor, given a functor

$$L \colon C \to E$$

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

$$J \colon A \to E$$

to be the inclusion of the full subcategory determined by $E(L(-),e)$ representable, and defining $R \colon A \to C$ accordingly, we have

$$L {\,\,}_J\!\dashv R$$

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.

nerves

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

$$N_F \colon B \to \mathbf{Set}^{A^{\mathop{op}}}$$

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:

$$\eta_a \colon y_A a \to (N_F F)(a)$$

at $a' \colon A$ is

$$F_{a,a'}\colon A(a,a') \to B(F a, F a')$$

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:

$$A(x,y) \simeq \mathbf{Set}^{A^{\mathop{op}}}(y_A x, y_A y)$$

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.

Related concepts

• relative monad

References

General

A comprehensive account of relative adjunctions (covering also adjoint functors in enriched category theory, and more generally formal category theory) may be found in:

• Nathanael Arkor, Dylan McDermott, The formal theory of relative monads, Journal of Pure and Applied Algebra 107676. (2024) [arXiv:2302.14014]

The original reference for relative adjunctions is:

• Friedrich Ulmer, Properties of dense and relative adjoint functors, Journal of Algebra, 8 1 (1968) 77-95 [doi:10.1016/0021-8693(68)90036-7]

Relative adjunctions were rediscovered in the context of relative monads in:

• Thorsten Altenkirch, James Chapman, Tarmo Uustalu, Monads need not be endofunctors, Logical Methods in Computer Science 11 1:3 (2015) 1–40 [arXiv:1412.7148, pdf, doi:10.2168/LMCS-11(1:3)2015]

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]

Examples

On the categorical semantics of dependent product types as relative right adjoints to context extension in comprehension categories:

• Michael Lindgren, Dependent products as relative adjoints, Stockholm (2021) [pdf]