nLab relative adjoint functor

Idea

The notion of relative adjoint functors with respect to a functor $J$ is a generalization of that of adjoint functor (to which it reduces for $J$ an identity functor). In generalization of the relation between adjunctions and monads, relative adjoint functors are related to relative monads.

Definition

Via hom-isomorphism

Fix a functor $J\colon B \to D$. Then, for a functor

$$R \colon C \longrightarrow D$$

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

$$L \colon B \longrightarrow C$$

and a natural isomorphism of the ofrm

$$Hom_C\big(L(-),-\big) \;\simeq\; Hom_D\big(J(-),R(-)\big) \,.$$

Dually, for $L \colon C \longrightarrow D$ to have a $J$-right adjoint $R \colon B \longrightarrow C$ means that there is a natural isomorphism of the form

$$Hom_D\big(L(-), J(-)\big) \;\simeq\; Hom_C\big(-, R(-)\big)$$

One writes

• $L {\,\,}_J\!\dashv R$ stands for $L$ being the $J$-left adjoint of $R$
• $L \dashv_J R$ stands for $R$ being the $J$-right adjoint of $L$

Via absolute lifting

Just as with regular adjoints, in the unenriched setting, relative adjoints can be defined in a more conceptual way in terms of absolute liftings. We have

1. $L {\,\,}_J\!\dashv R$ if $L = \mathop{Lift}_R J$, and this left lifting is absolute
2. $L \dashv_J R$ if $R = \mathop{Rift}_L J$, and this right lifting is absolute

However, this is not true for enriched functors.

Properties

asymmetry

The most important difference with regular adjunctions is the asymmetry of the concept. First, for $L {\,\,}_J\!\dashv R$ it makes no sense to ask for $R \dashv_J L$ (domains and comodomains do not typecheck). And secondly, and more importantly:

• $L$ is $J$-left adjoint to $R$: $R$ determines $L$
• $R$ is $J$-right adjoint to $L$: $L$ determines $R$

(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.

unit, counit

Asymmetry manifests itself here:

1. $L {\,\,}_J\!\dashv R$ yields a $J$-relative unit 2-cell $\iota\colon J \to R L$
2. while $L \dashv_J R$ gives a $J$-relative counit $\epsilon\colon L R \to J$

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

• $\iota\colon J \to R L$ given by $L = \mathop{Lift}_R J$
• $\epsilon\colon L R \to J$ given by $R = \mathop{Rift}_L J$

Alternatively, and just as with regular adjunctions, their components can be obtained from the natural hom-isomorphism: in the unit case, evaluating at $Lb$ we get a bijection

$$Hom_C(Lb, Lb) \simeq Hom_D(Jb, RLb)$$

and

$$\iota_b \colon J b \to R L b$$

is given by evaluating at $1_{Lb}$ the aforementioned bijection. A completely analogous procedure yields a description of the counit for $L \dashv_J R$.

Relative monads and comonads

Just as adjunctions give rise to monads and comonads, for relative adjoints

1. If $L {\,\,}_J\!\dashv R$, then $RL$ is a relative monad
2. If $L \dashv_J R$, then $LR$ 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.

Relative adjointness generalizes adjointness

The concept of relative adjoint functors is a generalization of the concept of adjoint functors: if a functor $R\colon C\to D$ has a left adjoint in the usual sense, then it also has a $J$-left adjoint for $J=id_D$.

Examples

fully faithful functors

A functor $F: A \to B$ is fully faithful iff it is representably fully faithful iff $1_A = \mathop{Lift}_F F$, and this lifting is absolute. Thus, $F$ fully faithful can be expressed as

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

partially defined adjoints

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

$$L \colon C \to D$$

it may happen that $Hom_D(L(-),d)$ is representable only for some $d$, but not for all of them. In that case, taking

$$J \colon B \to D$$

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

$$L \dashv_J R$$

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.

nerves

Take $A$ a locally small category, and $F\colon A \to B$ a locally left-small functor (one for which $B(Fa,b)$ is always small). The $A$-nerve induced by $F$ is the functor

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

given by $N_F(b)(a) = B(Fa,b)$. It is a fundamental fact that $F = \mathop{Lift}_{N_F} y_A$ and this lifting is absolute; or, in relative adjoint notation, $F {\,\,}_{y_A}\!\dashv N_F$. The universal 2-cell $\iota\colon y_A \to N_F F$ is given by the action of $F$ on morphisms:

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

at $a' \colon A$ is

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

Note that when specialized to $F = 1_A$, this reduces to the Yoneda lemma: first $N_{1_A} \simeq y_A$, and then $1_A = \mathop{Lift}_{y_A} y_A$ absolute in hom-isomorphism terms reads:

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

One of the axioms of a Yoneda structure on a 2-category abstract over this situation, by requiring the existence of $F$-nerves with respect to yoneda embeddings such that the 1-cell $F$ is an absolute left lifting as above; see Weber or Street–Walters .

Related concepts

• relative monad

References

General

• 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]

• 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]

• 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]

• 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]