nLab
relative adjoint functor

Idea

Relative adjoints with respect to a functor $J$ are a generalization of adjoints, where $J$ in the relative case plays the role of the identity in the standard setting: adjoints are the same as $\mathrm{Id}$-relative adjoints.

Definition

hom-isomorphism definition

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

(1)$$R:C\to D$$R \colon C \to D

has a left $J$-relative adjoint (or $J$-left adjoint) if there is a functor

(2)$$L:B\to C$$L \colon B \to C

and a natural isomorphism

(3)$${\mathrm{Hom}}_C(L(-),-)\simeq {\mathrm{Hom}}_D(J(-),R(-))$$Hom_C(L(-),-) \simeq Hom_D(J(-),R(-))

Dually, $L:C\to D$ has a $J$-right adjoint $R:B\to C$ if there’s a natural isomorphism

(4)$${\mathrm{Hom}}_D(L(-),J(-))\simeq {\mathrm{Hom}}_C(-,R(-))$$Hom_D(L(-), J(-)) \simeq Hom_C(-, R(-))

Notation

• $L{}_J⊣R$ stands for $L$ being the $J$-left adjoint of $R$
• $L{⊣}_{J}R$ stands for $R$ being the $J$-right adjoint of $L$

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

1. $L{}_J⊣R$ if $L={Lift}_{R}J$, and this left lifting is absolute
2. $L{⊣}_{J}R$ if $R={Rift}_{L}J$, and this right lifting is absolute

Properties

asymmetry

The most important difference with regular adjunctions is the asymmetry of the concept. First, for $L{}_J⊣R$ it makes no sense to ask for $R{⊣}_{J}L$ (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. $L{}_J⊣R$ yields a $J$-relative unit 2-cell $\iota :J\to RL$
2. while $L{⊣}_{J}R$ gives a $J$-relative counit $ϵ:LR\to J$

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

• $\iota :J\to RL$ given by $L={Lift}_{R}J$
• $ϵ:LR\to J$ given by $R={Rift}_{L}J$

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

(5)$${\mathrm{Hom}}_C(\mathrm{Lb},\mathrm{Lb})\simeq {\mathrm{Hom}}_D(\mathrm{Jb},\mathrm{RLb})$$Hom_C(Lb, Lb) \simeq Hom_D(Jb, RLb)

and

(6)$${\iota }_b:Jb\to RLb$$\iota_b \colon J b \to R L b

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

relative monads and comonads

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

1. If $L{}_J⊣R$, then $\mathrm{RL}$ is a relative monad?
2. If $L{⊣}_{J}R$, then $\mathrm{LR}$ 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.

Examples

partially defined adjoints

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

(7)$$L:C\to D$$L \colon C \to D

it may happen that ${\mathrm{Hom}}_D(L(-),d)$ is representable only for some $d$, but not for all of them. In that case, taking

(8)$$J:B\to D$$J \colon B \to D

be the inclusion of the full subcategory determined by ${\mathrm{Hom}}_D(L(-),d)$ representable, and defining $R:B\to C$ accordingly, we have

(9)$$L{⊣}_{J}R$$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.

fully faithful functors

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

(10)$$1{}_F⊣F$$1 {\,\,}_F\!\dashv F

nerves

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

(11)$$N_F:B\to {\mathrm{Set}}^{A^{op}}$$N_F \colon B \to \mathbf{Set}^{A^{\mathop{op}}}

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

(12)$${\iota }_a:y_{A}a\to (N_{F}F)(a)$$\iota_a \colon y_A a \to (N_F F)(a)

at $a\prime :A$ is

(13)$$F_{a,a\prime }:A(a,a\prime )\to B(\mathrm{Fa},\mathrm{Fa}\prime )$$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 $1_A={Lift}_{y_A}{y}_A$ absolute in hom-isomorphism terms reads:

(14)$$A(x,y)\simeq {\mathrm{Set}}^{A^{op}}(y_{A}x,y_{A}y)$$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 Mark Weber or the original Street-Walters papers cited in the references below.

References

• 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