category theory

## 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 $Id$-relative adjoints.

## Definition

### Via hom-isomorphism

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

$R \colon C \to D$

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

$L \colon B \to C$

and a natural isomorphism

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

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

$Hom_D(L(-), J(-)) \simeq Hom_C(-, R(-))$
• $L {\,\,}_J\!\dashv R$ 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 $L \dashv_J R$.

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.

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

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

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 $N_F(b)(a) = A(Fa,b)$. 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 .

• F. Ulmer, Properties of dense and relative adjoint functors, Journal of Algebra, Volume 8, Issue 1, 1968, Pages 77-95, pdf

• Ross Street, Bob Walters - Yoneda structures on 2-categories, Journal of Algebra, Volume 50, Issue 2, February 1978, Pages 350-379, article at mendeley

• Thorsten Altenkirch, James Chapman and Tarmo Uustalu, Monads need not be endofunctors In: Ong L. (eds) Foundations of Software Science and Computational Structures. FoSSaCS 2010. Lecture Notes in Computer Science, vol 6014. Springer, Berlin, Heidelberg, arXiv:1412.7148 [cs.PL] ,pdf

• Mark Weber - Yoneda structures from 2-toposes, Appl Categor Struct (2007) 15: 259. doi:10.1007/s10485-007-9079-2, pdf