nLab relative adjoint functor

Relative adjoint functors

Relative adjoint functors


Relative adjoints with respect to a functor JJ are a generalization of adjoints, where JJ in the relative case plays the role of the identity in the standard setting: adjoints are the same as IdId-relative adjoints.


Via hom-isomorphism

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

R:CD R \colon C \to D

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

L:BC L \colon B \to C

and a natural isomorphism

Hom C(L(),)Hom D(J(),R()) Hom_C(L(-),-) \simeq Hom_D(J(-),R(-))

Dually, L:CDL \colon C \to D has a JJ-right adjoint R:BCR \colon B \to C if there’s a natural isomorphism

Hom D(L(),J())Hom C(,R()) Hom_D(L(-), J(-)) \simeq Hom_C(-, R(-))
  • L JRL {\,\,}_J\!\dashv R stands for LL being the JJ-left adjoint of RR
  • L JRL \dashv_J R stands for RR being the JJ-right adjoint of LL

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

However, this is not true for enriched functors.



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

  • LL is JJ-left adjoint to RR: RR determines LL
  • RR is JJ-right adjoint to LL: LL determines RR

(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 JRL {\,\,}_J\!\dashv R yields a JJ-relative unit 2-cell ι:JRL\iota\colon J \to R L
  2. while L JRL \dashv_J R gives a JJ-relative counit ϵ:LRJ\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

  • ι:JRL\iota\colon J \to R L given by L=Lift RJL = \mathop{Lift}_R J
  • ϵ:LRJ\epsilon\colon L R \to J given by R=Rift LJR = \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 LbLb we get a bijection

Hom C(Lb,Lb)Hom D(Jb,RLb) Hom_C(Lb, Lb) \simeq Hom_D(Jb, RLb)


ι b:JbRLb \iota_b \colon J b \to R L b

is given by evaluating at 1 Lb1_{Lb} the aforementioned bijection. A completely analogous procedure yields a description of the counit for L JRL \dashv_J R.

Relative monads and comonads

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

  1. If L JRL {\,\,}_J\!\dashv R, then RLRL is a relative monad
  2. If L JRL \dashv_J R, then LRLR 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:CDR\colon C\to D has a left adjoint in the usual sense, then it also has a JJ-left adjoint for J=id DJ=id_D.


fully faithful functors

A functor F:ABF: A \to B is fully faithful iff it is representably fully faithful iff 1 A=Lift FF1_A = \mathop{Lift}_F F, and this lifting is absolute. Thus, FF fully faithful can be expressed as

1 FF 1 {\,\,}_F\!\dashv F
partially defined adjoints

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

L:CD L \colon C \to D

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

J:BD J \colon B \to D

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

L JR 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.


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

N F:BSet A op N_F \colon B \to \mathbf{Set}^{A^{\mathop{op}}}

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

ι a:y Aa(N FF)(a) \iota_a \colon y_A a \to (N_F F)(a)

at a:Aa' \colon A is

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

Note that when specialized to F=1 AF = 1_A, this reduces to the Yoneda lemma: first N 1 Ay AN_{1_A} \simeq y_A, and then 1 A=Lift y Ay A1_A = \mathop{Lift}_{y_A} y_A absolute in hom-isomorphism terms reads:

A(x,y)Set A op(y Ax,y Ay) 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 FF-nerves with respect to yoneda embeddings such that the 1-cell FF 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


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

Last revised on February 3, 2023 at 16:45:34. See the history of this page for a list of all contributions to it.