nLab relative adjoint functor

Relative adjoint functors

Relative adjoint functors


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


Via hom-isomorphism

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

R:CD R \colon C \longrightarrow D

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

L:BC L \colon B \longrightarrow C

and a natural isomorphism of the form

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

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

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

One writes

  • 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 .




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

Last revised on February 28, 2024 at 16:16:46. See the history of this page for a list of all contributions to it.