nLab relative adjoint functor

Relative adjoint functors

Relative adjoint functors

Idea

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.

Definition

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.

Properties

asymmetry

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)

and

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.

Examples

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.

nerves

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

References

  • 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

Last revised on June 2, 2022 at 11:17:35. See the history of this page for a list of all contributions to it.