adjunction in nLab

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This page is about adjunctions in general 2-categories. Specifically for the common case of adjunctions in Cat see at adjoint functors. For the notion of “adjunction of a set to a field” in field theory, see field extension.

Context

2-Category theory

2-category theory

Definitions

Transfors between 2-categories

Morphisms in 2-categories

Structures in 2-categories

Limits in 2-categories

Structures on 2-categories

Relation to monads

Examples

Idea

A pair of 1-morphisms in a 2-category form an adjunction if they are dual to each other (Lambek (1982), cf. here) in a precise sense.

There are two archetypical classes of examples:

If $A$ is a monoidal category and $\mathbf{B}A$ denotes the one-object 2-category whose single hom-category is $A$ (the delooping of $A$ ), then the notion of adjoint morphisms in $\mathbf{B}A$ coincides precisely with the notion of dual objects in $A$ , subsuming, in turn, classical examples such as dual vector spaces in the case that $A =$ FinDimVect.
Adjunctions in the 2-category Cat of categories are adjoint functors.
- Notice that essentially everything that makes category theory nontrivial and interesting, beyond groupoid-theory, is governed by the concept of adjoint functors. In particular universal constructions such as limits and colimits, Kan extensions, (co)ends are examples of adjunctions in Cat.
- Similarly, for $V$ any cosmos for enrichment, adjunctions in the 2-category VCat of $V$ -enriched categories are equivalently enriched adjoint functors. Already in simple cases such $V =$ truth values this subsumes classical concepts such as that of Galois connections.
- Remarkably, even adjunctions in the homotopy 2-category of $(\infty,1)$ -categories are equivalent to adjoint $\infty$ -functors, see the examples below.
These classes of examples make adjunctions a key notion in formal category theory.

Finally, the notion of adjunction may usefully be thought of as a generalization of the notion of equivalence in a 2-category: an adjoint 1-morphism need not be invertible (even up to 2-isomorphism) but it does have, in a precise sense, a left inverse from below or a right inverse from above.

If an adjoint 1-morphisms happens to be a genuine equivalence in a 2-category, then the adjunction is called an adjoint equivalence.

Definition

Properties

Relation to monads

See at monad – Relation between adjunctions and monads.

Examples

Adjoint functors

See also the proof here at adjoint functor.

Proof

Suppose given functors $L \,\colon\, C \to D$ , $R: D \to C$ and the structure of a pair of adjoint functors in the form of a natural isomorphism of hom-sets (here)

$\Psi_{c, d} \;\colon\; \hom_D(L(c), d) \cong \hom_C(c, R(d))$

Now the idea is that, in the spirit of the (proof of the) Yoneda lemma, we would like $\Psi$ to be determined by what it does to identity morphisms. With that in mind, define the adjunction unit $\eta \colon 1_C \to R L$ by the formula $\eta_c = \Psi_{c, L(c)}(1_{L(c)})$ . Dually, define the counit $\varepsilon \,\colon\, L R \to 1_D$ by the formula

$\varepsilon_d \,\coloneqq\, \Psi^{-1}_{R(d), d}(1_{R(d)}) \,.$

Then given $g \,\colon\, L(c) \to d$ , the claim is that

$\Psi_{c, d}(g) \,=\, (c \stackrel{\eta_c}{\to} R(L(c)) \stackrel{R(g)}{\to} R(d)) \,.$

This may be left as an exercise in the yoga of the Yoneda lemma, applied to $\hom_D(L(c), -) \to \hom_C(c, R(-))$ . By formal duality, given $f \,\colon\, c \to R(d)$ ,

$\Psi^{-1}_{c, d}(f) = (L(c) \stackrel{L(f)}{\to} L(R(d)) \stackrel{\varepsilon_d}{\to} d) \,.$

(We spell out the Yoneda-lemma proof of this dual form below.)

Finally, these operations should obviously be mutually inverse, but that can again be entirely encapsulated Yoneda-wise in terms of the effect on identity maps. Thus, if $\eta_c \coloneqq \Psi_{c, L(c)}(1_{L(c)})$ , via the recipe just given for $\Psi^{-1}$ we recover

$1_{L(c)} = (L(c) \stackrel{L(\eta_c)}{\to} L R L(c) \stackrel{\varepsilon_{L(c)}}{\to} L(c))$

and this is one of the famous triangle identities: $1_L = (L \stackrel{L \eta}{\to} L R L \stackrel{\varepsilon L}{\to} L)$ . Here, juxtaposition of functors and natural transformations denotes neither functor application, nor vertical composition, nor horizontal composition, but whiskering. By duality, we have the other triangle identity $1_R = (R \stackrel{\eta R}{\to} R L R \stackrel{R \varepsilon}{\to} R)$ . These two triangular equations are enough to guarantee that the recipes for $\Psi$ and $\Psi^{-1}$ indeed yield mutual inverses.

In conclusion it is perfectly sufficient to define an adjoint pair of functors in $Cat$ as given by unit and counit transformations $\eta: 1_C \to R L$ , $\varepsilon: L R \to 1_D$ , satisfying the triangle identities above.

Proof

(Yoneda-lemma argument)
We claim that $\Psi^{-1}_{c, d} \,\colon\, \hom_C(c, R(d)) \to \hom_D(L(c), d)$ can be defined by the formula

$\Psi^{-1}_{c, d}(f: c \to R(d)) = (L(c) \stackrel{L(f)}{\to} L(R(d)) \stackrel{\varepsilon_d}{\to} d)$

where $\varepsilon_d \coloneqq \Psi^{-1}_{R(d), d}(1_{R(d)})$ . This is by appeal to the proof of the Yoneda lemma applied to the transformation

$\Psi^{-1}_{-, d}: \hom_C(-, R(d)) \to \hom_D(L(-), d)$

For the naturality of $\Psi^{-1}$ in the argument $(-)$ would imply that given $f: c \to R(d)$ , we have a commutative square

$\array{ \hom_C(R(d), R(d)) & \stackrel{\Psi^{-1}_{R(d), d}}{\to} & \hom_D(L(R(d)), d) \\ \hom_C(f, R(d)) \big\downarrow & & \big\downarrow \hom_D(L(f), d) \\ \hom_C(c, R(d)) & \underset{\Psi^{-1}_{c, d}}{\to} & \hom_D(L(c), d) }$

Chasing the element $1_{R(d)}$ down and then across, we get $f: c \to R(d)$ and then $\Psi^{-1}_{c, d}(f)$ . Chasing across and then down, we get $\varepsilon_d$ and then $\varepsilon_d \circ L(f)$ . This completes the verification of the claim.

Enriched adjoint functors

Similarly one sees:

Adjoint $(\infty,1)$ -functors

(Riehl & Verity 2015, Rem. 4.4.5; Riehl & Verity 2022, Prop. F.5.6; see there for more).

References

Adjunctions in 2-categories were introduced (together with the notion of strict 2-categories itself) in:

Jean-Marie Maranda, pp. 762 in: Formal categories, Canadian Journal of Mathematics 17 (1965) 758-801 [doi:10.4153/CJM-1965-076-0, pdf]

Though see also the following, which uses more modern terminology:

Max Kelly, §2 in: Adjunction for enriched categories, in: Reports of the Midwest Category Seminar III, Lecture Notes in Mathematics 106, Springer (1969) [doi:10.1007/BFb0059145]

A thorough 2-categorical account is contained in:

Claude Auderset?, Adjonctions et monades au niveau des $2$ -catégories, Cahiers de topologie et géométrie différentielle 15.1 (1974): 3-20.
Stephen Schanuel and Ross Street, The free adjunction, Cahiers de topologie et géométrie différentielle catégoriques 27.1 (1986): 81-8

Review:

Saunders MacLane, §XII.4 of: Categories for the Working Mathematician, Graduate Texts in Mathematics 5 Springer (second ed. 1997) [doi:10.1007/978-1-4757-4721-8]
Steve Lack, §2.1 in: A 2-categories companion, in: Towards Higher Categories, The IMA Volumes in Mathematics and its Applications 152 Springer (2010) [arXiv:math.CT/0702535, doi:10.1007/978-1-4419-1524-5_4]
Niles Johnson, Donald Yau, Chapter 6 of: 2-Dimensional Categories, Oxford University Press 2021 (arXiv:2002.06055, doi:10.1093/oso/9780198871378.001.0001)

Fr the special case of adjoint functors see any text on category theory (and see the references at adjoint functor), for instance:

Francis Borceux, Vol 1, Section 3 of Handbook of Categorical Algebra
geometry of physics – categories and toposes – Adjunctions

For some early history and illustrative examples see

Joachim Lambek, The Influence of Heraclitus on Modern Mathematics, In Scientific Philosophy Today: Essays in Honor of Mario Bunge, edited by Joseph Agassi and Robert S Cohen, 111–21. Boston: D. Reidel Publishing Co. (1982) (doi:10.1007/978-94-009-8462-2_6)

(more along these lines at objective logic).

The fundamental role of adjoint functors in logic/type theory originates with the observaiton that substitution forms an adjoint triple with existential quantification and universal quantification:

William Lawvere, Adjointness in Foundations, (tac:16), Dialectica 23 (1969), 281-296
William Lawvere, Quantifiers and sheaves, Actes, Congrès intern, math., 1970. Tome 1, p. 329 à 334 (pdf)

Adjunctions in programming languages (though mainly again just adjoint functors):

Ralf Hinze, Generic Programming with Adjunctions, In: J. Gibbons (ed.) Generic and Indexed Programming Lecture Notes in Computer Science, vol 7470. Springer 2012 (pdf, slides doi:10.1007/978-3-642-32202-0_2)
Jeremy Gibbons, Fritz Henglein, Ralf Hinze, Nicolas Wu, Relational Algebra by Way of Adjunctions, Proceedings of the ACM on Programming Languages archive Volume 2 Issue ICFP, September 2018 Article No. 86 (pdf, doi:10.1145/3236781)

adjunction in nLab

Context

2-Category theory

Contents

Idea

Definition

Properties

Relation to monads

Examples

Adjoint functors

Proof

Proof

Enriched adjoint functors

Adjoint (∞,1)(\infty,1)-functors

References

Adjoint $(\infty,1)$ -functors