adjoint modality

[[!redirects adjoint cylinder]]

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### Context
#### Modalities, Closure and Reflection
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[[!include modalities - contents]]
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#### Duality
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[[!include duality - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The concept of [[adjunction]] as such expresses a [[duality]]. The stronger concept of an _adjoint cylinder_ or _adjoint [[modality]]_ is specifically an [[adjunction]] between [[idempotent monad|idempotent (co-)monads]] and is meant to express specifically a _duality between opposites_.

In terms of the corresponding [[adjoint triple]] of [[reflective subcategory|(co-)reflections]] and [[localizations]] the concept was suggested in ([Lawvere 91, p. 7](#Lawvere91), [Lawvere 94, p. 11](#Lawvere94)) to capture the phenomena of "Unity and Identity of Opposites" as they appear informally in [[Georg Hegel]]'s _[[Science of Logic]]_.
(One might therefore say the notion is meant to capture the idea of "dialectic", though there is some debate as to whether Hegel's  somewhat mythical "creation out of [[paradox]]" should really go by this term, see [this Wikipeda entry](#Wikipedia) ).

In terms of [[adjoint pairs]] of [[modal operators]] in the context of [[modal logic]]/[[modal type theory]] and thought of as [[Galois connections]] the concept appears in ([Reyes-Zolfaghari 91](#ReyesZolfaghari91)). Further developments along these lines include ([DJK 14](#DJK14)).

## Definition

In ([Lawvere 94](#Lawvere94)) an _adjoint cylinder_ is defined to be an [[adjoint triple]] such that the outer two adjoints are [[full and faithful functors]]. This means equivalently that the  induced [[adjoint pair]] on the codomain of these inclusions consists of an [[idempotent monad|idempotent]] [[monad]] and [[comonad]] ([[adjoint monads]]). One may also consider the situation where the middle functor of the adjoint triple is fully faithful, hence one has adjoint [[modal operators]]
either of the form

$$
  U \;\colon\; modality \dashv comodality
  \,,
$$

or of the form

$$
  U \;\colon\; comodality \dashv modality
  \,.
$$

A category equipped with an adjoint modality of the second form is called a _[[category of being]]_ in ([Lawvere 91](#Lawvere91)). If the category is a [[topos]] then this is also called a _[[level of a topos]]_.

Given any such, we may say that the "unity" expressed by the two opposites is exhibited by the canonical [[natural transformation]]

$$
  U X
  \;\colon\;
  \array{
    comodal X &\longrightarrow& X &\longrightarrow& modal X
    \\
    opposite\;1 && unity && opposite\;2
  }
$$

which is the composite of the [[counit of a comonad|counit of the comodality]] and the [[unit of a monad|unit of the modality]] ([Lawvere-Rosebrugh 03, p. 245](#LawvereRosebrugh03)).

If for two [[level of a topos|levels]] the next one contains the [[modal types]] of the [[idempotent comonad]] of the former, then [[Lawvere]] speaks of "[[Aufhebung]]" (see there for more).

One can consider longer sequences of such adjoints of co/modalities, but the longer they get, the less likely they are to be non-trivial. The longest that still has good nontrivial models seems to be [[adjoint triples]] of modalities. Of these there is then similarly either the form

$$
  modality \dashv comodality \dashv modality
$$

(the "Yin triple") as for instance in the definition of _[[cohesion]]_ and

$$
  comodality \dashv modality \dashv comodality
$$

(the "Yang triple") as for instance in the definition of _[[differential cohesion]]_.

Since adjoint triples are equivalently [[adjunctions]] of [[adjunctions]] ([Licata-Shulman, section 5](#LicataShulman)), it is suggestive to denote these as

$$
  \array{
    \lozenge &\dashv& \bigcirc
    \\
    \bot && \bot
    \\
    \bigcirc &\dashv& \Box
  }
$$

## Examples

### Simple illustrative examples
 {#SimpleIllustrativeExamples}

The following simple illustrative example of an adjunction of the form $\Box \dashv \bigcirc$ has been suggested in ([Lawvere 00](#Lawvere2000)).

+-- {: .num_example #EvenAndOddIntegersAdjointModality}
###### Example
**([[even number|even]] and [[even number|odd]] [[integers]])**

Regard the [[integers]] as a [[preordered set]] $(\mathbb{Z}, \leq)$ in the canonical way, and thus as a [[thin category]].

Consider the [[full subcategory]] inclusions

$$
  \array{
    (\mathbb{Z}, \leq )
    & \overset{even}{\hookrightarrow}&
    (\mathbb{Z},\leq)
    \\
    n &\mapsto & 2 n
  }
  \phantom{AAAAA}
  \array{
    (\mathbb{Z}, \leq )
    & \overset{odd}{\hookrightarrow}&
    (\mathbb{Z},\leq)
    \\
    n &\mapsto & 2 n + 1
  }
$$

of the [[even number|even]] and the [[even number|odd]] [[integers]], as well as the functor

$$
  \array{
    (\mathbb{Z}, \leq )
    & \overset{\lfloor-/2\rfloor}{\longrightarrow}&
    (\mathbb{Z},\leq)
    \\
    n &\mapsto& \lfloor n/2 \rfloor
  }
$$

which sends any $n$ to the [[floor]] $\lfloor n/2 \rfloor$ of $n/2$, hence to the largest integer which is smaller or equal to the [[rational number]] $n/2$.

These functors form an [[adjoint triple]]

$$
  even \;\dashv\; \lfloor -/2 \rfloor \;\dashv\; odd
$$

and hence induce an adjoint modality

$$
  Even \;\dashv\; Odd
$$

on $(\mathbb{Z}, \leq)$ with

1. $Even \coloneqq 2 \lfloor -/2 \rfloor$ sending any integer to its "even [[floor]] value"

1. $Odd \coloneqq 2 \lfloor -/2 \rfloor + 1$ sending any integer to its "odd [[ceiling]] value".

=--

+-- {: .proof}
###### Proof

Observe that for all $n \in \mathbb{Z}$ we have

$$
  2 \lfloor n/2 \rfloor
    \overset{ \epsilon_n }{\leq}
  n
    \overset{ \eta_n }{\leq}
  2 \lfloor n/2 \rfloor  + 1
  \,,
$$

where the first inequality is an equality precisely if $n$ is even, while the second is an equality precisely if $n$ is odd. Hence this provides a candidate [[unit of an adjunction|unit]] $\eta$ and [[counit of an adjunction|counit]] $\epsilon$.

Hence by [this characterization](adjoint+functor#UniversalArrow) of [[adjoint functors]]

1. the adjunction $\lfloor -/2 \rfloor \dashv odd$ is equivalent to the condition that

   for every $n \leq 2 k + 1$ we have $2 \lfloor n/2 \rfloor + 1  \leq 2 k + 1$;

1. the adjunction $even \dashv \lfloor -/2 \rfloor $ is equivalent to the condition that

   for every $2k \leq n$ we have $2k \leq 2 \lfloor n/2 \rfloor $,

which is readily seen to be the case

=--

In the same vein there is an example for an adjunction of the form $\bigcirc \dashv \Box$:

+-- {: .num_example }
###### Example

Consider the inclusion $\iota \colon (\mathbb{Z}, \lt) \hookrightarrow (\mathbb{R}, \lt)$ of the [[integers]] into the [[real numbers]], both regarded as [[linear orders]]. This inclusion has a [[left adjoint]] given by [[ceiling]] and a right adjoint given by [[floor]].

The composite $Ceiling \coloneqq \iota ceiling$ is an [[idempotent monad]] and the composite $Floor \coloneqq \iota floor$ is an [[idempotent comonad]] on $\mathbb{R}$. Both express a _moment of integrality_ in an real number, but in opposite ways, each real number $x\in \mathbb{R}$ sits in between its floor and celling

$$
  Floor(x) \leq x \leq Ceiling(x)
  \,.
$$

Indeed the moments form an [[adjunction]]

$$
  Ceiling \dashv Floor
  \,.
$$

=--

### Werden : Nichts $\dashv$ Sein
 {#Werden}

For $\mathbf{H}$ a [[topos]]/[[(∞,1)-topos]] consider the "initial topos", the [[terminal category]] $\ast \simeq Sh(\emptyset)$ ([[category of sheaves]] on the empty site).

There is then an [[adjoint triple]]

$$
  \mathbf{H}
   \stackrel
     {\overset{\vdash \varnothing}{\longleftarrow}}
     {\stackrel{\overset{}{\longrightarrow}}
     {\underset{\vdash \ast}{\longleftarrow}}}
  \ast
$$

given by including the [[initial object]] $\varnothing$ and the [[terminal object]] $\ast$ into $\mathbf{H}$, respectively.

In the [[type theory]] of $\mathbf{H}$ this corresponds to the [[adjoint pair]] of [[modalities]]

$$
  \varnothing \dashv \ast
$$

which are [[constant functor|constant]] on the [[initial object]]/[[terminal object]], respectively.

The induced [[unit of an adjunction]] transformation is

$$
  \array{
    \varnothing \longrightarrow X \longrightarrow \ast
  }
$$

hence the unique factorization of the unique function $\varnothing \longrightarrow \ast$ through any other [[type]].

Looking through ([Hegel 1812, vol 1, book 1, section 1, chapter 1](#Hegel1812)) one might call $\emptyset$ "nothing", call $\ast$ "being" and then call this unity of opposites "becoming". In particular in §174 of _[[Science of Logic]]_ it says

> there is nothing which is not an intermediate state between being and nothing

which seems to be well-captured by the above unity transformation.

### Quantity : discreteness $\dashv$ continuity
 {#Mengen}

The adjoint modality in a [[local topos]] is that given by
[[flat modality]] $\dashv$ [[sharp modality]]

$$
  \flat \dashv \sharp
  \,.
$$

Capturing [[discrete objects]]/[[codiscrete objects]].

The corresponding unity transformation is

$$
    \flat X \longrightarrow X \longrightarrow \sharp X
$$

According to ([Lawvere 94, p. 6](#Lawvere94)) this unity captures the duality that in a [[set]] all [[elements]] are distinct and yet indistinguishable, an apparent [[paradox]] that may be traced back to [[Georg Cantor]].

Looking through Hegel's [[Science of Logic]] at _[On discreteness and repulsion](#Science+of+Logic#OnDiscretenessAndRepulsion)_ one can see that matches with what Hegel calls

> (par 398) Quantity is the unity of these moments of continuity and discreteness

$$
  \array{
    \flat X &\longrightarrow& X &\longrightarrow& \sharp X
    \\
    {moment\;of \atop discreteness} && && {moment\;of \atop continuity}
  }
$$

### Continuum : repulsion $\dashv$ cohesion
 {#ContinuumRepulsionCohesion}

For $\mathbf{H}$ a [[cohesive topos]]/[[cohesive (∞,1)-topos]]
the [[shape modality]] $\dashv$ [[flat modality]] constitute an adjoint cylinder

$$
  ʃ \dashv \flat
  \,.
$$

The corresponding unity-transformation is the [points-to-pieces transform](cohesive%20topos#CanonicalComparison)

$$
  \array{
    \flat X \longrightarrow X \longrightarrow ʃ X
  }
$$

Looking through ([Hegel 1812, vol 1, book 1, section 2, chapter 1](#Hegel1812)) one might call $\flat$ "repulsion", call $ʃ$ "attraction"/"[[cohesion]]" and then call this unity of opposites "[[continuum]]". Indeed, by the discussion at _[[cohesive topos]]_, this does quite well capture the geometric notion of continuum geometry.

### Infinitesimal Continuum : infin. repulsion $\dashv$ infinit. cohesion
 {#ContinuumRepulsionCohesion}

For $\mathbf{H}$ equipped moreover with [[differential cohesion]],
there is the [[infinitesimal object|infinitesimal]] version of
[[shape modality]] $\dashv$ [[flat modality]]  namely the adjoint modality

[[infinitesimal shape modality]] $\dashv$ [[infinitesimal flat modality]]

$$
  \Im \dashv \&
  \,.
$$

The corresponding unity-transformation is the

$$
  \array{
    \& X \longrightarrow X \longrightarrow \Im X
  }
$$

maps from the [[coefficients]] for [[crystalline cohomology]] to the [[de Rham space]] of types $X$, where all infinitesimal neighbour points are identified.

In view of the above the unity exhibited here is clearly to be called the "infinitesimal continuum".

### Cohesive sets

The combination of the above two examples of [Continuum](#ContinuumRepulsionCohesion) and [Quantity](#Mengen) is an [[adjoint triple]] of [[modalities]]

$$
  ʃ \;\dashv\; \flat \;\dashv\; \sharp
$$

[[shape modality]] $\dashv$ [[flat modality]] $\dashv$ [[sharp modality]]

characteristic of a [[cohesive topos]].

### Skeleta and Co-Skeleta

[[simplicial skeleton]] $\dashv$ [[simplicial coskeleton]]

### Formal completion $\dashv$ Torsion approximation
 {#FormalCompletionAndTorsionApproximation}

For $A$ a [[commutative ring]] or more generally an [[E-∞ ring]] and $\mathfrak{a}\subset \pi_0 A$ a suitable ideal, then $\mathfrak{a}$-[[adic completion]] and $\mathfrak{a}$-[[torsion approximation]] form an adjoint modality on $A MMod$ the [[stable (∞,1)-category|stable]] [[(∞,1)-category of ∞-modules]] $A Mod_\infty$ over $A$.

  ($\mathfrak{a}$-adic completion) $\dashv$ ($\mathfrak{a}$-torsion approximation)

+-- {: .num_prop #CompletionTorsionAdjointModalityForModuleSpectra}
###### Proposition

Let $A$ be an [[E-∞ ring]] and let $\mathfrak{a} \subset \pi_0 A$ be a [[generators and relations|finitely generated]] ideal in its underlying [[commutative ring]].

Then there is an [[adjoint triple]] of [[adjoint (∞,1)-functors]]

$$
  \array{
    \underoverset{
      A Mod_{\mathfrak{a}comp}^{op}}
    {A Mod_{\mathfrak{a}tors}^{op}}
    {\simeq}
    &\stackrel{\overset{\Pi_{\mathfrak{a}}}{\longleftarrow}}{\stackrel{\hookrightarrow}{\underset{\flat_{\mathfrak{a}}}{\longleftarrow}}}&
    A Mod^{op}
  }
$$

where

* $A Mod$ is the [[stable (∞,1)-category|stable]] [[(∞,1)-category of modules]], i.e. of [[∞-modules]] over $A$;

* $A Mod_{\mathfrak{a}tor}$ and $A Mod_{\mathfrak{a} comp}$ are the [[full sub-(∞,1)-categories]] of $\mathfrak{a}$-[[torsion approximation|torsion]] and of $\mathfrak{a}$-[[completion of a module|complete]] $A$-[[∞-modules]], respectively;

* $(-)^{op}$ denotes the [[opposite (∞,1)-category]];

* the [[equivalence of (∞,1)-categories]] on the left is induced by the restriction of $\Pi_{\mathfrak{a}}$ and $\flat_{\mathfrak{a}}$.

=--

+-- {: .proof}
###### Proof

This is effectively the content of ([Lurie "Proper morphisms", section 4](#LurieProper)):

* the existence of $\Pi_{\mathfrak{a}}$ is corollary 4.1.16 and remark 4.1.17

* the existence of $\flat_{\mathfrak{a}}$ is lemma 4.2.2 there;

* the equivalence of sub-$\infty$-categories is proposition 4.2.5 there.

=--

See at _[[fracture theorem]]_ for more.

### Fermions and supergeometry

On [[super smooth infinity-groupoids]] there is an adjoint modality
deriving from the [adjoint triple relating plain algebra and superalgebra](super+algebra#AdjointsToInclusionOfPlainAlgebra).
The [[right adjoint]] deserves to be called the [[bosonic modality]] ("[[body]]"), hence its [[left adjoint]] the [[fermionic modality]].
This expresses the presence of [[supergeometry]]/[[fermions]], hence ultimately the [[Pauli exclusion principle]]. Following [PN§290](Science+of+Logic#290) this unity of opposites might hence be called "asunderness".

### Totally distributive categories

For $\mathcal{K}$ a [[totally distributive category]]
it induces on its [[category of presheaves]] an adjoint modality whose [[right adjoint]] is the [[Yoneda embedding]] $Y$ postcomposed with its [[left adjoint]] $X$.

### Recollements

See at _[[recollement]]_.

## Related concepts

* [[Aufhebung]]

* [[modal type theory]]

* [[adjoint logic]]

* [[category of being]]

* [[Galois connection]]

## References

### Traditional

The concept of _dialectical reasoning_ is usually attributed to

* [[Plato]], second part of the _[[Parmenides dialogue]]_ .

See

* [[Georg Hegel]], _[[Lectures on the History of Philosophy]] -- [Plato -- Dialectic -- Parmenides dialogue](Lectures+on+the+History+of+Philosophy#ParmenidesDialogue)_

Hegel in his _[History of Philosophy](https://www.marxists.org/reference/archive/hegel/works/hp/hpeleatics.htm)_ writes that dialectic begins with [[Zeno]] (one of the characters in that dialogue).

This is much amplified and expanded in

* {#Hegel1812} [[Georg Hegel]], _[[Science of Logic]]_, 1812

The origins of its proposed formalization in [[category theory]] are recalled in

* {#Lambek82} [[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)

See also

* {#Wandschneider99} [[Dieter Wandschneider]], _Dialektik als Letztbegründung der Logik_, in Koreanische Hegelgesellschaft (ed.), _Festschrift für Sok-Zin Lim_ Seoul 1999, 255–278 ([pdf](http://www.philosophie.rwth-aachen.de/global/show_document.asp?id=aaaaaaaaaabpltw))

* {#Wikipedia} Wikipedia, _[Hegelian dialectic](http://en.wikipedia.org/wiki/Hegelian_dialectic)_

### In terms of adjoint triples of (co-)reflections and localizations

Conceived of in terms of [[adjoint triples]] of [[reflective subcategory|(co-)reflections]] and [[localization of a category|localizations]] the concept appears in

* {#Lawvere91} [[William Lawvere]], _[[Some Thoughts on the Future of Category Theory]]_ in A. Carboni, M. Pedicchio, G. Rosolini, _Category Theory_  , [[Como|Proceedings of the International Conference held in Como]], Lecture Notes in Mathematics 1488, Springer (1991)

* {#Lawvere94} [[William Lawvere]], _[[Cohesive Toposes and Cantor's "lauter Einsen"]]_, Philosophia Mathematica (3) Vol. 2 (1994), pp. 5-15. ([[LawvereCohesiveToposes.pdf:file]])

* {#Lawvere94b} [[William Lawvere]], _[[Tools for the advancement of objective logic]]: closed categories and toposes_, in J. Macnamara and [[Gonzalo Reyes]] (Eds.), _The Logical Foundations of Cognition_, Oxford University Press 1993 (Proceedings of the Febr. 1991 Vancouver Conference "Logic and Cognition"),
pages 43-56, 1994.

* {#Lawvere96} [[William Lawvere]],  _[[Unity and Identity of Opposites in Calculus and Physics]]_,  Proceedings of ECCT 1994 Tours Conference,  Applied Categorical Structures, 4: 167-174 Kluwer Academic Publishers, (1996).

* {#Lawvere00} [[F. W. Lawvere]], _Adjoint Cylinders_, message to catlist November 2000. ([link](https://www.mta.ca/~cat-dist/archive/2000/00-11))

* {#LawvereRosebrugh03} [[William Lawvere]], [[Robert Rosebrugh]], p. 245 of: _[[Sets for Mathematics]]_,  Cambridge UP  2003 ([doi:10.1017/CBO9780511755460](https://doi.org/10.1017/CBO9780511755460), [book homepage](http://www.mta.ca/~rrosebru/setsformath/), [GoogleBooks](http://books.google.de/books?id=h3_7aZz9ZMoC&pg=PP1&dq=sets+for+mathematics), [pdf](http://patryshev.com/books/Sets%20for%20Mathematics.pdf))

### In terms of adjoint pairs of modal operators

In terms of [[adjoint pairs]] of [[modal operators]] and hence of [[Galois connections]], the concept appears in

* {#ReyesZolfaghari91} [[Gonzalo  Reyes]], H. Zolfaghari, _Topos-theoretic approaches to modality_, Lecture Notes in Mathematics 1488 (1991), 359-378.

* {#Reyes91} [[Gonzalo Reyes]], _A topos-theoretic approach to reference and modality_, Notre Dame J. Formal Logic Volume 32, Number 3 (1991), 359-391 ([Euclid](http://projecteuclid.org/euclid.ndjfl/1093635834))

with further developments in

* M. Sadrzadeh, R. Dyckho, _Positive logic with adjoint modalities: Proof
theory, semantics and reasoning about information_, Electronic Notes in Theoretical
Computer Science 249, 451-470, 2009, in _Proceedings of the 25th Conference on
Mathematical Foundations of Programming Semantics_ (MFPS 2009).

* {#Hermida10} [[Claudio Hermida]], section 3.3. of _A categorical outlook on relational modalities and simulations_, 2010 ([pdf](http://maggie.cs.queensu.ca/chermida/papers/sat-sim-IandC.pdf))

* {#DJK14} Wojciech Dzik, Jouni Järvinen, Michiro Kondo, _Characterising intermediate tense logics in terms of Galois connections_ ([arXiv:1401.7646](http://arxiv.org/abs/1401.7646))

Formalization specifically in [[modal type theory]] is in

* {#LicataShulman} [[Dan Licata]], [[Mike Shulman]], _Adjoint logic with a 2-category of modes_ ([pdf](http://dlicata.web.wesleyan.edu/pubs/ls15adjoint/ls15adjoint.pdf))

For an overview of the role of adjunctions in modal logic see:

* [[Matías Menni|M. Menni]], C. Smith, _Modes of Adjointness_ , J. Philos. Logic **43** no.3-4 (2014) pp.365-391.

### Examples

* {#LurieProper} [[Jacob Lurie]], section 4 of _[[Proper Morphisms, Completions, and the Grothendieck Existence Theorem]]_

[[!redirects adjoint cylinders]]

[[!redirects adjoint modality]]
[[!redirects adjoint modalities]]

[[!redirects opposite]]
[[!redirects opposites]]

[[!redirects unity of opposites]]
[[!redirects unities of opposites]]
[[!redirects unity and identity of opposites]]

[[!redirects dialectic]]
[[!redirects dialectics]]

[[!redirects duality of opposites]]
[[!redirects dualities of opposites]]


adjoint monad

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A [[monad]] $(T,\mu,\eta)$ is [[adjoint functor|adjoint]] to a [[comonad]] $(G,\delta,\epsilon)$, if its underlying endofunctor $T$ is _[[left adjoint]]_ to the underlying [[1-morphism]] $G$ of the comonad, and $\delta$ and $\epsilon$ are conjugate/adjoint/[[mate]] 2-cells to $\mu$ and $\eta$ in the sense explained below.

## Construction

In fact given a [[monad]] $\mathbf{T} = (T,\mu^T,\eta^T)$ which has a _[[right adjoint]]_ $G$, automatically $G$ is a part of a comonad $\mathbf{G} = (G,\delta^G,\epsilon^G)$ where $\delta^G$ and $\epsilon^G$ are in some sense dual to $\mu^T$ and $\eta^T$.

Thus there is a bijective correspondence between monads having a right adjoint and comonads having a left adjoint (what [[Alexander Rosenberg]] calls **duality**). This is a little more than a consequence of two general facts:

1. If $T\dashv G$ then $T^k \dashv G^k$ for every [[natural number]] $k$.

2. Given two [[adjunctions]] $S\dashv T$ and $S'\dashv T'$ where $S,S': B\to A$, then there is a [[bijection]] between the [[natural transformations]] $\phi:S'\Rightarrow S$ and natural transformations $\psi:T\Rightarrow T'$ such that

$$
\array{
  A (S,-) &\to& B(-,T)
  \\
  {}^{\mathllap{A(\phi,-)}}\downarrow &&\downarrow {}^{\mathrlap{B(-,\psi)}}
  \\
  A(S',-)&\to & B(-,T')
}
$$

where the horizontal arrows are the natural bijections given by the adjunctions. [Eilenberg & Moore 1965](#EilenbergMoore65) would write $\phi\dashv\psi$ and talk about "adjointness for morphisms" (of functors), which is of course relative to the given adjunctions among functors. MacLane calls the correspondence *conjugation* (p 99-102 in *[[Categories for Working Mathematician]]*).  It is a special case, of a general construction of [[mates]].

If $\eta,\eta'$ and $\epsilon,\epsilon'$ are their
[[unit of an adjunction|unit]] and counit of course the upper arrow is $(S M\stackrel{f}\to N)\mapsto T f\circ \eta_M$ and the lower arrow $(S'M\stackrel{g}\to N)\mapsto T'g\circ\eta'_M$. Thus the condition renders as

$$T'(f\circ\phi_M)\circ\eta'_M = \psi_N\circ T f\circ\eta_M$$

or $T'f\circ T'\phi_M\circ\eta'_M = T'f\circ \psi_{SM}\circ\eta_M$. Given $\phi$, the uniqueness of $B(-,\psi)$ is clear from the above [[diagram]], as the horizontal arrows are [[isomorphism|invertible]]. $B(-,\psi)$ determines $\psi$, namely $\psi_N = B(-,\psi)(id_N)$. For the existence of $\psi$ (given $\phi$) satisfying the above equation, one proposes that $\psi$ is the composition
$\psi  = T'\epsilon \circ T'\phi T \circ \eta'T$, i.e.

$$
T\stackrel{\eta' T}\longrightarrow T'S' T\stackrel{T'\phi T}\longrightarrow T' S T \stackrel{T'\epsilon}\longrightarrow T'
$$
and checks that it works. The inverse is similarly given by the composition
$$
S'\stackrel{S'\eta}\longrightarrow S' T S\stackrel{S'\psi S}\longrightarrow S'T'S\stackrel{\epsilon' S}\longrightarrow S
$$

This correspondence now enables in our special case to dualize $\mu^T$ to $\delta^G$, and similarly unit to the counit.

## Examples

\begin{example}\label{AdjointMonadsInducedFromAdjointTriples}
**(adjoint monads induced from adjoint triples of adjoint functors)**
\linebreak
Every [[adjoint triple]] (of [[adjoint functors]]) $F^*\dashv F_* \dashv F^!$ [induces](monad#RelationBetweenAdjunctionsAndMonads) an [[adjoint pair]] $F_* F^*\dashv F_* F^!$. The endofunctor $F_* F^*$ is underlying a monad induced by the adjunction $F^*\dashv F_*$ and $F_* F^!$ is underlying a comonad induced by the adjuntion $F_*\dashv F^!$. This pair of a monad and a comonad are adjoint.
\end{example}

(See also at *[[adjoint modality]]*.)

## Properties

### General

\begin{proposition}
\label{IsomorphismOfEMCategories}
 Given an [[adjoint functor|adjoint pair]] of a [[monad]] and comonad

$$
  \lozenge \,\dashv\, \Box
$$

on some category $\mathcal{C}$, then there is an [[equivalence of categories|equivalence]] between their [[Eilenberg-Moore categories]] of [[algebra over a monad|algebras]] over $\lozenge$ and [[coalgebra over a comonad|coalgebras]] over $\Box$, compatible with their [[forgetful functors]] to $\mathcal{C}$:

\begin{tikzcd}[column sep=-1pt]
  \mathrm{EM}(\lozenge)
  \ar[dr, "{ U }"{swap}]
  \ar[rr, "{\sim}"]
  &&
  \mathrm{EM}(\Box)
  \ar[dl, "{ U }"]
  \\
  &
  \mathcal{C}
\end{tikzcd}
\end{proposition}
This is due to [Eilenberg & Moore 1965](#EilenbergMoore65), where it is implied by the last part of Prop. 3.3. In the more explicit form above the statement may be found in [MacLane & Moerdijk 1992, Thm. 2 on p. 249](#MacLaneMoerdijk92),

### General adjoint (co)algebras $End(A)$

Given a [[small category]] $A$, the [[endofunctor category]] $End(A)$ (of [[endofunctors]] and [[natural transformations]] between them, with [[vertical composition]] as [[composition]]) is [[monoidal category|monoidal]] with respect to the composition as the tensor product of objects (endofunctor) and Godement product ([[horizontal composition]]) as the [[tensor product]] of morphisms (natural transformations). Hence we can consider [[operads]] and [[algebras over operads]], as well as, dually, coalgebras over cooperads; or some other framework for general algebras and coalgebras (or even props).

In any case, given an adjunction $T\dashv G$, operations $T^n\to T$ dualize to cooperations $G\to G^n$, and more generally multioperations $T^k\to T^l$ dualize to the multioperations $G^l\to G^k$. We would like to sketch the proof that the identities for operations on $T$, correspond to the identities on cooperations on $G$ (and more generally there is a duality among the identities for multioperations). This is essentially the consequence of

__Lemma.__ ([[Zoran Škoda|Zoran]]) Given the adjunction $T\dashv G$ with unit $\eta$ and counit $\epsilon$, and the sequence

$$
T^k \stackrel{\alpha}\longrightarrow T^l\stackrel\beta\longrightarrow T^s
$$

the composition $\alpha^*\circ\beta^*$ of the dual (in the above sense) sequence

$$
G^k \stackrel{\alpha^*}\longleftarrow G^l\stackrel{\beta^*}\longleftarrow G^s
$$

equal to the dual $(\beta\circ\alpha)^*$ of $\beta\circ\alpha$,

*Proof*. [[Mike Shulman]] notices that this is a special case of known contravariant functoriality of [[mate]]s, but here is a direct proof.

We need to prove that the composition

$$
G^s\stackrel{\eta_l G^s}\to G^l T^l G^s\stackrel{G^l\beta G^s}\to G^l T^s G^s\stackrel{G^l \epsilon_s}\to G^l\stackrel{\eta_k G^l}\to G^k T^k G^l\stackrel{G^k\alpha G^l}\to G^k T^l G^l\stackrel{G^k\epsilon_l}\to G^k
$$

equals the composition

$$
G^s\stackrel{\eta_k G^l}\to G^k T^k G^s\stackrel{G^k\alpha G^l}\to G^k T^l G^s\stackrel{G^k\beta G^s}\to G^k T^s G^s\stackrel{G^k\epsilon_s}\to G^k.
$$

Note that in the two compositions there is an opposite order between the expressions involving $\alpha$ and those involving $\beta$. But anyway, their equality reduces to a naturality calculation (which in particular exchanges the order of $\alpha$ and $\beta$ in effect):

$$\array{
G^s &\stackrel{\eta_l G^s}\to & G^l T^l G^s &\stackrel{G^l\beta G^s}\to & G^l T^s G^s&\stackrel{G^l \epsilon_s}\longrightarrow& G^l\\
\eta^k G^s\downarrow &&\downarrow \eta_k G^l T^l G^s&&\downarrow \eta_k G^l T^s G^s&& \downarrow \eta_k G^l \\
G^k T^k G^s &\stackrel{G^k T^k\eta_l G^s}\longrightarrow &G^k T^k G^l T^l G^s &\stackrel{G^k T^k G^l \beta G^s}\longrightarrow & G^k T^k G^l T^s G^s &\stackrel{G^k T^k G^l \epsilon_s}\longrightarrow & G^k T^k G^l\\
G^k \alpha G^s \downarrow && \downarrow G^k \alpha G^l T^l G^s&&\downarrow G^k \alpha G^k T^s G^s&&\downarrow G^k \alpha G^l\\
G^k T^l G^s &\stackrel{G^k T^l\eta_l G^s}\longrightarrow & G^k T^l G^l T^l G^s &\stackrel{G^k T^l G^l \beta G^s}\longrightarrow & G^k T^l G^l T^s G^s &\stackrel{G^k T^l G^l\epsilon_s}\longrightarrow & G^k T^l G^l\\
G^k\beta G^s \downarrow &&\downarrow{\rho} &&\downarrow G^k \epsilon_l T^s G^s &&\downarrow G^k\epsilon_l \\
G^k T^s G^s &=&G^k T^s G^l &=&G^k T^s G^s&\stackrel{G^k \epsilon_s}\longrightarrow& G^k
}$$

where $\rho := G^k \beta G^s \circ G^k \epsilon_l G^s = G^k\epsilon_l T^s G^s\circ G^k T^l G^l \beta G^s$. The commutativity of all small squares in the diagram is evident, except the lower left corner. This one follows by one of the triangle identities for the adjunction $T^l\dashv G^l$. Namely,

$$
G^k \beta G^s = G^k \beta G^s \circ (G^k \epsilon_l T^l G^s\circ G^k T^l \eta_l G^s) = \rho \circ G^k T^l \eta_l G^s
$$

## Examples

### Adjoint modalities

An [[adjoint modality]] is an example of a pair of adjoint monads.

## References

Discussion of adjoint monads originates with

* {#EilenbergMoore65} [[Samuel Eilenberg]], [[John Moore]], Section 3 in: _Adjoint functors and triples_, Illinois J. Math. **9** 3 (1965) 381-398 [[doi:10.1215/ijm/1256068141](https://doi.org/10.1215/ijm/1256068141)]

(there called "adjoint triples", sticking with the old term "triple" for "monad", a terminology that now clashes with the modern use of *[[adjoint triples]]* of [[adjoint functors]]).

Textbook account:

* {#MacLaneMoerdijk92} [[Saunders Mac Lane]], [[Ieke Moerdijk]], pp. 248 in: _[[Sheaves in Geometry and Logic|Sheaves in Geometry and Logic --- A First Introduction to Topos Theory]]_, Springer (1992) [[doi:10.1007/978-1-4612-0927-0](https://dx.doi.org/10.1007/978-1-4612-0927-0)]

Discussion in the context of [[ambidextrous adjunctions]] and [[Frobenius monads]]:

* {#Street04} [[Ross Street]], *Frobenius monads and pseudomonoids*, J. Math. Phys. **45** 3930 (2004) [[doi:10.1063/1.1788852](https://doi.org/10.1063/1.1788852)]

* {#Lauda05} [[Aaron Lauda]], *Frobenius algebras and ambidextrous adjunctions*, Theory and Applications of Categories **16** 4 (2006) 84-122 [[arXiv:math/0502550](http://arxiv.org/abs/math/0502550), [tac:16-04](http://www.tac.mta.ca/tac/volumes/16/4/16-04abs.html)]

[[!redirects adjoint monads]]
[[!redirects adjoint comonad]]
[[!redirects adjoint comonads]]


adjoint morphism

A [[morphism]] participating in an [[adjunction]].

[[!redirects adjoint morphisms]]


adjoint operator

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Functional analysis
+-- {: .hide}
[[!include functional analysis - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Definition

Let $A: H\to H$  be an [[unbounded operator]] on a [[Hilbert space]] $H$. An unbounded operator $A^*$ is its __[[adjoint operator|adjoint]]__ if

* $(A x|y) = (x|A^*y)$ for all $x\in dom(A)$ and $y\in dom(A^*)$; and

* every $B$ satisfying the above property for $A^*$ is a restriction of $A$.

On [[finite-dimensional Hilbert spaces]], adjoint operators always exists, in [[matrix]]-components with respect to any [[orthonormal linear basis]] given by passage to the [[complex conjugation|complex conjugate]] [[transpose matrix]].

On infinite-dimensional Hilbert spaces an adjoint operator does not need to exist, in general.

## History
 {#History}

Recounted by [MacLane 1988](#MacLane88), [p. 330](http://www.ams.org/publicoutreach/math-history/hmath1-maclane25.pdf#page=8):

> Two of [[John von Neumann|von Neummann]]'s papers on this topic [[[Hilbert spaces]]] had been accepted in the Mathematische Annalen, a journal of Springer Verlag. [[Marshall Stone]] had seen the manuscripts, and urged von Neumann to observe that his treatment of linear operators $T$ on a Hilbert space could be much more effective if he were to use the notion of an adjoing $T^ast$ to the linear transformation $T$ --- one for which the now familiar equation

> $\;\;\;\;\; \langle T a, b \rangle \;=\; \langle a, T^\ast b \rangle$

> would hold for all suitable $a$ and $b$. Von Neumann saw the point immediately, as was his wont, and wishes to withdraw the papers before publication. They were already set up in type; Springer finally agreed to cancel them on the condition that von Neumann write for them a book on the subject --- which he soon did [[1932](#vonNeumann1932)].

> This story (told to me by [[Marshall Stone]]) illustrates the important conceptual advance represented by the definition of adjoint operators. &lbrack...] I have written elsewhere [[1970](#MacLane70)] that it is a step toward the subsequent description of a [[functor]] $G$ [[right adjoint]] to a functor $F$, in terms of [a natural isomorphism](adjoint+functor#InTermsOfHomIsomorphism)

> $\;\;\;\;\; hom(F a, b) \;\simeq\; hom(a, G b)$

> between [[hom-sets]] in suitable [[categories]].

(Cf. discussion at [adjoint functor -- idea](adjoint+functor#Idea).)

## Related concepts

* [[self-adjoint operator]]

## References

The notion of adjoint operators is originally due to [[Marshall Stone]], see also the history section [above](#History), as recounted in

* {#MacLane70} [[Saunders MacLane]], *The Influence of M. H. Stone on the Origins of Category Theory*, in *Functional Analysis and Related Fields*, Springer (1970) [[doi:10.1007/978-3-642-48272-4_12](https://doi.org/10.1007/978-3-642-48272-4_12)]

* {#MacLane88} [[Saunders Mac Lane]]: §5 in: *Concepts and Categories in Perspective*, in: P. Duren, *A century of mathematics in America* Part 1, AMS (1988) 323-365. [[pdf](http://www.ams.org/samplings/math-history/hmath1-maclane25.pdf), [ISBN:0-8218-0124-4](https://www.ams.org/publicoutreach/math-history/hmath1-index)]

Original discussion in print is due to:

* {#vonNeumann1932} [[John von Neumann]]:

  *Mathematische Grundlagen der Quantenmechanik*, Springer (1932, 1971) [[doi:10.1007/978-3-642-96048-2](https://link.springer.com/book/10.1007/978-3-642-96048-2)]

  *Mathematical Foundations of Quantum Mechanics* Princeton University Press (1955) [[doi:10.1515/9781400889921](https://doi.org/10.1515/9781400889921), [Wikipedia entry](https://en.wikipedia.org/wiki/Mathematical_Foundations_of_Quantum_Mechanics)]

Lecture notes:

* [[Bergfinnur Durhuus]], *Operators on Hilbert Space* [[pdf](https://web.math.ku.dk/~durhuus/MatFys/MatFys4.pdf), [[Durhuus-OperatorsOnHilbertSpace.pdf:file]]]

[[!redirects adjoint operators]]


adjoint quadruple

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

An **adjoint quadruple** is a sequence of three [[adjunctions]]

$$
  f_! \dashv f^* \dashv f_* \dashv f^!
$$

between a [[quadruple]] of [[morphisms]].  That is, it is an [[adjoint string]] of length 4.

## Properties

### General

Every adjoint quadruple

$$
  (f_! \dashv f^* \dashv f_* \dashv f^!)
  :
  C
    \stackrel{\overset{f_!}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\stackrel{\overset{f_*}{\to}}{\underset{f^!}{\leftarrow}}}}
  D
$$

induces an [[adjoint triple]] on $C$

$$
  (f^* f_! \dashv f^* f_* \dashv f^! f_*)
  :
  C \to C
  \,,
$$

(hence a [[monad]] [[left adjoint]] to a [[comonad]] left adjoint to a monad) and an adjoint triple

$$
  (f_! f^* \dashv f_* f^* \dashv f_* f^!) : D \to D
$$

on $D$.

Since moreover every [[adjoint triple]] $(F \dashv G \dashv H)$ induces an [[adjoint functor|adjoint pair]] $(G F \dashv G H)$ and an adjoint pair $(F G \dashv H G)$, the adjoint quadruple above induces four adjoint pairs, such as

$$
  (f^* f_* f^* f_! \dashv f^* f_* f^! f_*) : C \to C
  \,.
$$

$\,$

### Canonical natural transformation

Let

$$
  (p_! \dashv p^* \dashv p_*\dashv p^!)
    \;\colon\;
  \mathcal{E}
    \longrightarrow
  \mathcal{S}
$$

be an  [[adjoint quadruple]] of [[adjoint functor|adjoint functors]] such that $p^*$ and $p^!$ are [[full and faithful functor|full and faithful functors]]. We record some general properties of such a setup.

We write

$$
  \eta \;\colon\; id \to p^* p_!
$$

etc. for [[unit of an adjunction|units]] and

$$
  \epsilon \;\colon\; p_! p^* \to id
$$

etc. for counits.

+-- {: .num_prop #TheCanonicalMorphisms}
###### Proposition/Definition

We have [[commuting diagrams]], [[natural transformation|natural]] in $X \in \mathcal{E}$, $S \in \mathcal{S}$

$$
  \array{
    p_*X  &\underoverset{\simeq}{\epsilon_{p^* X}^{-1}}{\longrightarrow}& p_! p^* p_*X
    \\
    {}^{\mathllap{p_*(\eta_X)}}\downarrow
    &\searrow^{\mathrlap{\theta_X}}&
    \downarrow^{\mathrlap{p_!(\epsilon_X)}}
    \\
    p_* p^* p_! X &\stackrel{\eta_{p_!X}^{-1}}{\longrightarrow}& p_! X
  }
$$

and

$$
  \array{
    p^* S &\stackrel{\eta_{p^* S}}{\longrightarrow}& p^! p_* p^* S
    \\
    {}^{\mathllap{p^* \epsilon_S^{-1}}}\downarrow
    &\searrow^{\mathrlap{\phi_X}}& \downarrow^{\mathrlap{p^!(\eta_S^{-1})}}
    \\
    p^* p_* p^!S  &\stackrel{{\epsilon}_{p_!S }}{\longrightarrow}& p^!S
  }
  \,.
$$

where the diagonal morphisms

$$
  \theta_X : p_* X \to p_! X
$$

and

$$
  \phi_S : p^* S \to p^! S
$$

are defined to be the equal composites of the sides of these diagrams.

=--

This appears as ([Johnstone 11, lemma 2.1, corollary 2.2](#Johnstone11)).

+-- {: .num_prop #TheEpiAndTheMono}
###### Proposition

The following conditions are equivalent:

* for all $X \in \mathcal{E}$ the morphism $\theta_X : p_*X \to p_! X$ is an [[epimorphism]];

* for all $S \in \mathcal{S}$,, the morphism $\phi_S : p^*S \to p^! S$
  is a [[monomorphism]];

* $p_*$ is [[faithful functor|faithful]] on morphisms of the form $A \to p^* S$.

=--

This appears as ([Johnstone 11, lemma 2.3](#Johnstone11)).

+-- {: .proof}
###### Proof

By the above definition, $\phi_S$ is a [[monomorphism]] precisely if $\eta_{p^* S} : p^* S \to p^! p_* p^* S$ is. This in turn is so (see [[monomorphism]]) precisely if the first [[function]] in

$$
    \mathcal{E}(A,p^* X)
     \stackrel{(\eta_{p^* X}) \circ (-)}{\longrightarrow}
    \mathcal{E}(A, p^! p_* p^* S)
     \stackrel{\simeq}{\longrightarrow}
    \mathcal{S}(p_* A, p_* p^* S)
$$

and hence the composite is a monomorphism in [[Set]].

By definition of [[adjunct]] and using the $(p_* \dashv p^!)$-[[zig-zag identity]], this is equal to the action of $p_*$ on morphisms

$$
  (\eta_{p^* X}) \circ (-)  :
  (A \to p^* S) \mapsto p_*(A \to p^* S)
  \,.
$$

Similarly, by the above definition the morphism $\theta_X$ is an epimorphism precisely if $p_!(\epsilon_X) : p_! p^* p_* X \to p_! X$ is so, which is the case precisely if the top morphism in

$$
  \array{
    \mathcal{S}(p_! X, S)
      &\stackrel{(-) \circ p_!(\epsilon_X)}{\longrightarrow} &
    \mathcal{S}(p_! p^* p_* X, S)
    \\
    {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\simeq}}
    \\
    && \mathcal{E}(p^* p_* X, p^* S)
    \\
    {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\simeq}}
    \\
    \mathcal{E}(X, p^* S) &\stackrel{p_*}{\longrightarrow}& \mathcal{S}(p_* X, p_* p^* S)
  }
$$

and hence the bottom morphism is a monomorphism in [[Set]],
where again the commutativity of this diagram follows from the
definition of [[adjunct]] and the
$(p_! \dashv p^*)$-[[zig-zag identity]].

=--

$\,$

## Examples

### Via Kan extension of adjoint pairs
 {#ViaKanExtensionOfAdjointPairs}

A rich source of adjoint quadruples arises form [[adjoint pairs]] between [[small categories]] by left/right [[Kan extension]] to their [[categories of presheaves]].

More interesting examples of adjoint quadruples tend to arise from these presheaf constructions when the quadruple ([[corestriction|co]])[[restriction|restricts]] to sub-[[categories of sheaves]].

We spell out two proofs of this fact, the first using [[coend]]-calculus in the generality of [[enriched category theory]], the second using more elementary [[colimit]]-notation.

\begin{prop}\label{KanExtensionOfAdjointPairIsAdjointQuadruple}
**([[Kan extension]] of [[adjoint pair]] is [[adjoint quadruple]])**
\linebreak
For $\mathcal{V}$ a [[symmetric monoidal category|symmetric]] [[closed monoidal category]] with all [[limits]] and [[colimits]], let $\mathcal{C}$, $\mathcal{D}$ be two [[small category|small]] $\mathcal{V}$-[[enriched categories]]and let

$$
  \mathcal{C}
    \underoverset
      {\underset{p}{\longrightarrow}}
      {\overset{q}{\longleftarrow}}
      {\bot}
  \mathcal{D}
$$

be a $\mathcal{V}$-[[enriched adjunction]]. Then there are $\mathcal{V}$-[[enriched natural isomorphisms]]

$$
  (q^{op})^\ast \;\simeq\; Lan_{p^{op}}
  \;\colon\;
  [\mathcal{C}^{op},\mathcal{V}]
    \longrightarrow
  [\mathcal{D}^{op},\mathcal{V}]
$$

$$
  (p^{op})^\ast \;\simeq\; Ran_{q^{op}}
  \;\colon\;
  [\mathcal{D}^{op},\mathcal{V}]
    \longrightarrow
  [\mathcal{C}^{op},\mathcal{V}]
$$

between the precomposition on [[enriched presheaves]] with one functor and the left/right [[Kan extension]] of the other.

By essential uniqueness of [[adjoint functors]] ([this Prop.](adjoint+functor#UniquenessOfAdjoints)), this means that the two [[Kan extension]] [[adjoint triples]] of $q$ and $p$

$$
  \array{
    Lan_{q^{op}} &\dashv& (q^{op})^\ast &\dashv& Ran_{q^{op}}
    \\
    && Lan_{p^{op}} &\dashv& (p^{op})^\ast &\dashv& Ran_{p^{op}}
  }
$$

merge into an [[adjoint quadruple]]

$$
  \array{
    Lan_{q^{op}} &\dashv& (q^{op})^\ast &\dashv& (p^{op})^\ast &\dashv& Ran_{p^{op}}
  }
  \;\colon\;
  [\mathcal{C}^{op},\mathcal{V}]
    \leftrightarrow
  [\mathcal{D}^{op}, \mathcal{V}]
$$

\end{prop}
\begin{proof}
For every [[enriched presheaf]] $F \;\colon\; \mathcal{C}^{op} \to \mathcal{V}$ we have a sequence of $\mathcal{V}$-[[enriched natural isomorphism]] as follows

$$
  \begin{aligned}
    (Lan_{p^{op}} F)(d)
      & \simeq
    \int^{ c \in \mathcal{C} } \mathcal{D}(d,p(c)) \otimes F(c)
    \\
    & \simeq
    \int^{ c \in \mathcal{C} } \mathcal{C}(q(d),c) \otimes F(c)
    \\
    & \simeq
    F(q(d))
    \\
    & = \left( (q^{op})^\ast F\right) (d)
    \,.
  \end{aligned}
$$

Here the first step is the [[coend]]-formula for [[left Kan extension]] ([here](Kan+extension#PointwiseByCoEnds)), the second step is the [[enriched adjunction]]-isomorphism for $q \dashv p$ and the third step is the [[co-Yoneda lemma]].

This shows the first statement. By essential uniqueness of adjoints ([this Prop.](adjoint+functor#UniquenessOfAdjoints)), the other statements follow.
\end{proof}

The following is the same argument without using coend-calculus. This argument applies verbatim also, for instance, in [[(infinity,1)-category theory|$\infty$-category theory]] using results from standard sources:

\begin{prop}
\label{AdjointPairInducesAdjointQuadrupleUnderKanExtension}
Given a pair of [[adjoint functors]] between [[small categories]]
\begin{tikzcd}
    \mathcal{S}_2
    \ar[
      rr,
      shift left=7pt,
      "{ \ell }"{above}
    ]
    &&
    \mathcal{S}_1
    \ar[
      ll,
      shift left=7pt,
      "{ r }"{below}
    ]
    \ar[
      ll,
      phantom,
      "{ \scalebox{.6}{$\bot$} }"
    ]
\end{tikzcd}
the induced operations of pre-composition on [[categories of presheaves]] are adjoint to each other, $\ell^\ast \,\dashv\, r^\ast$, and their [[adjoint triples]] of [[Kan extensions]] overlap:

\begin{tikzcd}[column sep=large]
      \mathrm{PSh}(\mathcal{S}_2)
      \;\;
      \ar[
        rr,
        shift left=-7pt,
        "{ \ell_\ast \,\simeq\, r^\ast    }"{description}
      ]
      \ar[
        rr,
        shift left=32pt-7pt,
        "{ \ell_! }"{description, pos=.4}
      ]
      &&
      \;\;
      \mathrm{PSh}(\mathcal{S}_1)
      \ar[
        ll,
        shift right=16pt-7pt,
        "{ \ell^\ast \,\simeq\, r_! }"{description}
      ]
      \ar[
        ll,
        shift right=-16pt-7pt,
        "{ r_\ast }"{description, pos=.4}
      ]
      \ar[
        ll,
        phantom,
        shift right=8pt-7pt,
        "{\scalebox{.6}{$\bot$}}"
      ]
      \ar[
        ll,
        phantom,
        shift right=24pt-7pt,
        "{\scalebox{.6}{$\bot$}}"
      ]
      \ar[
        ll,
        phantom,
        shift right=-8pt-7pt,
        "{\scalebox{.6}{$\bot$}}"
      ]
\end{tikzcd}
\end{prop}
\begin{proof}

{#ProofOfAdjointPairInducesAdjointQuadrupleUnderKanExtension}
We already know that each functor $f$ by itself induces an [[adjoint triple]] $f_! \dashv f^\ast \dashv f_\ast$, by [[Kan extension]]. Due to essential uniqueness of adjoints ([this Prop.](adjoint+functor#UniquenessOfAdjoints)) it is hence sufficient to show that these two adjoint triples "overlap", in that ($\ell^\ast \simeq r_!$ and equivalently) $\ell_\ast \simeq r^\ast$, hence equivalently that $\ell^\ast \dashv r^\ast$.

Now the [hom-isomorphism](adjoint+functor#InTermsOfHomIsomorphism) which is characteristic of the latter adjunction $\ell^\ast \dashv r^\ast$ may be obtained as the following sequence of [[natural bijections]]:

$$
  \begin{aligned}
      &
      \mathrm{PSh}(\mathcal{S}_1)
      \big(
        X_1
        ,\,
        r^\ast(X_2)
      \big)
      \\
      &
      \;\simeq\;
      \mathrm{PSh}(\mathcal{S}_1)
      \Big(
         \underset{
           \underset{
             s_1 \to X_1
           }{\longrightarrow}
         }{\lim}
         \,
        y(s_1)
        \,
        ,\,
        r^\ast
        \big(
         \underset{
           \underset{
             s_2 \to X_2
           }{\longrightarrow}
         }{\lim}
         \,
        y(s_2)
        \big)
      \Big)
      \\
      &
      \;\simeq\;
      \underset{
        \underset{
          s_1 \to X_1
        }{\longleftarrow}
      }{\lim}
      \mathrm{PSh}(\mathcal{S}_1)
      \Big(
        y(s_1)
        ,\,
         \underset{
           \underset{
             s_2 \to X_2
           }{\longrightarrow}
         }{\lim}
         \,
        r^\ast
        \big(
          y(s_2)
        \big)
      \Big)
      \\
      &
      \;\simeq\;
      \underset{
        \underset{
          s_1 \to X_1
        }{\longleftarrow}
      }{\lim}
      \,
         \underset{
           \underset{
             s_2 \to X_2
           }{\longrightarrow}
         }{\lim}
      \mathrm{PSh}(\mathcal{S}_1)
      \Big(
        y(s_1)
        ,\,
        r^\ast
        \big(
          y(s_2)
        \big)
      \Big)
      \\
      & \;\simeq\;
      \underset{
        \underset{
          s_1 \to X_1
        }{\longleftarrow}
      }{\lim}
      \;
      \underset{
        \underset{
          s_2 \to X_2
        }{\longrightarrow}
      }{\lim}
      \mathcal{S}_2
      \big(
        r(s_1)
        ,\,
        s_2
      \big)
      \\
      & \;\simeq\;
      \underset{
        \underset{
          s_1 \to X_1
        }{\longleftarrow}
      }{\lim}
      \;
      \underset{
        \underset{
          s_2 \to X_2
        }{\longrightarrow}
      }{\lim}
      \mathrm{PSh}(\mathcal{S}_2)
      \Big(
        y\big(r(s_1)\big)
        ,\,
        y(s_2)
      \Big)
      \\
      & \;\simeq\;
      \underset{
        \underset{
          s_1 \to X_1
        }{\longleftarrow}
      }{\lim}
      \mathrm{PSh}(\mathcal{S}_2)
      \Big(
        y\big(r(s_1)\big)
        ,\,
        \underset{
          \underset{
            s_2 \to X_2
          }{\longrightarrow}
        }{\lim}
        y(s_2)
      \Big)
      \\
      & \;\simeq\;
      \mathrm{PSh}(\mathcal{S}_2)
      \Big(
        \underset{
          \underset{
            s_1 \to X_1
          }{\longrightarrow}
        }{\lim}
        y\big(r(s_1)\big)
        ,\,
        \underset{
          \underset{
            s_2 \to X_2
          }{\longrightarrow}
        }{\lim}
        y(s_2)
      \Big)
      \\
      & \;\simeq\;
      \mathrm{PSh}(\mathcal{S}_2)
      \Big(
        \underset{
          \underset{
            s_1 \to X_1
          }{\longrightarrow}
        }{\lim}
        \mathcal{S}_2
        \big(
          (-)
          ,\,
          r(s_1)
        \big)
        ,\,
        \underset{
          \underset{
            s_2 \to X_2
          }{\longrightarrow}
        }{\lim}
        y(s_2)
      \Big)
      \\
      & \;\simeq\;
      \mathrm{PSh}(\mathcal{S}_2)
      \Big(
        \underset{
          \underset{
            s_1 \to X_1
          }{\longrightarrow}
        }{\lim}
        \mathcal{S}_2
        \big(
          \ell(-)
          ,\,
          s_1
        \big)
        ,\,
        \underset{
          \underset{
            s_2 \to X_2
          }{\longrightarrow}
        }{\lim}
        y(s_2)
      \Big)
      \\
      & \;\simeq\;
    PSh(\mathcal{S}_2)
      \Big(
        \underset{
          \underset{
            s_1 \to X_1
          }{\longrightarrow}
        }{\lim}
        \ell^\ast
        \big(
          y(s_1)
        \big)
        ,\,
        \underset{
          \underset{
            s_2 \to X_2
          }{\longrightarrow}
        }{\lim}
        y(s_2)
      \Big)
      \\
      & \;\simeq\;
    PSh(\mathcal{S}_2)
      \Big(
        \ell^\ast
        \big(
        \underset{
          \underset{
            s_1 \to X_1
          }{\longrightarrow}
        }{\lim}
        y(s_1)
        \big)
        ,\,
        \underset{
          \underset{
            s_2 \to X_2
          }{\longrightarrow}
        }{\lim}
        y(s_2)
      \Big)
      \\
      & \;\simeq\;
    PSh(\mathcal{S}_2)
      \Big(
        \ell^\ast
        (
          X_1
        )
        ,\,
        X_2
      \Big)
  \end{aligned}
$$

Here we used repeatedly

* the [[co-Yoneda lemma]] in the form

  $$
    X
    \;\simeq\;
    \underset{
      \underset{y(s) \to X}{\longrightarrow}
    }{\lim}
    \,y(s)
    \,,
  $$

  expressing a [[presheaf]] $X \,\in\, PSh(S)$ as a [[colimit]] of [[representable functors]] $y(s)$ (the colimit is over the [[comma category]] $y \downarrow X$, but that does not even matter in the proof above),

* the fact that any [[hom-functor preserves limits|hom-functor sends colimits in its first argument to limits]],

* the strong [[Yoneda lemma]] which says that $PSh(\mathcal{S})\big(y(s), X\big) \,\simeq\, X(s)$,

* the fact that [[colimits of presheaves are computed objectwise]].

As before, the adjunction $\ell^\ast \dashv r_!$ implies the overlapping adjoint triples by essential uniqueness of [[adjoint functors]] ([this Prop.](adjoint+functor#UniquenessOfAdjoints)).
\end{proof}

### Cohesion
 {#CohesiveToposes}

\begin{prop}\label{KanExtensionOfFiniteProductPreservingReflectionIsCohesiveAdjointQuadruple}
**(Kan extension of finite product preserving reflection is cohesive adjoint quadruple)**
\linebreak
Let

\begin{tikzcd}
  \mathcal{S}_1
  \ar[
    r,
    shift left=7pt,
    "{ p }"{above},
    "{ \mathclap{\widehat{\times}} }"{description, pos=.0}
  ]
  &
  \mathcal{S}_2
  \ar[
    l,
    shift left=7pt,
    hook',
    "{ i }"{below}
  ]
  \ar[
    l,
    phantom,
    "{\scalebox{.6}{$\bot$}}"{description}
  ]
\end{tikzcd}

be a pair of [[adjoint functors]] between [[small categories]] that have [[finite products]] (or at least after passing to their [[free coproduct completion]]), such that

1. the [[right adjoint]] $i$ is [[fully faithful functor|fully faithful]],

1. the [[left adjoint]] $p$ [[preserves limits|preserves]] [[finite products]] (or at least its coproduct-preserving extension to [[free coproduct completions]] does).

Then the induced adjoint quadruple of Kan extensions from Prop. \ref{AdjointPairInducesAdjointQuadrupleUnderKanExtension} is [[cohesive topos|cohesive]] in that

1. the two reverse functors are [[fully faithful functor|fully faithful]].

1. the leftmost adjoint $p_!$ [[preserved limit|preserves]] [[finite products]];

\begin{tikzcd}
  \mathrm{PSh}(\mathcal{S}_1)
      \ar[
        rr,
        "{ p_\ast \,\simeq\, i^\ast    }"{description}
      ]
      \ar[
        rr,
        shift left=40pt,
        "{ p_! }"{description, pos=.39},
        "\mathclap{\times}"{description, pos=0}
      ]
      &&
  \mathrm{PSh}(\mathcal{S}_1)
      \ar[
        ll,
        hook',
        shift right=20pt,
        "{ p^\ast \,\simeq\, i_! }"{description}
      ]
      \ar[
        ll,
        hook',
        shift right=-20pt,
        "{ i_\ast }"{description, pos=.39}
      ]
      \ar[
        ll,
        phantom,
        shift right=10pt,
        "{\scalebox{.6}{$\bot$}}"
      ]
      \ar[
        ll,
        phantom,
        shift right=30pt,
        "{\scalebox{.6}{$\bot$}}"
      ]
      \ar[
        ll,
        phantom,
        shift right=-10pt,
        "{\scalebox{.6}{$\bot$}}"
      ]
\end{tikzcd}

\end{prop}
\begin{proof}
  The preservation of finite products by the leftmost adjoint follows by Prop. \ref{LeftKanExtensionOfFinProdPreservingIsFinProdPreserving} below.

The fully faithfulness of $i_!$ follows by Prop. \ref{LeftKanExtensionOfFullyFaithfulIsFullyFaithful} below. This implies that also $i_\ast$ is fully faithful, by [this Prop.](adjoint+triple#FullyFaithful).
\end{proof}

\begin{example}
Consider a [[site]] $\mathcal{S}$ with [[finite products]], in particular with a [[terminal object]]. Then the inclusion of the [[full subcategory]] on this  terminal object, which is the [[terminal category]] $\ast$ is an adjunction of the form

$$
  \mathcal{S}
  \underoverset
    {\underset{}{\hookleftarrow}}
    {\overset{}{\longrightarrow}}
    {\;\;\;\bot\;\;\;}
   \ast
  \,.
$$

If $\mathcal{S}$ is a *[[cohesive site]]* then the
induced adjoint quadruple from Prop. \ref{KanExtensionOfFiniteProductPreservingReflectionIsCohesiveAdjointQuadruple} ([[corestriction|co]])[[restriction|restricts]] to the [[category of sheaves]] $Sh(\mathcal{S}) \xhookrightarrow{\;} PSh(\mathcal{S})$ and exhibits it as a [[cohesive topos]].
\end{example}

The proof of Prop. \ref{KanExtensionOfFiniteProductPreservingReflectionIsCohesiveAdjointQuadruple} as spelled out below applies verbatim also in [[(infinity,1)-category theory|$\infty$-category theory]]. Here, it has interesting examples even without passage to [[(infinity,1)-sheaves|$\infty$-sheaves]],  when the [[(infinity,1)-site|$\infty$-sites]] are higher than [[1-sites]]:

\begin{example}\label{COhesionOfGlobalOverGEquivariantHomotopyTheory}
**([[cohesion of global- over G-equivariant homotopy theory]])**
\linebreak
  For $G$ a group, there is an adjunction of [[(2,1)-categories]]

\begin{tikzcd}
  \mathrm{Sngrt}_{/\prec G}
  \ar[
    r,
    shift left=7pt,
    "{ \tau_0 }"{above},
    "{ \mathclap{\widehat{\times}} }"{description, pos=.0}
  ]
  &
  G \mathrm{Orbt}
  \ar[
    l,
    shift left=7pt,
    hook',
    "{ i }"{below}
  ]
  \ar[
    l,
    phantom,
    "{\scalebox{.6}{$\bot$}}"{description}
  ]
\end{tikzcd}

between the [[slice (infinity,1)-category|slice]] of the [[global orbit category]] over the object corresponding to $G$, and the $G$-[[orbit category]]. The adjoint quadruple of [[(infinity,1)-functors|$\infty$-functors]] between [[(infinity,1)-categories of (infinity,1)-presheaves|$\infty$-categories of $\infty$-presheaves]] induced by this  via Prop. \ref{KanExtensionOfFiniteProductPreservingReflectionIsCohesiveAdjointQuadruple} exhibits slices of [[global homotopy theory]] as being [[cohesive (infinity,1)-topos|cohesive]] over $G$-[[equivariant homotopy theory]]

\begin{tikzcd}
  \phantom{AAAAAAA}
  \overset{
    \mathclap{
    \raisebox{8pt}{
      \tiny
      \color{blue}
      \begin{tabular}{c}
        global equivariant
        \\
        homotopy theory sliced
        \\
        over $G$-orbi-singularity
        \\
        \phantom{A}
      \end{tabular}
    }\;\;\;\;
    }
  }{
    \;\;(\mathrm{Glo} \mathrm{Grpd}_\infty)_{/\prec G}\;\;
  }
      \ar[
        rr,
        "{ \mathrm{FixLoc}    }"{description}
      ]
      \ar[
        rr,
        shift left=40pt,
        "{  }"{description, pos=.35},
        "\mathclap{\times}"{description, pos=0}
      ]
      &&
     \overset{
       \mathclap{
       \;\;
       \raisebox{3pt}{
         \tiny
         \color{blue}
         \begin{tabular}{c}
            $G$-equivariant
            \\
            homotopy theory
            \\
            \phantom{A}
         \end{tabular}
       }
       }
     }{
       \;\; G \mathrm{Grpd}_\infty \;\;
      }
     \phantom{AAAAAAA}
      \ar[
        ll,
        hook',
        shift right=20pt,
        "{ \mathrm{OrbSp} }"{description}
      ]
      \ar[
        ll,
        hook',
        shift right=-20pt,
        "{  }"{description, pos=.35}
      ]
      \ar[
        ll,
        phantom,
        shift right=10pt,
        "{\scalebox{.5}{$\bot$}}"
      ]
      \ar[
        ll,
        phantom,
        shift right=30pt,
        "{\scalebox{.5}{$\bot$}}"
      ]
      \ar[
        ll,
        phantom,
        shift right=-10pt,
        "{\scalebox{.5}{$\bot$}}"
      ]
\end{tikzcd}

For more on this see at *[[cohesion of global- over G-equivariant homotopy theory]]*.
\end{example}

This example is a special case of the following general class:
\begin{example}\label{nTruncationsOfFullSubcategoriesOfInfinityToposes}
  For $n \,\in\, \mathbb{N}$ and
  for $\mathcal{S} \xhookrightarrow{\;\;} \mathbf{H} $ any [[small (infinity,1)-category|small]] [[full sub-(infinity,1)-category|full sub-$\infty$-category]] of an [[(infinity,1)-topos|$\infty$-topos]] which is

1. has all [[finite products|finite]] [[homotopy products]], these being computed in $\mathbf{H}$,

1. is closed under [[n-truncated object of an (infinity,1)-category|$n$-truncation]]

then the $n$-truncation reflection restricts

\begin{tikzcd}
  \mathcal{S}
  \ar[
    r,
    shift left=7pt,
    "{ \tau_n }"{above},
    "{ \mathclap{\times} }"{description, pos=.0}
  ]
  &
  \mathcal{S}_{\tau_n}
  \ar[
    l,
    shift left=7pt,
    hook',
    "{ i }"{below}
  ]
  \ar[
    l,
    phantom,
    "{\scalebox{.6}{$\bot$}}"{description}
  ]
\end{tikzcd}

and it preserves finite products (by [this Prop.](n-truncated+object+of+an+infinity1-category#nTruncationInToposPreservesFiniteProducts)).

Moreover, this adjunction [[corestriction|(co)]][[restriction|restricts]] to [[connected objects]] $X \,\in\, \mathcal{S}_{cn} \xhookrightarrow{\;} \mathcal{S}$ (i.e. those for which $\mathcal{S}(X,-)$ preserves coproducts):

\begin{tikzcd}
  \mathcal{S}_{\mathrm{cn}}
  \ar[
    r,
    shift left=7pt,
    "{ \tau_n }"{above},
    "{ \mathclap{\widehat{\times}} }"{description, pos=.0}
  ]
  &
  \mathcal{S}_{\mathrm{cn},\tau_n}
  \ar[
    l,
    shift left=7pt,
    hook',
    "{ i }"{below}
  ]
  \ar[
    l,
    phantom,
    "{\scalebox{.6}{$\bot$}}"{description}
  ]
\end{tikzcd}

If all objects of $\mathcal{S}$ are coproducts of connected ones then coroduct-preserving extension of the (co)restriced left adjoint is the original left adjoint and hence preserves finite products.

Hence the Kan extensions according to Prop. \ref{KanExtensionOfFiniteProductPreservingReflectionIsCohesiveAdjointQuadruple} exhibit the [[(infinity,1)-category of (infinity,1)-sheaves|$\infty$-category of $\infty$-presheaves]] $PSh_\infty(\mathcal{S})$ as being [[cohesive (infinity,1)-topos|cohesive]] over $PSh_\infty(\mathcal{S}_{\tau_n})$.

\end{example}

\linebreak

We now spell out the proof of the lemmas used in the proof of Prop. \ref{KanExtensionOfFiniteProductPreservingReflectionIsCohesiveAdjointQuadruple}.

\begin{lemma}\label{LeftKanExtensionIsOriginalFunctorOnRepresentables}
  Given a functor $\mathcal{S}_1 \xrightarrow{\;f\;} \mathcal{S}_2$ between [[small categories]], its [[left Kan extension]]
$f_! \;\colon\; PSh(\mathcal{S}_1) \xrightarrow{\;\;} PSh(\mathcal{S}_1)$ restricts to $f$ on [[representable functor|representables]], in that for $s_1 \,\in\, \mathcal{S}_1$ we have a [[natural isomorphism]]
$$
  f_!\big( y(s_1) \big)
  \;\simeq\;
  y\big(
    f(s_1)
  \big)
  \,.
$$
\end{lemma}
\begin{proof}
For $X \,\in\, PSh(\mathcal{S}_2)$
we have the following sequence of [[natural isomorphism]]:
$$
   \begin{aligned}
     PSh(\mathcal{S}_2)
     \left(
       f_!
       \left(
         y(s_1)
       \right)
       ,\,
       X
     \right)
         & \;\simeq\;
     PSh(\mathcal{S}_1)
     \left(
       y(s_1)
       ,\,
       f^\ast(X)
     \right)
     \\
     &
     \;\simeq\;
     X\left( f(s_1) \right)
     \\
     &
     \;\simeq\;
     PSh(\mathcal{S}_2)
     \left(
       y\left(f(s_1)\right)
       ,\,
       X
     \right)
     \,.
   \end{aligned}
$$
The first line is the defining adjointness of $f_!$, the second line the [[Yoneda lemma]] over $\mathcal{S}_1$ and the definition of $f^\ast$, while the last line is the Yoneda lemma over $\mathcal{S}_2$.

Since the composite of these isomorphisms is natural, the [[Yoneda lemma]] over $PSh(\mathcal{S}_2)^{op}$ (which is [[large category|large]] but [[locally small category|locally small]], so that the lemma does apply) implies the claim.
\end{proof}
\begin{lemma}\label{LeftKanExtensionOfBinProductPreservingIsBinProductPres}
  Let $\mathcal{S}_1$ and $\mathcal{S}_2$ be [[small categories]] with [[binary products]] and $\mathcal{S}_1 \xrightarrow{\;f\;} \mathcal{S}_2$ a [[functor]] which [[preserved limit|preserves]] these, in that for $s, s' \,\in\, \mathcal{S}_1$ there is a [[natural isomorphism]] $f(s \times s') \,\simeq\, f(X_1) \times f(X_2)$. Then also the [[left Kan extension]] $f_!$ [[preserved limit|preserves]] [[binary products]], in that for $X, X' \,\in\, PSh(\mathcal{S}_1)$ there is a [[natural isomorphism]]
$$
  f_!(X \times X')
  \;\simeq\;
  f_!(X) \times f_!(X')
  \,.
$$
\end{lemma}
\begin{proof}
This is the composite of the following sequence of [[natural isomorphisms]]:
$$
  \begin{aligned}
    f_!(X \times X')
       & \;\simeq\;
    f_!
    \Big(\!
      \big(
        \underset{
          \underset{ s \to X}{\longrightarrow}
        }{\lim}
        \,
        y(s)
      \big)
      \times
      \big(
        \underset{
          \underset{ s' \to X}{\longrightarrow}
        }{\lim}
        \,
        y(s')
      \big)
    \!\Big)
    \\
    & \;\simeq\;
    f_!
    \Big(
        \underset{
          \underset{ s \to X}{\longrightarrow}
        }{\lim}
        \;
        \underset{
          \underset{ s' \to X}{\longrightarrow}
        }{\lim}
        \,
      \left(
        y(s)
        \times
        y(s')
      \right)
    \!\!\Big)
    \\
    & \;\simeq\;
        \underset{
          \underset{ s \to X}{\longrightarrow}
        }{\lim}
        \;
        \underset{
          \underset{ s' \to X}{\longrightarrow}
        }{\lim}
    \,
    f_!
      \big(
        y(s)
        \times
        y(s')
      \big)
    \\
    & \;\simeq\;
        \underset{
          \underset{ s \to X}{\longrightarrow}
        }{\lim}
        \;
        \underset{
          \underset{ s' \to X}{\longrightarrow}
        }{\lim}
    \,
    f_!
      \big(
         y(s \times s')
      \big)
    \\
    & \;\simeq\;
        \underset{
          \underset{ s \to X}{\longrightarrow}
        }{\lim}
        \;
        \underset{
          \underset{ s' \to X}{\longrightarrow}
        }{\lim}
    \,
    y\big(  f(s \times s') \big)
    \\
    & \;\simeq\;
        \underset{
          \underset{ s \to X}{\longrightarrow}
        }{\lim}
        \;
        \underset{
          \underset{ s' \to X}{\longrightarrow}
        }{\lim}
    \,
    y\big(  f(s) \times f(s') \big)
    \\
    & \;\simeq\;
        \underset{
          \underset{ s \to X}{\longrightarrow}
        }{\lim}
        \;
        \underset{
          \underset{ s' \to X}{\longrightarrow}
        }{\lim}
    \,
    y\big(  f(s) \big)
      \times
    y\big(f(s') \big)
    \\
    & \;\simeq\;
        \underset{
          \underset{ s \to X}{\longrightarrow}
        }{\lim}
        \;
        \underset{
          \underset{ s' \to X}{\longrightarrow}
        }{\lim}
    \,
    f_!
      \left(
        y(s)
      \right)
   \times
    f_!
      \left(
        y(s')
      \right)
    \\
    & \;\simeq\;
    \Big(\,
        \underset{
          \underset{ s \to X}{\longrightarrow}
        }{\lim}
    \,
    f_!
      \left(
        y(s)
      \right)
   \!\!\Big)
   \times
   \Big(\,
        \underset{
          \underset{ s' \to X}{\longrightarrow}
        }{\lim}
    \,
    f_!
      \left(
        y(s')
      \right)
  \!\!\Big).
  \end{aligned}
$$
Here

* the first line is the [[co-Yoneda lemma]], expressing the [[presheaves]] as [[colimits]] of [[representable functor|representables]].

* The second line uses that in a [[topos]] (like [[Set]] or whatever the [[base topos]] may be) [[universal colimits|colimits are pullback-stable]] and in particular distribute over [[products]].

* The third line uses that $f_!$ is a [[left adjoint]] and that [[left adjoints preserve colimits]].

* Then inside the two colimits we use

  1. that the [[Yoneda embedding]] preserves [[limits]], in particular [[products]],

  1. Lem. \ref{LeftKanExtensionIsOriginalFunctorOnRepresentables}, to evaluate $f_!$ on representables as $f$,

  1. the assumption that $f$ preserves products.

* The last step is the first two steps in reverse.

\end{proof}

The following
Lem. \ref{LeftKanExtensionOfBinProductPreservingOnCoprodComplIsBinProductPres}
is an immediate variant of
Lem. \ref{LeftKanExtensionOfBinProductPreservingIsBinProductPres}
obtained by relaxing the assumptions slightly to a form that is often still readily checked:
\begin{lemma}
\label{LeftKanExtensionOfBinProductPreservingOnCoprodComplIsBinProductPres}
  Let
  \begin{tikzcd}
    \mathcal{S}_1
    \ar[
      rr,
      "{f}"{above},
      "{\mathclap{\widehat{\times}}}"{description, pos=0}
    ]
    &&
    \mathcal{S}_2
  \end{tikzcd}
  be a [[functor]] between [[small categories]] such that

1. their [[free coproduct completions]] $PSh_{\sqcup}(-)$ have [[binary products]],

1. the unique coproduct-[[preserved colimit|preserving]] extension $f_!$ of $f$ to these completions [[preserved limit|preserves]] [[binary products]], in that for $s, s' \,\in, \mathcal{S}_1$ there is a [[natural isomorphism]]:

   $$
     f_!\big( y(s) \times y(s') \big)
       \,\simeq\,
     f_!\big(y(s)\big) \times f_!\big( y(s') \big)
   $$

Then also the left [[Kan extension]] $f_! \,\colon\, PSh(\mathcal{S}_1) \xrightarrow{\;} PSh(\mathcal{S}_2)$ to the full [[categories of presheaves]] preserves products.
\end{lemma}
\begin{proof}
  The proof starts and ends as the proof of Prop. \ref{LeftKanExtensionOfBinProductPreservingIsBinProductPres},
but the main step in between is now more immediate, as it just needs to invoke the assumption:
$$
  \begin{aligned}
    f_!(X \times X')
       & \;\simeq\;
    f_!
    \Big(\!
      \big(
        \underset{
          \underset{ s \to X}{\longrightarrow}
        }{\lim}
        \,
        y(s)
      \big)
      \times
      \big(
        \underset{
          \underset{ s' \to X}{\longrightarrow}
        }{\lim}
        \,
        y(s')
      \big)
    \!\Big)
    \\
    & \;\simeq\;
    f_!
    \Big(
        \underset{
          \underset{ s \to X}{\longrightarrow}
        }{\lim}
        \;
        \underset{
          \underset{ s' \to X}{\longrightarrow}
        }{\lim}
        \,
      \left(
        y(s)
        \times
        y(s')
      \right)
    \!\!\Big)
    \\
    & \;\simeq\;
        \underset{
          \underset{ s \to X}{\longrightarrow}
        }{\lim}
        \;
        \underset{
          \underset{ s' \to X}{\longrightarrow}
        }{\lim}
    \,
    f_!
      \big(
        y(s)
        \times
        y(s')
      \big)
    \\
    & \;\simeq\;
        \underset{
          \underset{ s \to X}{\longrightarrow}
        }{\lim}
        \;
        \underset{
          \underset{ s' \to X}{\longrightarrow}
        }{\lim}
    \,
    f_!
      \left(
        y(s)
      \right)
   \times
    f_!
      \left(
        y(s')
      \right)
    \\
    & \;\simeq\;
    \Big(\,
        \underset{
          \underset{ s \to X}{\longrightarrow}
        }{\lim}
    \,
    f_!
      \left(
        y(s)
      \right)
   \!\!\Big)
   \times
   \Big(\,
        \underset{
          \underset{ s' \to X}{\longrightarrow}
        }{\lim}
    \,
    f_!
      \left(
        y(s')
      \right)
  \!\!\Big).
  \end{aligned}
$$
\end{proof}

In conclusion so far:
\begin{prop}\label{LeftKanExtensionOfFinProdPreservingIsFinProdPreserving}
  If a pair of [[small categories]] $\mathcal{S}_1$, $\mathcal{S}_2$ has [[finite products]] and a [[functor]] $\mathcal{S}_1 \xrightarrow{\;f\;} \mathcal{S}_2$ [[preserved limit|preserves]] these, then so does its [[left Kan extension]] $f_! \,\colon\, PSh(\mathcal{S}_1) \xrightarrow{\;} PSh(\mathcal{S}_2)$.

More generally this is the case if the [[free coproduct completions]] $PSh_{\sqcup}(\mathcal{S}_i)$ have [[finite products]] and the unique coproduct-preserving extension preserves these.
\end{prop}
\begin{proof}
  We need to show that $f_!$ preserves (1) the [[terminal object]] and (2) [[binary products]]. With the given assumption on $f$, the first follows with Lem. \ref{LeftKanExtensionIsOriginalFunctorOnRepresentables} while the second follows with Lem. \ref{LeftKanExtensionOfBinProductPreservingIsBinProductPres} or Lem. \ref{LeftKanExtensionOfBinProductPreservingOnCoprodComplIsBinProductPres}, respectively.
\end{proof}

\begin{proposition}\label{LeftKanExtensionOfFullyFaithfulIsFullyFaithful}
The left Kan extension $f_!$ of a [[fully faithful functor]] $f$ between [[small categories]] is itself fully faithful:
\begin{tikzcd}
    \mathcal{S}_1
    &
    \mathcal{S}_2
    \ar[
      l,
      hook',
      "{i}"{above}
    ]
  &
  \;\;\;\;
  \Rightarrow
  \;\;\;\;
  &
    \mathrm{PSh}(\mathcal{S}_1)
    &
    \mathrm{PSh}(\mathcal{S}_2)
    \ar[
      l,
      hook',
      "{i_!}"{above}
    ]
\end{tikzcd}
\end{proposition}
\begin{proof}
  We need to show for $X, X' \,\in\, PSh(\mathcal{S}_2)$ the morphism
$$
  PSh(\mathcal{S}_2)
  (X,\, X')
  \xrightarrow{ \; (i_!)_{X, X'} \; }
  PSh(\mathcal{S}_`)
  \big(
    i_!(X)
    ,\,
   i_!( X')
  \big)
$$
is an equivalence. But since $i_!$ is a left adjoint and since $X$ and $X'$ are colimits of representables, this morphism is the unique one which reduces to
\[
  \label{ComponentIsomorphismOfAFullyFaithfulFunctor}
  \mathcal{S}_2
  (s,\, s')
  \xrightarrow{ \; i_{s, s'} \; }
  \mathcal{S}_1
  \big(
    i(s)
    ,\,
   i( s')
  \big)
\]
on representables, where this is an isomorphism by the assumption that $i$ is fully faithful. If follows that $(i_!)_{X, X'}$, is the compostite of the following isomorphisms, and hence an isomorphism:
$$
  \begin{aligned}
    &
    PSh(\mathcal{S}_1)
    \big(
      i_!(X_1)
      ,\,
      i_!(X_2)
    \big)
    \\
    &
    \;\simeq\;
    PSh(\mathcal{S}_1)
    \Big(
      i_!
      \big(
      \underset{
        \underset{s \to X}{\longrightarrow}
      }{\lim}
        y(s)
      \big)
      ,\,
      i_!
      \big(
        \underset{
          \underset{s' \to X'}{\longrightarrow}
        }{\lim}
        y(s')
      \big)
    \Big)
    \\
    &
    \;\simeq\;
    PSh(\mathcal{S}_1)
    \Big(
      \underset{
        \underset{s \to X}{\longrightarrow}
      }{\lim}
        i_!
        \big(
          y(s)
        \big)
      ,\,
      \underset{
        \underset{s' \to X'}{\longrightarrow}
      }{\lim}
        i_!
        \big(
          y(s')
        \big)
    \Big)
    \\
    &
    \;\simeq\;
    PSh(\mathcal{S}_1)
    \Big(
      \underset{
        \underset{s \to X}{\longrightarrow}
      }{\lim}
        y
        \big(
          i
          (s)
        \big)
      ,\,
      \underset{
        \underset{s' \to X'}{\longrightarrow}
      }{\lim}
        y
        \big(
          i(s')
        \big)
    \Big)
    \\
    &
    \;\simeq\;
    \underset{
    }{\lim}
      \underset{
        \underset{s' \to X'}{\longrightarrow}
      }{\lim}
    PSh(\mathcal{S}_1)
    \Big(
        y
        \big(
          i
          (s)
        \big)
      ,\,
        y
        \big(
          i(s')
        \big)
    \Big)
    \\
    &
    \;\simeq\;
    \underset{
      \underset{s \to X}{\longleftarrow}
    }{\lim}
      \underset{
        \underset{s' \to X'}{\longrightarrow}
      }{\lim}
    \mathcal{S}_1
    \big(
      i(s)
      ,\,
      i(s')
    \big)
    \\
    &
    \;\simeq\;
    \underset{
      \underset{s \to X}{\longleftarrow}
    }{\lim}
      \underset{
        \underset{s' \to X'}{\longrightarrow}
      }{\lim}
    \mathcal{S}_2
    \big(
      s
      ,\,
      s'
    \big)
    \\
    &
    \;\simeq\;
    \underset{
      \underset{s \to X}{\longleftarrow}
    }{\lim}
      \underset{
        \underset{s' \to X'}{\longrightarrow}
      }{\lim}
    PSh(\mathcal{S}_2)
    \big(
      y(s)
      ,\,
      y(s')
    \big)
    \\
    &
    \;\simeq\;
    PSh(\mathcal{S}_2)
    \Big(
      \underset{
        \underset{s \to X}{\longrightarrow}
      }{\lim}
      y(s)
      ,\,
      \underset{
        \underset{s' \to X'}{\longrightarrow}
      }{\lim}
      y(s')
    \Big)
    \\
    &
    \;\simeq\;
    PSh(\mathcal{S}_2)
    \big(
      X
      ,\,
      X'
    \big)
  \end{aligned}
$$
Here the first and last steps are the [[co-Yoneda lemma]] and the preservation of its colimits as in the proofs before. In the middle steps we are using Lem. \ref{LeftKanExtensionIsOriginalFunctorOnRepresentables} to evaluate $i_!$ on representables and then (eq:ComponentIsomorphismOfAFullyFaithfulFunctor) inside the (co)limits.
\end{proof}

## Related concepts

* [[adjunction]], [[adjoint functor]]

* [[adjoint triple]], [[adjoint string]]

## References

The above prop. \ref{TheCanonicalMorphisms} is from:

* {#Johnstone11} [[Peter Johnstone]], _Remarks on punctual local connectedness_, Theory and Applications of Categories, Vol. 25, 2011, No. 3, pp 51-63.  ([tac](http://www.tac.mta.ca/tac/volumes/25/3/25-03abs.html))

[[!redirects adjoint quadruples]]


adjoint representation

Every [[Lie group]] $G$ has a canonical [[representation]] on the [[vector space]] underlying its [[Lie algebra]] $\mathfrak{g}$, given by the [[derivative]] of its [[adjoint action]] at the neutral element. This is called the _adjoint representation_.

The [[associated bundles]] via the adjoint representation are called [[adjoint bundles]].


adjoint string

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### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

# Adjoint strings

* table of contents
{: toc}

## Definition

In [[category theory]],
an **adjoint string of length $n$**, **adjoint chain of length $n$**, **adjoint sequence of length $n$**, or an **adjoint $n$-tuple**, is a sequence of $(n-1)$ [[adjunctions]] between $n$ [[functors]] (or more generally [[morphisms]] in a [[2-category]]):

$$f_1 \dashv f_2 \dashv \cdots \dashv f_n $$

## Special cases

* An adjoint $2$-tuple is just an ordinary [[adjunction]].
* An adjoint $3$-tuple is an [[adjoint triple]].
* An adjoint $4$-tuple is an [[adjoint quadruple]].

## Examples

1. There is an adjoint $5$-tuple between $[Set^{op}, Set]$ and $Set$. Indeed, given a [[locally small category]] $B$, and the [[Yoneda embedding]], $y: B \to [B^{op}, Set]$, then $y$ being the rightmost functor of an adjoint $5$-tuple entails that $B$ is equivalent to [[Set]]; see [Rosebrugh-Wood](#RWSets).

1. For any category $C$, there is a functor $ids: C\to Ar(C)$ from $C$ to its [[arrow category]] that assigns the identity morphism of each object.  This functor always has both a left and a right adjoint which assign the codomain and domain of an arrow respectively; thus we have an adjoint triple $cod \dashv ids \dashv dom$.  If $C$ has an initial object $0$, then $cod$ has a further left adjoint $I$ assigning to each object $x$ the morphism $0\to x$; and dually if $C$ has a terminal object $1$ then $dom$ has a further right adjoint $T$ assigning to $x$ the morphism $x\to 1$.  Thus if $C$ has an initial and terminal object, we have an adjoint $5$-tuple.

1. Continuing from the last example, if $C$ is moreover a [[pointed category]] with [[pullbacks]] and [[pushouts]], then $I$ has a further left adjoint that constructs the [[cokernel]] of a morphism $x\to y$, i.e. the pushout of $y \leftarrow x \to 0$; and $T$ has a further right adjoint that constructs the [[kernel]] of a morphism $x \to y$, namely the pullback of $x\to y \leftarrow 0$.  Thus we have an adjoint $7$-tuple.  In fact, the existence of such an adjoint $7$-tuple characterizes pointed categories among categories with finite limits and colimits.

1. The previous two examples apply also to [[derivators]], and the extension of the analogous adjoint $5$-tuple to a $7$-tuple again characterizes the [[pointed derivators]].  Moreover, the [[stable derivators]] are characterized by the extension of this $7$-tuple to a doubly-infinite adjoint string with period 6 ([GrothShul17](#GrothShul17)).

1. Let $[n]$ denote the [[totally ordered]] $(n+1)$-element set, regarded as a category. For each positive integer $n$, we have $n+1$ order-preserving injections from $[n-1]$ to $[n]$, and $n$ order-preserving surjections from $[n]$ to $[n-1]$. Regarded as functors, these injections and surjections interleave to form an adjoint chain of length $2n + 1$.  These categories, functors, and adjunctions form the [[simplex category]] [[simplex category#As2Categories|regarded as a locally posetal 2-category]]; see below.

1. Let $C$ be a category with a [[terminal object]] but no [[initial object]]. Then there are functors
  $$
    \array{
      \delta_i \colon [n+1,C] \to [n,C] & 0\leq i \leq n;
      \\
      \sigma_i\colon [n,C] \to [n+1,C] &  0\leq i \leq n
    }
  $$
such that
  $$
    \delta_0 \dashv \sigma_0 \dashv \cdots \dashv \delta_n \dashv \sigma_n
  $$
is a maximal string of adjoint functors (all but $\sigma_n$ are obtained by applying $[-, C]$ to the simplex category example, and $\sigma_n$ exploits the presence of the terminal object of $C$).

1. Generalizing the simplex category example: if $P$ is a [[lax idempotent monad]] with unit $u: 1 \to P$ and multiplication $m: P P \to P$ (so that $m \dashv u P$), then there is an adjoint string
$$P^{n-1} m \dashv P^{n-1} u P \dashv P^{n-2}m P \dashv \ldots \dashv m P^{n-1} \dashv u P^n$$
of length $2 n + 1$, back and forth between $P^{n+1}$ and $P^n$. The example of $[n]$ and $[n+1]$ above is based on the fact that the [[simplex category]] $\Delta$, regarded as a locally posetal [[bicategory]], is the [[walking structure|walking]] lax idempotent monoid.

1. Given an [[ambidextrous adjunction]] (and in particular a [[self-adjoint functor]]), $F \dashv G$ and $G \dashv F$, we of course get an infinite adjoint string
$$\ldots \dashv F \dashv G \dashv F \dashv \ldots$$
of period 2.

## References

A study of adjoint strings, in particular showing that cyclic chains of any length, and adjoint chains of any length exist, may be found in:

* {#Booth72} [[Peter I. Booth]], *Sequences of adjoint functors*, Archiv der Mathematik **23** (1972)  489-493 [[doi:10.1007/BF01304920](https://doi.org/10.1007/BF01304920)]

See also:

* {#Hoffmann80} [[Rudolf-E. Hoffmann]], _Sequences of adjoints for Ens-valued functors_, manuscripta mathematica **32** (1980) [[EuDML;154724](https://eudml.org/doc/154724)]

Characterizing the [[category of sets]] as that whose [[Yoneda embedding]] extends to the [[left adjoint|left]] to an [[adjoint quintuple]] of [[adjoint functors]]:

* {#RWSets} [[Robert Rosebrugh]], [[Richard J. Wood]], *An adjoint characterization of the category of sets*. PAMS **122** 2 (1994) 409-413 [[jstor:2161031](https://www.jstor.org/stable/2161031)]

On [[adjoint quadruples]] with a [[fully faithful functor|fully faithful]] [[right adjoint]]:

* [[Bob Rosebrugh]], [[Richard J. Wood]], *Distributive Adjoint Strings*, Theory and Applications of Categories, **1** 6 (1995) 119-145 [[tac:1-06](http://www.tac.mta.ca/tac/volumes/1995/n6/1-06abs.html)]

See also:

* {#GrothShul17} [[Moritz Groth]], [[Mike Shulman]], _Generalized stability for abstract homotopy theories_, [arXiv:1704.08084](https://arxiv.org/abs/1704.08084).

[[!redirects adjoint string]]
[[!redirects adjoint strings]]
[[!redirects adjoint chain]]
[[!redirects adjoint chains]]
[[!redirects adjoint sequence]]
[[!redirects adjoint sequences]]
[[!redirects adjoint n-tuple]]
[[!redirects adjoint n-tuples]]
[[!redirects distributive adjoint string]]
[[!redirects distributive adjoint strings]]

[[!redirects adjoint quintuple]]
[[!redirects adjoint quintuples]]


adjoint triangle theorem

+-- {: .rightHandSide}
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###Context###
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
=--
=--

# The adjoint triangle theorem
* automatic table of contents
{: toc}

## Idea

The **adjoint triangle theorem** in [[category theory]] gives conditions under which, given a [[pair]] of [[functors]] and an [[adjoint functor]], further adjoints exist.

Depending on the specific assumptions, the theorem has several variants. The following gives the most common formulation going back to [Dubuc (1968)](#dubuc68).

## Statement ##

+-- {: .num_theorem #ATT}
###### Theorem

Suppose that $U \colon B\to C$ is a [[functor]] which has a [[left adjoint]] $F \,\colon\, C\to B$ with the property that the diagram
$$
  F U F U
  \;
  \underoverset
    {\epsilon F U}
    {F U \epsilon}
    {\rightrightarrows}
   \;
   F U \xrightarrow{\epsilon} 1_B
$$
is a pointwise [[coequalizer]] (i.e. $U$ is of [[descent type]]).  Then for $A$ a category with [[reflexive coequalizer|coequalizers of reflexive pairs]], a functor $R \colon A\to B$ has a [[left adjoint]] if and only if the composite $U R$ does.

=--
+-- {: .proof}
###### Proof

The direction "only if" is obvious since adjunctions compose.  For "if", let $F'$ be a left adjoint of $U R$, and define $L:B\to A$ to be the pointwise coequalizer of
$$ F' U F U \xrightarrow{F' U \epsilon} F' U $$
and
$$
  F' U F U \xrightarrow{F' U \theta U}
  F' U R F' U \xrightarrow{\epsilon' F' U}
  F' U
$$
where $\theta:F \to R F'$ is the [[mate]] of the equality $U R = U R$ under the adjunctions $F\dashv U$ and $F'\dashv U R$.  One then verifies that this works.

=--

+-- {: .num_remark #Monadic}
###### Remark
The hypotheses on $U$ are satisfied whenever it is [[monadic functor|monadic]].
=--

+-- {: .num_remark #RegMono}
###### Remark
In fact, it suffices to assume that each counit $\epsilon : F U b \to b$ is a [[regular epimorphism]], rather than it is the coequalizer of a specific given pair of maps.  See [(Street-Verity), Lemma 2.1](#StreetVerity).
=--

## Ramifications

Similarly, the [[adjoint lifting theorem]] states conditions on a square of functors in order to ensure the existence of certain adjoints. Since a triangle can be viewed as a square with 'two sides composed', it is possible to deduce the adjoint lifting theorem from the adjoint triangle theorem as a corollary.

It is also possible to derive the [[monadicity theorem]] from the adjoint triangle theorem [Dubuc (1968)](#dubuc68).

## Related entries

* [[adjoint functor theorem]]

* [[adjoint lifting theorem]]

## References ##

* [[Michael Barr]], [[Charles Wells]], section 3.7, pp.131 in: *[[Toposes, Triples, and Theories]]*, Springer (1985), Reprints in Theories and Applications of Categories **12** (2005) 1-287 [[tac:tr12](http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.html)]

* {#dubuc68} [[Eduardo Dubuc]], _Adjoint triangles_, pp.69-81 in LNM **61** Springer Heidelberg 1968. [[doi:10.1007/BFb0077118](https://doi.org/10.1007/BFb0077118)]

* I. B. Im, [[Max Kelly|G. M. Kelly]], _Adjoint-Triangle Theorems for Conservative Functors_, Bulletin of the Australian Mathematical Society **36** 1 (1987) pp.133-136. [[doi:10.1017/S000497270002637X](https://doi.org/10.1017/S000497270002637X)]

* [[John Power]], _A unified approach to the lifting of adjoints_, Cahiers de Topologie et Géométrie Différentielle Catégoriques **29** 1 (1988) 67-77. ([numdam](http://www.numdam.org/item/CTGDC_1988__29_1_67_0))

* {#StreetVerity} [[Ross Street]], [[Dominic Verity]], _The comprehensive factorization and torsors_, Theory and Applications of Categories **23** 3 (2010) 42-75. ([TAC](http://www.tac.mta.ca/tac/volumes/23/3/23-03abs.html))

* [[Walter Tholen]], _Adjungierte Dreiecke, Colimites und Kan-Erweiterungen_, Mathematische Annalen **217** (1975) pp.121-129. ([gdz](http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002311682))

Generalizations of the adjoint triangle theorem to [[2-categories]] are considered in

* [[Fernando Lucatelli Nunes]], _On biadjoint triangles_, Theory and Applications of Categories **31** 9 (2016) 217-256. [TAC](http://tac.mta.ca/tac/volumes/31/9/31-09abs.html)

* [[Fernando Lucatelli Nunes]], _On lifting of biadjoints and lax algebras_, General Algebraic Structures with Applications **9** 1 (2018) 29-58. [[doi:10.29252/CGASA.9.1.29](https://doi.org/10.29252/CGASA.9.1.29), [arXiv:1607.03087](https://arxiv.org/abs/1607.03087)]

[[!redirects adjoint triangle]]
[[!redirects adjoint triangles]]


adjoint triple

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

> This entry is about the notion of _adjoint triple_ involving three functors. This is not to be confused with the notion of [[adjoint monads]], which were also sometimes called adjoint triples, with "triple" then being a synonym for "monad". However, an adjoint triple in the sense here does induce an [[adjoint monad]]!

#Contents#
* table of contents
{:toc}

## Definition

+-- {: .num_defn #Defn}
###### Definition

An __adjoint triple__ (of [[functors]] between [[categories]] or generally of [[1-morphisms]] in a [[2-category]])

$$
  (F \dashv G \dashv H) \colon C \to D
$$

is a [[triple]] of [[functors]]/morphisms $F,H \colon C \to D$ and $G \colon D \to C$ together with [[adjunction]] data $F\dashv G$ and $G\dashv H$.   That is, it is an [[adjoint string]] of length 3.

=--

+-- {: .num_prop #AsAdjunctionOfAdjunctions}
###### Proposition

An adjoint triple $(F\dashv G\dashv H)$, def. \ref{Defn} is equivalently an [[adjoint pair]] in the 2-category whose morphisms are adjoint pairs in the original 2-category, hence an adjunction of adjunctions

$$
  (F \dashv G) \dashv (G \dashv H)
  \,.
$$

=--

This fact plays an important role in [Licata--Shulman, 5.1](#LicataShulman).  Relatedly, it also appears in the characterization of certain kinds of [[geometric morphism]] (e.g. the [[local geometric morphism|local]] ones) in terms of adjunctions in the 2-category [[Topos]].

It may be suggestive to denote this like so

$$
  \array{
     F &\dashv& G
     \\
     \bot && \bot
     \\
     G &\dashv& H
  }
$$

such that the two adjoint pairs appear horizontally, while the second order adjunction between them runs vertically.

## Properties
 {#Properties}

+-- {: .num_note #GIsBicontinuous}
###### Note

The two adjunctions imply of course that $G$ preserves all [[limit]]s and [[colimit]]s that exist in $D$.

=--

+-- {: .num_note #AdjointPairFromAdjointTriple}
###### Note

Every adjoint triple

$$
  (F \dashv G \dashv H) \colon C \to D
$$

gives rise to an [[adjunction|adjoint pair]]

$$
  (G F \dashv G H) \colon C \to C
$$

consisting of the [[monad]] $G F$ [[left adjoint]] to the [[comonad]] $G H$ on $C$;
as well as to an adjoint pair

$$
   (F G \dashv H G) \colon D \to D
$$

consisting of the comonad $F G$ left adjoint to the monad $H G$ on $D$.

=--

See *[[adjoint monad]]* for more.

In general there is a duality (an [[dual equivalence|antiequivalence of categories]]) between the categories of monads having right adjoints and of comonads having left adjoints.

Note also (see [there](adjoint+monad#IsomorphismOfEMCategories)) that the [[algebra over a monad|algebras]] over a left-[[adjoint monad]] are identified with the [[coalgebra over a comonad|coalgebras]] for its right-adjoint comonad (duel to [Eilenberg & Moore 1965](adjoint+monad#EilenbergMoore65), see eg. [MacLane & Moerdijk 1992, Theorems V.8.1 and V.8.2](#SGL)).

### Fully faithful adjoint triples
 {#FullyFaithFulAdjointTriples}

+-- {: .num_prop #FullyFaithful}
###### Proposition

For an adjoint triple $F\dashv G\dashv H$ we have that $F$ is [[full and faithful functor|fully faithful]] if and only if $H$ is fully faithful.

=--

(In this case either [[adjoint pair]] is an [[idempotent adjunction]], see Prop. \ref{Idempotent} below.)

+-- {: .proof}
###### Proof

By a basic [[adjoint functor#FullyFaithfulAndInvertibleAdjoints|property]] of [[adjoint functors]] ([this Prop.](adjoint+functor#FullyFaithfulAndInvertibleAdjoints)), we have that

* the [[left adjoint]] $F$ being full and faithful is equivalent to the [[unit of an adjunction|unit]] $Id \to G F$ being a [[natural isomorphism]];

* the [[right adjoint]] $H$ being full and faithful is equivalent to the counit $G H \to Id$ being a [[natural isomorphism]].

Moreover, by Note \ref{AdjointPairFromAdjointTriple} and the fact that adjoints are unique up to isomorphism, we have that $G F$ is isomorphic to the identity precisely if $G H$ is.

Finally, by a standard fact about [[adjoint functors]] (see for instance ([Elephant, Lemma A1.1.1](#Elephant))), we have that $G H$ is naturally isomorphic to the identity precisely if it is so by the [[unit of an adjunction|counit]].

=--

The preceeding proposition is [[folklore]]; perhaps its earliest appearance in print is ([DT, Lemma 1.3](#DyckhoffTholen)).  A slightly shorter proof is in ([KL, Prop. 2.3](#KellyLawvere)).  Both proofs explicitly exhibit an inverse to the counit $G H \to Id$ or the unit $Id \to G F$ given an inverse to the other (which could be extracted by [[beta-reduction|beta-reducing]] the above, slightly more abstract argument). It also appears in ([SGL, Lemma 7.4.1](#SGL)).

In the situation of Proposition \ref{FullyFaithful}, we say that $F\dashv G \dashv H$ is a **fully faithful adjoint triple**.  This is often the case when $D$ is a category of "spaces" structured over $C$, where $F$ and $H$ construct "discrete" and "codiscrete" spaces respectively.

For instance, if $G\colon D\to C$ is a [[topological concrete category]], then it has both a left and right adjoint which are fully faithful.  Not every fully faithful adjoint triple is a topological concrete category (among other things, $G$ need not be [[faithful functor|faithful]]), but they do exhibit certain similar phenomena.  In particular, we have the following.

+-- {: .num_prop #FinalLifts}
###### Proposition
Suppose $(F \dashv G \dashv H) \colon C \to D$ is an adjoint triple in which $F$ and $H$ are fully faithful, and suppose that $C$ is [[cocomplete category|cocomplete]].  Then $G$ admits [[final lift|final lifts]] for [[small category|small]] $G$-structured [[sinks]].
=--
+-- {: .proof}
###### Proof
Let $\{G(S_i) \to X\}$ be a small sink in $C$, and consider the diagram in $D$ consisting of all the $S_i$, all the counits $\varepsilon\colon F G(S_i) \to S_i$ (where $F$ is the left adjoint of $G$), and all the images $F G(S_i) \to F(X)$ of the morphisms making up the sink.  The colimit of this diagram is preserved by $G$ (since it has a right adjoint as well).  But the image of the diagram consists essentially of just the sink itself (since $F$ is fully faithful, $G(\varepsilon)$ is an isomorphism), and its colimit is $X$; hence the colimit of the original diagram is a lifting of $X$ to $D$ (up to isomorphism).  It is easy to verify that this lifting has the correct universal property.
=--

Thus, we can talk about objects of $D$ having the [[weak structure]] or [[strong structure]] induced by any small collection of maps.

+-- {: .num_cor #Fibration}
###### Corollary
In the situation of Proposition \ref{FinalLifts}, $G$ is a ([[Street fibration|Street]]) [[Grothendieck fibration|opfibration]].  If it is also an [[isofibration]], then it is a Grothendieck opfibration.
=--
+-- {: .proof}
###### Proof
A final lift of a singleton sink is precisely an opcartesian arrow.
=--

Dually, of course, if $C$ is complete, then $G$ admits initial lifts for small $G$-structured cosinks and is a fibration.

In particular, the proposition and its corollary apply to a [[cohesive topos]], and (suitably categorified) to a [[cohesive (∞,1)-topos]].

### Idempotent adjoint triples

+-- {: .num_prop #Idempotent}
###### Proposition
For an adjoint triple $F\dashv G\dashv H$, the adjunction $F\dashv G$ is an [[idempotent adjunction]] if and only if the adjunction $G\dashv H$ is so.
=--
+-- {: .proof}
###### Proof
The monad $G F$ is left adjoint to the comonad $H G$, with the structure maps being [[mates]].  Therefore, by a standard fact, the category of $G F$-algebras and the category of $H G$-coalgebras are isomorphic over their common base.  However, $F\dashv G$ is idempotent precisely when $G F$ is an [[idempotent monad]], hence precisely when the forgetful functor of the category of $G F$-algebras is fully faithful, and dually for $G\dashv H$.  Since the categories of algebras are isomorphic respecting their forgetful functors, one forgetful functor is fully faithful if and only if the other is.
=--

A special case of this situation is Prop. \ref{FullyFaithful} above.

## Examples

### Special cases

* If one of the two [[adjoint pairs]] induced from an adjoint triple involving identities, then the other exhibits an _[[adjoint cylinder]]_ / _[[unity of opposites]]_.

* An adjoint triple $F\dashv G\dashv H$ is **Frobenius** if $F$ is naturally isomorphic to $H$. See [[Frobenius functor]].

* An *[[affine morphism]]* is an adjoint triple of functors in which the middle term is [[conservative functor|conservative]]. For example, any [[affine morphism of schemes]] induce an affine triples of functors among the categories of [[quasicoherent module]]s.

* An adjoint triple of functors among $A_\infty$- or [[triangulated functor]]s with certain additional structure is called **spherical** . See e.g. ([Anno](#Anno)). The main examples come from [[Serre functor]]s in a [[Calabi-Yau category]] context.

* A context of [[six operations]] $(f_! \dashv f^!)$, $(f^\ast \dashv f_\ast)$ induces an adjoint triple when either $f^! \simeq f^\ast$ or $f_! = f_\ast$. This is called a  _[[Wirthmüller context]]_ or a  _[[Grothendieck context]]_, respectively.

### Specific examples

* Given any [[ring]] [[homomorphism]] $f^\circ: R\to S$ (in commutative case dual to an [[affine morphism]] $f: Spec S\to Spec R$ of [[affine schemes]]), there is an adjoint triple $f^!\dashv f_*\dashv f^*$ where $f^*: {}_R Mod\to {}_S Mod$ is an [[extension of scalars]], $f_*: {}_S Mod\to {}_R Mod$ the restriction of scalars and $f^! : M\mapsto Hom_R ({}_R S, {}_R M)$ its [[coextension of scalars|right adjoint]]. This triple is affine in the above sense.

* If $T$ is a [[lax-idempotent 2-monad]], then a $T$-algebra $A$ has an adjunction $a : T A \rightleftarrows A : \eta_A$.  If this extends to an adjoint triple with a further left adjoint to $a$, then $A$ is called a [[continuous algebra]].

## Related concepts

* [[adjoint quadruple]], [[adjoint string]]

* [[cohesive topos]]

* [[ambidextrous adjunction]]

* [[affine morphism]], [[affine localization]]

* [[Quillen adjoint triple]]

## References

Some remarks on adjoint triples are in

* {#Johnstone} [[Peter Johnstone]], _Remarks on punctual local connectedness_, [tac/25-03](http://www.tac.mta.ca/tac/volumes/25/3/25-03abs.html).

The [[modal type theory]] of adjoint triples is discussed in

* {#LicataShulman} [[Dan Licata]] and [[Mike Shulman]], _Adjoint logic with a 2-category of modes_ ([pdf](http://dlicata.web.wesleyan.edu/pubs/ls15adjoint/ls15adjoint.pdf)).

On spherical triples see

* {#Anno} Rina Anno, _Spherical functors_, [arxiv/0711.4409](https://arxiv.org/abs/0711.4409).

Generalities are in

* {#Elephant} [[Peter Johnstone]], _[[Sketches of an Elephant]]_.

Proofs of the folklore Proposition \ref{FullyFaithful} can be found in

* {#DyckhoffTholen} Roy Dyckhoff and [[Walter Tholen]], "Exponentiable morphisms, partial products, and pullback complements", JPAA 49 (1987), 103--116.

* {#KellyLawvere} [[G.M. Kelly]] and [[F.W. Lawvere]], "On the complete lattice of essential localizations", Bulletin de la Société Mathématique de Belgique, Série A, v. 41 no 2 (1989) 289-319 [[[Kelly-Lawvere_EssentialLocalizations.pdf:file]]]

* {#SGL} [[Saunders Mac Lane]], [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic|Sheaves in Geometry and Logic --- A First Introduction to Topos Theory]]_, Springer (1992), [[doi:10.1007/978-1-4612-0927-0](https://dx.doi.org/10.1007/978-1-4612-0927-0)]

Several lemmas concerning adjoint pairs and adjoint triples are included in

* [[Alexander Rosenberg]], _Noncommutative schemes_, Compos. Math. __112__ (1998) 93--125, [doi](https://dx.doi.org/10.1023/A:1000479824211)

together with geometric consequences. Note a somewhat nonstandard usage of terminology continuous functor (also flatness in the paper includes having right adjoint).

[[!redirects adjoint triples]]
[[!redirects fully faithful adjoint triple]]
[[!redirects fully faithful adjoint triples]]
[[!redirects adjunction of adjunctions]]
[[!redirects adjunction between adjunctions]]


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