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duality

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## 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 (co-)monads and is meant to express specifically a duality between opposites.

In terms of the corresponding adjoint triple of (co-)reflections and localizations the concept was suggested in (Lawvere 91, p. 7, Lawvere 94, p. 11) 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 ).

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). Further developments along these lines include (DJK 14).

## Definition

In (Lawvere 94) 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 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). 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 the comodality and the unit of the modality (Lawvere-Rosebrugh 03, p. 245).

If for two 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), it is suggestive to denote these as

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

## Examples

### Simple illustrative examples

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

###### Example

(even and 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 and the 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”

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

###### 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 $\eta$ and counit $\epsilon$.

Hence by this characterization 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 $2 \lfloor n/2 \rfloor + 1 \leq 2 k + 1$;

2. 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 $\bigcirc \dashv \Box$:

###### 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 : Sein $\dashv$ Nichts

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

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

given by including the initial object $\emptyset$ and the terminal object $\ast$ into $\mathbf{H}$.

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

$\emptyset \dashv \ast$

which are constant on the initial object/terminal object, respectively.

The induced unity transformation is

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

hence the unique factorization of the unique function $\emptyset \longrightarrow \ast$ through any other type.

Looking through (Hegel 1812, vol 1, book 1, section 1, chapter 1) 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

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) 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 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

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

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

Looking through (Hegel 1812, vol 1, book 1, section 2, chapter 1) 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

For $\mathbf{H}$ equipped moreover with differential cohesion, there is the 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 and Quantity 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

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 of ∞-modules $A Mod_\infty$ over $A$.

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

###### Proposition

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

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

where

• $A Mod$ is the stable (∞,1)-category 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 and of $\mathfrak{a}$-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

This is effectively the content of (Lurie “Proper morphisms”, section 4):

• 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. 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 this unity of opposties 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.

The concept of dialectical reasoning is usually attributed to

See

Hegel in his History of Philosophy writes that dialectic begins with Zeno (one of the characters in that dialogue).

This is much amplified and expanded in

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

• 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)

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

Conceived of in terms of adjoint triples of (co-)reflections and localizations the concept appears in

### 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

• Gonzalo Reyes, H. Zolfaghari, Topos-theoretic approaches to modality, Lecture Notes in Mathematics 1488 (1991), 359-378.

• Gonzalo Reyes, A topos-theoretic approach to reference and modality, Notre Dame J. Formal Logic Volume 32, Number 3 (1991), 359-391 (Euclid)

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).

• Claudio Hermida, section 3.3. of A categorical outlook on relational modalities and simulations, 2010 (pdf)

• Wojciech Dzik, Jouni Järvinen, Michiro Kondo, Characterising intermediate tense logics in terms of Galois connections (arXiv:1401.7646)

Formalization specifically in modal type theory is in

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

• M. Menni, C. Smith, Modes of Adjointness , J. Philos. Logic 43 no.3-4 (2014) pp.365-391.