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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{adjoint modality} [[!redirects adjoint cylinder]] \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{modalities_closure_and_reflection}{}\paragraph*{{Modalities, Closure and Reflection}}\label{modalities_closure_and_reflection} [[!include modalities - contents]] \hypertarget{duality}{}\paragraph*{{Duality}}\label{duality} [[!include duality - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{SimpleIllustrativeExamples}{Simple illustrative examples}\dotfill \pageref*{SimpleIllustrativeExamples} \linebreak \noindent\hyperlink{Werden}{Werden : Sein $\dashv$ Nichts}\dotfill \pageref*{Werden} \linebreak \noindent\hyperlink{Mengen}{Quantity : discreteness $\dashv$ continuity}\dotfill \pageref*{Mengen} \linebreak \noindent\hyperlink{ContinuumRepulsionCohesion}{Continuum : repulsion $\dashv$ cohesion}\dotfill \pageref*{ContinuumRepulsionCohesion} \linebreak \noindent\hyperlink{ContinuumRepulsionCohesion}{Infinitesimal Continuum : infin. repulsion $\dashv$ infinit. cohesion}\dotfill \pageref*{ContinuumRepulsionCohesion} \linebreak \noindent\hyperlink{cohesive_sets}{Cohesive sets}\dotfill \pageref*{cohesive_sets} \linebreak \noindent\hyperlink{skeleta_and_coskeleta}{Skeleta and Co-Skeleta}\dotfill \pageref*{skeleta_and_coskeleta} \linebreak \noindent\hyperlink{FormalCompletionAndTorsionApproximation}{Formal completion $\dashv$ Torsion approximation}\dotfill \pageref*{FormalCompletionAndTorsionApproximation} \linebreak \noindent\hyperlink{fermions_and_supergeometry}{Fermions and supergeometry}\dotfill \pageref*{fermions_and_supergeometry} \linebreak \noindent\hyperlink{totally_distributive_categories}{Totally distributive categories}\dotfill \pageref*{totally_distributive_categories} \linebreak \noindent\hyperlink{recollements}{Recollements}\dotfill \pageref*{recollements} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \noindent\hyperlink{traditional}{Traditional}\dotfill \pageref*{traditional} \linebreak \noindent\hyperlink{in_terms_of_adjoint_triples_of_coreflections_and_localizations}{In terms of adjoint triples of (co-)reflections and localizations}\dotfill \pageref*{in_terms_of_adjoint_triples_of_coreflections_and_localizations} \linebreak \noindent\hyperlink{in_terms_of_adjoint_pairs_of_modal_operators}{In terms of adjoint pairs of modal operators}\dotfill \pageref*{in_terms_of_adjoint_pairs_of_modal_operators} \linebreak \noindent\hyperlink{examples_2}{Examples}\dotfill \pageref*{examples_2} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} The concept of [[adjunction]] as such expresses a [[duality]]. The stronger concept of an \emph{adjoint cylinder} or \emph{adjoint [[modality]]} is specifically an [[adjunction]] between [[idempotent monad|idempotent (co-)monads]] and is meant to express specifically a \emph{duality between opposites}. In terms of the corresponding [[adjoint triple]] of [[reflective subcategory|(co-)reflections]] and [[localizations]] the concept was suggested in (\hyperlink{Lawvere91}{Lawvere 91, p. 7}, \hyperlink{Lawvere94}{Lawvere 94, p. 11}) to capture the phenomena of ``Unity and Identity of Opposites'' as they appear informally in [[Georg Hegel]]`s \emph{[[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 \hyperlink{Wikipedia}{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 (\hyperlink{ReyesZolfaghari91}{Reyes-Zolfaghari 91}). Further developments along these lines include (\hyperlink{DJK14}{DJK 14}). \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} In (\hyperlink{Lawvere94}{Lawvere 94}) an \emph{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 \begin{displaymath} U \;\colon\; modality \dashv comodality \,, \end{displaymath} or of the form \begin{displaymath} U \;\colon\; comodality \dashv modality \,. \end{displaymath} A category equipped with an adjoint modality of the second form is called a \emph{[[category of being]]} in (\hyperlink{Lawvere91}{Lawvere 91}). If the category is a [[topos]] then this is also called a \emph{[[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]] \begin{displaymath} U X \;\colon\; \itexarray{ comodal X &\longrightarrow& X &\longrightarrow& modal X \\ opposite\;1 && unity && opposite\;2 } \end{displaymath} which is the composite of the [[counit of a comonad|counit of the comodality]] and the [[unit of a monad|unit of the modality]]. 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 \begin{displaymath} modality \dashv comodality \dashv modality \end{displaymath} (the ``Yin triple'') as for instance in the definition of \emph{[[cohesion]]} and \begin{displaymath} comodality \dashv modality \dashv comodality \end{displaymath} (the ``Yang triple'') as for instance in the definition of \emph{[[differential cohesion]]}. Since adjoint triples are equivalently [[adjunctions]] of [[adjunctions]] (\hyperlink{LicataShulman}{Licata-Shulman, section 5}), it is suggestive to denote these as \begin{displaymath} \itexarray{ \lozenge &\dashv& \bigcirc \\ \bot && \bot \\ \bigcirc &\dashv& \Box } \end{displaymath} \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \hypertarget{SimpleIllustrativeExamples}{}\subsubsection*{{Simple illustrative examples}}\label{SimpleIllustrativeExamples} The following simple illustrative example of an adjunction of the form $\Box \dashv \bigcirc$ has been suggested in (\hyperlink{Lawvere2000}{Lawvere 00}). \begin{example} \label{EvenAndOddIntegersAdjointModality}\hypertarget{EvenAndOddIntegersAdjointModality}{} \textbf{([[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 \begin{displaymath} \itexarray{ (\mathbb{Z}, \leq ) & \overset{even}{\hookrightarrow}& (\mathbb{Z},\leq) \\ n &\mapsto & 2 n } \phantom{AAAAA} \itexarray{ (\mathbb{Z}, \leq ) & \overset{odd}{\hookrightarrow}& (\mathbb{Z},\leq) \\ n &\mapsto & 2 n + 1 } \end{displaymath} of the [[even number|even]] and the [[even number|odd]] [[integers]], as well as the functor \begin{displaymath} \itexarray{ (\mathbb{Z}, \leq ) & \overset{\lfloor-/2\rfloor}{\longrightarrow}& (\mathbb{Z},\leq) \\ n &\mapsto& \lfloor n/2 \rfloor } \end{displaymath} 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]] \begin{displaymath} even \;\dashv\; \lfloor -/2 \rfloor \;\dashv\; odd \end{displaymath} and hence induce an adjoint modality \begin{displaymath} Even \;\dashv\; Odd \end{displaymath} on $(\mathbb{Z}, \leq)$ with \begin{enumerate}% \item $Even \coloneqq 2 \lfloor -/2 \rfloor$ sending any integer to its ``even [[floor]] value'' \item $Odd \coloneqq 2 \lfloor -/2 \rfloor + 1$ sending any integer to its ``odd [[ceiling]] value''. \end{enumerate} \end{example} \begin{proof} Observe that for all $n \in \mathbb{Z}$ we have \begin{displaymath} 2 \lfloor n/2 \rfloor \overset{ \epsilon_n }{\leq} n \overset{ \eta_n }{\leq} 2 \lfloor n/2 \rfloor + 1 \,, \end{displaymath} 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 \href{adjoint+functor#UniversalArrow}{this characterization} of [[adjoint functors]] \begin{enumerate}% \item the adjunction $\lfloor -/2 \rfloor \dashv odd$ is equivalent to the condition that for every $n \leq 2 k + 1$ we have $2 \lfloor n/2 \rfloor + 1 \leq 2 k + 1$; \item the adjunction $even \dashv \lfloor -/2 \rfloor$ is equivalent to the condition that for every $2k \leq n$ we have $2k \leq 2 \lfloor n/2 \rfloor$, \end{enumerate} which is readily seen to be the case \end{proof} In the same vein there is an example for an adjunction of the form $\bigcirc \dashv \Box$: \begin{example} \label{}\hypertarget{}{} 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 \emph{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 \begin{displaymath} Floor(x) \leq x \leq Ceiling(x) \,. \end{displaymath} Indeed the moments form an [[adjunction]] \begin{displaymath} Ceiling \dashv Floor \,. \end{displaymath} \end{example} \hypertarget{Werden}{}\subsubsection*{{Werden : Sein $\dashv$ Nichts}}\label{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]] \begin{displaymath} \mathbf{H} \stackrel{\overset{\vdash \ast}{\longleftarrow}}{\stackrel{\overset{}{\longrightarrow}}{\underset{\vdash \emptyset}{\longleftarrow}}} \ast \end{displaymath} 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]] \begin{displaymath} \emptyset \dashv \ast \end{displaymath} which are constant on the [[initial object]]/[[terminal object]], respectively. The induced unity transformation is \begin{displaymath} \itexarray{ \emptyset \longrightarrow X \longrightarrow \ast } \end{displaymath} hence the unique factorization of the unique function $\emptyset \longrightarrow \ast$ through any other [[type]]. Looking through (\hyperlink{Hegel1812}{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 \S{}174 of \emph{[[Science of Logic]]} it says \begin{quote}% there is nothing which is not an intermediate state between being and nothing \end{quote} which seems to be well-captured by the above unity transformation. \hypertarget{Mengen}{}\subsubsection*{{Quantity : discreteness $\dashv$ continuity}}\label{Mengen} The adjoint modality in a [[local topos]] is that given by [[flat modality]] $\dashv$ [[sharp modality]] \begin{displaymath} \flat \dashv \sharp \,. \end{displaymath} Capturing [[discrete objects]]/[[codiscrete objects]]. The corresponding unity transformation is \begin{displaymath} \flat X \longrightarrow X \longrightarrow \sharp X \end{displaymath} According to (\hyperlink{Lawvere94}{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 \emph{\hyperlink{Science+of+Logic#OnDiscretenessAndRepulsion}{On discreteness and repulsion}} one can see that matches with what Hegel calls \begin{quote}% (par 398) Quantity is the unity of these moments of continuity and discreteness \end{quote} \begin{displaymath} \itexarray{ \flat X &\longrightarrow& X &\longrightarrow& \sharp X \\ {moment\;of \atop discreteness} && && {moment\;of \atop continuity} } \end{displaymath} \hypertarget{ContinuumRepulsionCohesion}{}\subsubsection*{{Continuum : repulsion $\dashv$ cohesion}}\label{ContinuumRepulsionCohesion} For $\mathbf{H}$ a [[cohesive topos]]/[[cohesive (∞,1)-topos]] the [[shape modality]] $\dashv$ [[flat modality]] constitute an adjoint cylinder \begin{displaymath} ʃ \dashv \flat \,. \end{displaymath} The corresponding unity-transformation is the \href{cohesive%20topos#CanonicalComparison}{points-to-pieces transform} \begin{displaymath} \itexarray{ \flat X \longrightarrow X \longrightarrow ʃ X } \end{displaymath} Looking through (\hyperlink{Hegel1812}{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 \emph{[[cohesive topos]]}, this does quite well capture the geometric notion of continuum geometry. \hypertarget{ContinuumRepulsionCohesion}{}\subsubsection*{{Infinitesimal Continuum : infin. repulsion $\dashv$ infinit. cohesion}}\label{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]] \begin{displaymath} \Im \dashv \& \,. \end{displaymath} The corresponding unity-transformation is the \begin{displaymath} \itexarray{ \& X \longrightarrow X \longrightarrow \Im X } \end{displaymath} 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''. \hypertarget{cohesive_sets}{}\subsubsection*{{Cohesive sets}}\label{cohesive_sets} The combination of the above two examples of \hyperlink{ContinuumRepulsionCohesion}{Continuum} and \hyperlink{Mengen}{Quantity} is an [[adjoint triple]] of [[modalities]] \begin{displaymath} ʃ \;\dashv\; \flat \;\dashv\; \sharp \end{displaymath} [[shape modality]] $\dashv$ [[flat modality]] $\dashv$ [[sharp modality]] characteristic of a [[cohesive topos]]. \hypertarget{skeleta_and_coskeleta}{}\subsubsection*{{Skeleta and Co-Skeleta}}\label{skeleta_and_coskeleta} [[simplicial skeleton]] $\dashv$ [[simplicial coskeleton]] \hypertarget{FormalCompletionAndTorsionApproximation}{}\subsubsection*{{Formal completion $\dashv$ Torsion approximation}}\label{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) \begin{prop} \label{CompletionTorsionAdjointModalityForModuleSpectra}\hypertarget{CompletionTorsionAdjointModalityForModuleSpectra}{} 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]] \begin{displaymath} \itexarray{ \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} } \end{displaymath} where \begin{itemize}% \item $A Mod$ is the [[stable (∞,1)-category|stable]] [[(∞,1)-category of modules]], i.e. of [[∞-modules]] over $A$; \item $A Mod_{\mathfrak{a}tor}$ 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; \item $(-)^{op}$ denotes the [[opposite (∞,1)-category]]; \item the [[equivalence of (∞,1)-categories]] on the left is induced by the restriction of $\Pi_{\mathfrak{a}}$ and $\flat_{\mathfrak{a}}$. \end{itemize} \end{prop} \begin{proof} This is effectively the content of (\hyperlink{LurieProper}{Lurie ``Proper morphisms'', section 4}): \begin{itemize}% \item the existence of $\Pi_{\mathfrak{a}}$ is corollary 4.1.16 and remark 4.1.17 \item the existence of $\flat_{\mathfrak{a}}$ is lemma 4.2.2 there; \item the equivalence of sub-$\infty$-categories is proposition 4.2.5 there. \end{itemize} \end{proof} See at \emph{[[fracture theorem]]} for more. \hypertarget{fermions_and_supergeometry}{}\subsubsection*{{Fermions and supergeometry}}\label{fermions_and_supergeometry} On [[super smooth infinity-groupoids]] there is an adjoint modality deriving from the \href{super+algebra#AdjointsToInclusionOfPlainAlgebra}{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 \href{Science+of+Logic#290}{PN\S{}290} this unity of opposties might hence be called ``asunderness''. \hypertarget{totally_distributive_categories}{}\subsubsection*{{Totally distributive categories}}\label{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]] $X$. \hypertarget{recollements}{}\subsubsection*{{Recollements}}\label{recollements} See at \emph{[[recollement]]}. \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[Aufhebung]] \item [[modal type theory]] \item [[adjoint logic]] \item [[category of being]] \item [[Galois connection]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} \hypertarget{traditional}{}\subsubsection*{{Traditional}}\label{traditional} The concept of \emph{dialectical reasoning} is usually attributed to \begin{itemize}% \item [[Plato]], second part of the \emph{[[Parmenides dialogue]]} . \end{itemize} See \begin{itemize}% \item [[Georg Hegel]], \emph{[[Lectures on the History of Philosophy]] -- \href{Lectures+on+the+History+of+Philosophy#ParmenidesDialogue}{Plato -- Dialectic -- Parmenides dialogue}} \end{itemize} Hegel in his \emph{\href{https://www.marxists.org/reference/archive/hegel/works/hp/hpeleatics.htm}{History of Philosophy}} writes that dialectic begins with [[Zeno]] (one of the characters in that dialogue). This is much amplified and expanded in \begin{itemize}% \item [[Georg Hegel]], \emph{[[Science of Logic]]}, 1812 \end{itemize} The origins of its proposed formalization in [[category theory]] are recalled in \begin{itemize}% \item [[Joachim Lambek]], \emph{The Influence of Heraclitus on Modern Mathematics}, In \emph{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) \end{itemize} See also \begin{itemize}% \item [[Dieter Wandschneider]], \emph{Dialektik als Letztbegr\"u{}ndung der Logik}, in Koreanische Hegelgesellschaft (ed.), \emph{Festschrift f\"u{}r Sok-Zin Lim} Seoul 1999, 255--278 (\href{http://www.philosophie.rwth-aachen.de/global/show_document.asp?id=aaaaaaaaaabpltw}{pdf}) \item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Hegelian_dialectic}{Hegelian dialectic}} \end{itemize} \hypertarget{in_terms_of_adjoint_triples_of_coreflections_and_localizations}{}\subsubsection*{{In terms of adjoint triples of (co-)reflections and localizations}}\label{in_terms_of_adjoint_triples_of_coreflections_and_localizations} Conceived of in terms of [[adjoint triples]] of [[reflective subcategory|(co-)reflections]] and [[localization of a category|localizations]] the concept appears in \begin{itemize}% \item [[William Lawvere]], \emph{[[Some Thoughts on the Future of Category Theory]]} in A. Carboni, M. Pedicchio, G. Rosolini, \emph{Category Theory} , [[Como|Proceedings of the International Conference held in Como]], Lecture Notes in Mathematics 1488, Springer (1991) \item [[William Lawvere]], \emph{[[Cohesive Toposes and Cantor's ``lauter Einsen'']]}, Philosophia Mathematica (3) Vol. 2 (1994), pp. 5-15. ([[LawvereCohesiveToposes.pdf:file]]) \item [[William Lawvere]], \emph{[[Tools for the advancement of objective logic]]: closed categories and toposes}, in J. Macnamara and [[Gonzalo Reyes]] (Eds.), \emph{The Logical Foundations of Cognition}, Oxford University Press 1993 (Proceedings of the Febr. 1991 Vancouver Conference ``Logic and Cognition''), pages 43-56, 1994. \item [[William Lawvere]], \emph{[[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). \item [[F. W. Lawvere]], \emph{Adjoint Cylinders}, message to catlist November 2000. (\href{https://www.mta.ca/~cat-dist/archive/2000/00-11}{link}) \end{itemize} \hypertarget{in_terms_of_adjoint_pairs_of_modal_operators}{}\subsubsection*{{In terms of adjoint pairs of modal operators}}\label{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 \begin{itemize}% \item [[Gonzalo Reyes]], H. Zolfaghari, \emph{Topos-theoretic approaches to modality}, Lecture Notes in Mathematics 1488 (1991), 359-378. \item [[Gonzalo Reyes]], \emph{A topos-theoretic approach to reference and modality}, Notre Dame J. Formal Logic Volume 32, Number 3 (1991), 359-391 (\href{http://projecteuclid.org/euclid.ndjfl/1093635834}{Euclid}) \end{itemize} with further developments in \begin{itemize}% \item M. Sadrzadeh, R. Dyckho, \emph{Positive logic with adjoint modalities: Proof theory, semantics and reasoning about information}, Electronic Notes in Theoretical Computer Science 249, 451-470, 2009, in \emph{Proceedings of the 25th Conference on Mathematical Foundations of Programming Semantics} (MFPS 2009). \item [[Claudio Hermida]], section 3.3. of \emph{A categorical outlook on relational modalities and simulations}, 2010 (\href{http://maggie.cs.queensu.ca/chermida/papers/sat-sim-IandC.pdf}{pdf}) \item Wojciech Dzik, Jouni J\"a{}rvinen, Michiro Kondo, \emph{Characterising intermediate tense logics in terms of Galois connections} (\href{http://arxiv.org/abs/1401.7646}{arXiv:1401.7646}) \end{itemize} Formalization specifically in [[modal type theory]] is in \begin{itemize}% \item [[Dan Licata]], [[Mike Shulman]], \emph{Adjoint logic with a 2-category of modes} (\href{http://dlicata.web.wesleyan.edu/pubs/ls15adjoint/ls15adjoint.pdf}{pdf}) \end{itemize} For an overview of the role of adjunctions in modal logic see: \begin{itemize}% \item [[Matías Menni|M. Menni]], C. Smith, \emph{Modes of Adjointness} , J. Philos. Logic \textbf{43} no.3-4 (2014) pp.365-391. \end{itemize} \hypertarget{examples_2}{}\subsubsection*{{Examples}}\label{examples_2} \begin{itemize}% \item [[Jacob Lurie]], section 4 of \emph{[[Proper Morphisms, Completions, and the Grothendieck Existence Theorem]]} \end{itemize} [[!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]] \end{document}