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\section*{co-Heyting boundary}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory}

[[!include topos theory - 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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{remark}{Remark}\dotfill \pageref*{remark} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


[[!redirects coHeyting boundary]] [[!redirects Co-Heyting boundary]] [[!redirects boundary operator]]

\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In a [[co-Heyting algebra]] it is possible to define an equivalent to the [[boundary]] operation in [[topology]]. In applications this affords an intrinsic view of [[mereology]] without consideration of the embedding of bodies into an ambient space.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Let $a$ be an element of a [[co-Heyting algebra]] $L$ with subtraction $\backslash$ and [[co-Heyting negation]] $\sim$. The \textbf{(co-Heyting) boundary} of $a$ is defined as $\partial a :=a\wedge\sim a$.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{quote}%

\end{quote}
\begin{itemize}%
\item $\partial (a\wedge b) = (\partial a \wedge b)\vee (a\wedge\partial b)\quad$. (\textbf{Leibniz rule})


\item $\partial (a\vee b)\vee\partial (a\wedge b) =\partial a\vee\partial b\quad$.


\item The boundaries $x=\partial a$ can be characterized as those $x$ with $\partial x = x$ (in particular, $\partial^2=\partial$), or, alternatively, those $x$ with $\sim x=1$, showing that boundary parts are precisely the (intuitively) \textbf{thin parts}.


\item Every part is the sum of its \emph{regular core} and its boundary: $a=\sim\sim a\vee\partial a$. This suggests to view $\partial a$ as the \emph{irregular} part of $a$.



\end{itemize}
\hypertarget{remark}{}\subsection*{{Remark}}\label{remark}

As the lattice of [[subtopos|subtoposes]] of a given [[topos]] $\mathcal{E}$ comes naturally with a co-Heyting structure it becomes (in principle) possible to define the boundary $\partial\mathcal{A}$ of a subtopos $\mathcal{A}$ in this lattice and then in turn the boundary $\partial T'$ of the [[geometric theory]] $T'$ that $\mathcal{A}$ [[classifying topos|classifies]] which is an extension of the theory $T$ classified by $\mathcal{E}$ (\hyperlink{Law91a}{Lawvere 1991}, \hyperlink{Cara09}{Caramello 2009}).\footnote{The boundary operation is one in Lawvere's arsenal of (mereological) tools for the study of logical theories in the context of a topos and its subtoposes (besides e.g. the [[Aufhebung|Aufhebungs relation]]). For a more complete picture of the toolbox see \hyperlink{Law02}{Lawvere (2002)}.} 

When a subtopos $\mathcal{A}$ is complemented in the lattice of subtoposes, as occurs e.g. for [[open subtopos|open]], [[closed subtopos|closed]] or [[locally closed subtopos|locally closed]] subtoposes, its boundary vanishes: $\partial\mathcal{A}=0$ since in a co-Heyting algebra the complement $\neg a$ necessarily coincides with $\sim a$ (cf. at [[co-Heyting negation]]).

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[boundary]]
\item [[co-Heyting negation]]
\item [[co-Heyting algebra]]
\item [[Heyting algebra]]
\item [[Heyting category]]
\item [[mereology]]
\item [[subtopos]]
\item [[Aufhebung]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Closed boundaries of subtoposes are defined in an exercise of \hyperlink{SGA4}{SGA4} (cf. at [[open subtopos]] for some of the details). The concept for co-Heyting algebras seems to stem from Lawvere (\hyperlink{Law76}{1976},\hyperlink{Law86}{1986}, \hyperlink{Law91a}{1991}) although the 1927 \hyperlink{Zar27}{article of M. Zarycki} already studies properties and axiomatic potential of the boundary operator in topology. \hyperlink{RRZ04}{La Palme Reyes, Reyes\&Zolfaghari (2004)} has an introductory exposition in the context of bi-Heyting algebras. For boundaries of geometric theories \hyperlink{Cara09}{Caramello (2009)} is essential reading although they don't appear there explicitly. For mereological applications of the concept see \hyperlink{Law86}{Lawvere (1986)}, \hyperlink{SW97}{Stell\&Worboys (1997)}, \hyperlink{Pagl09}{Pagliani (2009)} and \hyperlink{Mor13}{Mormann (2013)}.

\begin{itemize}%
\item [[M. Artin]], [[A. Grothendieck]], [[J. L. Verdier]], \emph{Th\'e{}orie des Topos et Cohomologie Etale des Sch\'e{}mas ([[SGA4]])}, LNM \textbf{269} Springer Heidelberg 1972. (expos\'e{} IV, exercise 9.4.8, pp.461-462)


\item [[Olivia Caramello|O. Caramello]], \emph{Lattices of theories} , arXiv:0905.0299v1 (2009). (\href{http://arxiv.org/pdf/0905.0299v1}{pdf})


\item [[G. M. Kelly]], F. W. Lawvere, \emph{On the Complete Lattice of Essential Localizations} , Bull.Soc.Math. de Belgique \textbf{XLI} (1989) pp.261-299.


\item C. Kennett, [[Emily Riehl|E. Riehl]], M. Roy, M. Zaks, \emph{Levels in the toposes of simplicial sets and cubical sets} , JPAA \textbf{215} no.5 (2011) pp.949-961. (\href{http://arxiv.org/abs/1003.5944}{preprint})


\item M. La Palme Reyes, [[Gonzalo E. Reyes|G. E. Reyes]], H. Zolfaghari, \emph{Generic Figures and their Glueings} , Polimetrica Milano 2004.


\item [[William Lawvere]], \emph{Variable Quantities and Variable Structures in Topoi} , pp.101-131 in Heller, Tierney (eds.), \emph{Algebra, Topology and Category Theory: a Collection of Papers in Honor of Samuel Eilenberg} , Academic Press New York 1976.


\item [[William Lawvere|F. W. Lawvere]], \emph{Introduction} , pp.1-16 in \emph{Categories in Continuum Physics} , Springer LNM \textbf{1174} 1986.


\item F. W. Lawvere, \emph{Intrinsic Co-Heyting Boundaries and the Leibniz Rule in Certain Toposes} , pp.279-281 in Springer LNM \textbf{1488} (1991).


\item F. W. Lawvere, \emph{Tools for the Advancement of Objective Logic: Closed Categories and Toposes}, pp.43-56 in: J. Macnamara, G. E. Reyes (eds.), \emph{The Logical Foundations of Cognition} , Oxford UP 1994.


\item F. W. Lawvere, \emph{Linearization of graphic toposes via Coxeter groups} , JPAA \textbf{168} (2002) pp.425-436.


\item [[Matías Menni|M. Menni]], C. Smith, \emph{Modes of Adjointness} , J. Philos. Logic \textbf{43} no.3-4 (2014) pp.365-391.


\item T. Mormann, \emph{Heyting Mereology as a Framework for Spatial Reasoning} , Axiomathes \textbf{23} no.1 (2013) pp.237-264. (\href{http://philpapers.org/go.pl?id=MORCMA-3&proxyId=&u=http%3A%2F%2Fphilpapers.org%2Farchive%2FMORCMA-3.pdf}{draft})


\item P. Pagliani, \emph{Intrinsic co-Heyting boundaries and information incompleteness in Rough Set Analysis} , pp.123-130 in Polkowski, Skowron (eds.), \emph{RSCTC 1998} , Springer LNCS \textbf{1424} (2009).


\item M. van Lambalgen, \emph{Logical constructions suggested by vision} , Proceedings of ITALLC98, CLSI 2000. (\href{http://staff.science.uva.nl/~michiell/docs/ITALLCCSLI.pdf}{draft})


\item C. Rauszer, \emph{Semi-Boolean algebras and their applications to intuitionistic logic with dual operations}, Fund. Math. \textbf{83} no.3 (1974) pp.219-249. (\href{http://matwbn.icm.edu.pl/ksiazki/fm/fm83/fm83120.pdf}{pdf})


\item [[Gonzalo E. Reyes|G. E. Reyes]], H. Zolfaghari, \emph{Bi-Heyting Algebras, Toposes and Modalities} , J. Phi. Logic \textbf{25} (1996) pp.25-43.


\item J.G. Stell, M.F. Worboys, \emph{The algebraic structure of sets of regions} , pp.163-174 in Hirtle, Frank (eds.), \emph{Spatial Information Theory}, Springer LNCS \textbf{1329} (1997).


\item M. Zarycki, \emph{Quelque notions fondamentales de l'Analysis Situs au point de vue de l'Alg\`e{}bre de la Logique} , Fund. Math. \textbf{IX} (1927) pp.3-15. (\href{http://matwbn.icm.edu.pl/ksiazki/fm/fm9/fm912.pdf}{pdf})


\item H. Zolfaghari, \emph{Topos et Modalit\'e{}s} , Th\`e{}se de doctorat Universit\'e{} de Montr\'e{}al 1991.



\end{itemize}


\end{document}