nLab co-Heyting boundary



Category theory

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topos theory



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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.


Let aa be an element of a co-Heyting algebra LL with subtraction \\backslash and co-Heyting negation \sim. The (co-Heyting) boundary of aa is defined as a:=aa\partial a :=a\wedge\sim a.


  • (ab)=(ab)(ab)\partial (a\wedge b) = (\partial a \wedge b)\vee (a\wedge\partial b)\quad. (Leibniz rule)

  • (ab)(ab)=ab\partial (a\vee b)\vee\partial (a\wedge b) =\partial a\vee\partial b\quad.

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

  • Every part is the sum of its regular core and its boundary: a=aaa=\sim\sim a\vee\partial a. This suggests to view a\partial a as the irregular part of aa.


As the lattice of subtoposes of a given topos \mathcal{E} comes naturally with a co-Heyting structure it becomes (in principle) possible to define the boundary 𝒜\partial\mathcal{A} of a subtopos 𝒜\mathcal{A} in this lattice and then in turn the boundary T\partial T' of the geometric theory TT' that 𝒜\mathcal{A} classifies which is an extension of the theory TT classified by \mathcal{E} (Lawvere 1991, Caramello 2009).1

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


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

  • M. Artin, A. Grothendieck, J. L. Verdier, Théorie des Topos et Cohomologie Etale des Schémas (SGA4), LNM 269 Springer Heidelberg 1972. (exposé IV, exercise 9.4.8, pp.461-462)

  • O. Caramello, Lattices of theories , arXiv:0905.0299v1 (2009). (pdf)

  • G. M. Kelly, F. W. Lawvere, On the Complete Lattice of Essential Localizations , Bull.Soc.Math. de Belgique XLI (1989) 261-299 [pdf]

  • C. Kennett, E. Riehl, M. Roy, M. Zaks, Levels in the toposes of simplicial sets and cubical sets , JPAA 215 no.5 (2011) pp.949-961. (preprint)

  • M. La Palme Reyes, G. E. Reyes, H. Zolfaghari, Generic Figures and their Glueings , Polimetrica Milano 2004.

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

  • F. W. Lawvere, Introduction , pp.1-16 in Categories in Continuum Physics , Springer LNM 1174 1986.

  • F. W. Lawvere, Intrinsic Co-Heyting Boundaries and the Leibniz Rule in Certain Toposes , pp.279-281 in Springer LNM 1488 (1991).

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

  • F. W. Lawvere, Linearization of graphic toposes via Coxeter groups , JPAA 168 (2002) pp.425-436.

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

  • T. Mormann, Heyting Mereology as a Framework for Spatial Reasoning , Axiomathes 23 no.1 (2013) pp.237-264. (draft)

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

  • M. van Lambalgen, Logical constructions suggested by vision , Proceedings of ITALLC98, CLSI 2000. (draft)

  • C. Rauszer, Semi-Boolean algebras and their applications to intuitionistic logic with dual operations, Fund. Math. 83 no.3 (1974) pp.219-249. (pdf)

  • G. E. Reyes, H. Zolfaghari, Bi-Heyting Algebras, Toposes and Modalities , J. Phi. Logic 25 (1996) pp.25-43.

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

  • M. Zarycki, Quelque notions fondamentales de l’Analysis Situs au point de vue de l’Algèbre de la Logique , Fund. Math. IX (1927) pp.3-15. (pdf)

  • H. Zolfaghari, Topos et Modalités , Thèse de doctorat Université de Montréal 1991.

  1. 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 Aufhebungs relation). For a more complete picture of the toolbox see Lawvere (2002).

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