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Preliminary version

Covariant derivation

Anders Kock and Gonzalo Reyes

Cahiers de topologie et géométrie différentielle catégoriques, tome 47, no 1 (2006), p. 2-28.

Cet article expose une théorie synthétique des distributions (qui ne sont pas nécessairement de support compact). On compare cette théorie avec la théorie classique de Schwartz. Cette comparaison s’effectue par un plongement plein de la catégorie des espaces vectoriels convenables (et leurs applications lisses) dans certains gros topos, modèles de la géométrie différentielle synthétique.

Distributions and heat equation in SDG

Generic figures and their glueings, A constructive approach to functor categories by
Marie La Palme Reyes, Gonzalo E. Reyes and Houman Zolfaghari.  Originally published in 2004 by Polimetrica, corso Milano 26, Monza (MI), but now out of print. The entire book can be found by clicking the link below.

This book is a “missing link” between the elementary textbook of Lawvere and Schanuel “Conceptual Mathematics” and the much more advanced textbooks such as MacLane and Moerdijk “Sheaves in Geometry and Logic”. The book introduces (and limits itself) to presheaves toposes, i.e., readily visualizable categories whose objects result from glueing simpler ones, the “generic figures”. Fundamental differences between toposes and the category of sets appear at this level already. Six easy to visualize examples accompany the reader through the whole book, illuminating new material, exemplifying general results and suggesting new developments. The book is aimed (via relevant examples) at a beginner mathematician or scientist or philosopher who would like to take advantage of the richness of presheaf toposes to prepare himself or herself either for further study or applications of the theory described.

generic figures


Porta H. and G.E. Reyes (1980). Variétés à bord et topos lisses. Exposé 7, Séminaire de Géométrie différentielle synthétique, Université de Montréal, 1980. [[Re-typed by the second author in 2004]]

Le but de cet article est de plonger la catégorie des variétés à bord dans le topos de Dubuc (Cahiers topos).

Variétés à bord et topos lisses

Galli, A., Reyes G.E. and M. Sagastume (2003). Strong amalgamation, Beck-Chevalley for equivalence relations and interpolation in Algebraic Logic. Fuzzy sets and systems 138, 3-23.

We extend Makkai’s proof of strong amalgamation (push-outs of monos along arbitrary maps are monos) from the category of Heyting algebras to a class which includes the categories of symmetric bounded distributive lattices, symmetric Heyting algebras, Heyting modal S4-algebras, Heyting modal bi-S4-algebras, and Lukasiewicz n-valued algebras. We also extend and improve Pitt’s proof that strong amalgamation implies Beck-Chevalley for filters of Heyting algebras to exact categories with certain push-outs. As a consequence, a form of the Interpolation Lemma for some non-classical calculi is proved.

Strong Amalgamation

Kock A. and G.E. Reyes

Theory and Applications of Categories, Vol. 11, No. 14, 2003, pp. 321–336.

ABSTRACT. We take some first steps in providing a synthetic theory of distributions. In particular, we are interested in the use of distribution theory as foundation, not just as tool, in the study of the wave equation.

Some calculus with extensive quantities

Kock A. and G. E. Reyes. Distributions in the Cahiers topos. 34 pages. Version pré́liminaire (February 2002) (SDG)

Kock A. and G.E. Reyes (2001) Addendum: Aspects of fractional exponent functors.
Theory Appl. Categ, vol. 8. (Electronic)

Kock A. and G.E. Reyes (2001). Some differential equations in SDG. arXiv:math.CT/0104164 17 April 2001

This paper has been superseded. Its contents have been improved and re-written by the authors as three separate papers: “Some calculus with extensive quantities: wave equation”, “Categorical distribution theory; heat equation” and “Ordinary differential equations and their exponentials”. These papers may be unloaded here. I enclose it because it gives a bird’s view on our project.

Galli, A., Reyes G.E. and M. Sagastume (2000). Completeness theorems via the double dual functor. Studia Logica 64, pp 61-81.

The aim of this paper is to apply properties of the double dual endofunctor on the category of bounded distributive lattices and some extensions thereof to obtain completeness of certain non-classical propositional logics in a unified way. In particulart, we obtain completeness theorems for Moisil calculus, n-valued Lukasiewicz calculus and Nelson calculus. Furthermore we show some conservativeness results by these methods.

Completeness theorems via the double dual