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Kock.A. and G.E. Reyes (2004) Categorical distribution theory; heat equation.arXiv:math.CT/0407242 13 Dec 2004
The theory of distributions with compact support has been developed “synthetically”, i.e., in the context of a cartesian closed category with a ring object having suitable properties, “the smooth reals” and in which everything is smooth. Topos models which contain both classical smooth manifolds and infinitesimal structures have been provided to relate it to the classical theory and the wave equation has been shown to have solutions in these models (see “Some calculus with extensive quantities: wave equation” by the authors). In this paper, we provide a similar theory for distributions which are not necessarily of compact support. As a test case, we show that the Cahiers topos is a model for this theory and admits a fundamental (distributional) solution of the heat equation on the unlimited line.
Generic figures and their glueings, A constructive approach to functor categories.
Polimetrica, corso Milano 26, Monza (MI). Marie La Palme Reyes, Gonzalo E. Reyes and Houman Zolfaghari
Abstract
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 develpments. The book is aimed (via appropiate 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.
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).
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.
Kock A. and G.E. Reyes (2003). Some calculus with extensive quantities: wave equation. arXiv: math.CT/0303297 v1 24 Mar 2003
This paper is a contribution to a synthetic theory of distributions. The sense in which we understand “synthetic” in this context is that we place ourselves in a setting (category) with a ring object having suitable properties, “the smooth reals” and where everything is smooth (differentiable). A main assumption about the category in which we work is that it is cartesian closed, meaning that function “spaces”, and hence some of the methods of functional analysis, are available. The blunt assumption of smoothness allows a simplification of the theory in the sense that distributions are functionals which are linear, continuity comes for free in this smooth universe. We provide topos models which contain both classical smooth manifolds and infinitesimal structures that relate the theory to the classical one. As a pilot project, we show how to construct the fundamental solution of the wave equation: the description of the evolution of a point (Dirac) distribution over time.
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.
Kock A. and G. E. Reyes (1999). A note on frame distributions. Cahiers de
topologie.Géom. Différentielle. Catég. 40, No. 2, 127-140.
