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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.

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 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

Reyes G.E. A mathematical analysis of Masaccio’s Trinity. Preliminary version (February 2004). Published in Categories and Types in Logic, Language, and Physics. Editors C. Casadio et al. Springer LNCS 8222. 2014.

The aim of this note is to study several questions of a mathematical nature suggested by this fresco: (1) How accurate is the use of perspective? (2) What are the dimensions of the chapel? (3) What are the dimensions of the coffers of the vaulted ceiling of the chapel? (4) Where is the point of view situated with respect to the fresco? (5) Where are the different characters situated inside the chapel? (6) What are the “real” heights of the characters portrayed? Questions (1)-(4) admit answers that may be computed starting from the data of the fresco, by using some rules of perspective and simple mathematical facts. This is not true for the others. Nevertheless, we will show that under some reasonable hypotheses estimates may be made. A pictorial reproduction of the Trinity may be unloaded by clicking the next document in Varia: “Masaccio Trinity in the WEB”.

A mathematical analysis of Masaccio’s Trinity

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

Reyes G.E. and A. Royer (2003) On the law of motion in Special Relativiyt. arXiv: physics/0302065 v1 19 Feb 2003

Newton’s law of motion for a particle of a given mass subject to a force at a given time may formulated either as “force = rate of change of the quantity of motion” or, since the mass is constant, as “force = mass times acceleration”, where velocity and acceleration are relative to an inertial frame. This law may be interpreted in either of two ways: (1) The force acting on the particle at the given time during an infinitesimal lapse of time imparts to the laboratory a boost, while the particle maintains its velocity relative to the new frame. (2) The force acting on the particle at a given time during an infinitesimal lapse of time imparts to the particle a boost relative to its proper frame which moves with the same velocity relative to the laboratory. We show that the relativistic law of motion admits both interpretations, the first of which is in fact equivalent to this law. As a consequence, we show that the relativistic law of motion may also be formulated as “force = mass times acceleration” in analogy with Newton’s law, but with a relativistic mass and a relativistic acceleration defined in terms of the relativistic addition law of velocities, rather than ordinary mass and ordinary vectorial addition of velocities that lead to the classical acceleration and to Newton’s law.

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

Royer A. and G.E. Reyes. Lorentz transformation matrices in (3,1) block form. 18 pages. Version préliminaire (Août 2001)

Reyes G.E. and A. Royer . Forces and the equation of motion in special relativity. 29 pages. Version préliminaire (Août 2001)

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