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This document is a re-writing of the book of Makkai-Reyes with the same title. It is due to Francisco Marmolejo, who not only re-wrote it but made several corrections.

Makkai-Reyes book

Preliminary version

General Relativity: Affine connections, parallel transport and sprays

G.E. Reyes, Embedding manifolds with boundary in smooth toposes, Cahiers de Top. et Geom. Diff. Catég., (48), 83-103, (2007). We embed fully and faithfully the category of manifolds with boundary in some “smooth” toposes. We show that the embedding preserves open coverings and some products. Furthermore, it sends a Weil’s prolongation of a manifold by a Weil algebra into the embedded manifold raised to the associated infinitesimal space of the algebra. Our main tool is to “double” the manifold with boundary to obtain one without boundary.

Reyes G.E. A model of SDG in which only trivial distributions with compact support have a density. The aim of this note is to show that in the classifier of the theory of real analytic (i.e. C ω-)rings, linear functionals defined on the exponential whose base is the ring of “reals” (i.e., the generic model) and whose exponent is a finite power of this ring and having values in the reals, are trivial in a sense to be specified, provided that they have a density. This was conjectured by Anders Kock.

A model of SDG in which only trivial distributions with compact support have a density

Preliminary version

Metrics, connections and curvature

Preliminary version

Lie derivatives, Lie brackets and vector fields over curves

This is an improved version (in English) of the Porta/Reyes preprint(in French) in the preceeding entry. Its aim is to embedd manifolds with boundary in smooth toposes defined by closed ideals. To appear in Cahiers de Topologie et Géométrie différentielle catégoriques.

Embedding manifolds with boundary in smooth toposes

Preliminary version

Curvature and Torsion of a connection

A. Kock and G. E. Reyes, Ordinary differential equations and their exponentials, Central European J. of Math. 4 (2006), 64-81

Vector fields or, equivalently, ordinary differential equations have long been considered, heuristically, to be the same as “infinitesimal (pointed) actions” or “infinitesimal flows”, but it is only with the development of Synthetic Differential Geometry (SDG) that we have the tools to formulate these notions and prove their equivalence in a rigourous mathematical way. We exploit this fact to define the exponential of two ordinary differential equations as the exponential of the corresponding infinitesimal actions. The resulting action is seen to be the same as a partial differential equation whose solutions may be obtained by conjugation from the solutions of the differential equations that make up the exponential. Furthermore, we show that this method of conjugation is equivalent, under some conditions, to the method of change of variables, widely used to solve differential equations.

Ordinary differential equations and their exponentials

Preliminary version

The Riemann-Christoffel tensor