Category Archive

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.

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This paper has been superseded.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.

Kock A. and G. E. Reyes (1999). A note on frame distributions. Cahiers de
topologie.Géom. Différentielle. Catég. 40, No. 2, 127-140.

Kock A. and G. E. Reyes (1999). Aspects of fractional exponent functor. Theory Appl. Categ. 5, No. 10, 251-265. (Electronic)

We prove that certain categories arising from atoms in a Grothendieck topos are themselves Grothendieck toposes. We also investigate enrichments of these categories over the base topos; there are in fact often two distinct enrichments.

Kock A. and G. E. Reyes . Fractional exponent functors and categories of differential
equations. 33 pages. Version préliminaire (November 1998) (SDG)