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Smooth Infinitesimal Analysis

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If you have a question about this talk, please contact Filip Bár.

This talk provides an introduction to Smooth Infinitesimal Analysis (SIA) using the Kock-Lawvere axiom. We will consider first the one-dimensional case and then, after introducing the notion of Kock-Lawvere module (called ‘Euclidean R-module’ in Lavendhomme), we will move on to arbitrary dimensions.

K-L modules are stable under exponentiation and we shall see that in the smooth world a Gâteaux-differential is in fact already a Fréchet differential. In particular, our calculus works on spaces of functionals considered in the calculus of variations in the same way as in finite dimensions.

Finally, we will introduce a compatible preoder on our ring R and the integration axiom. The usual rules to calculate integrals will be obtained very easily from this and the K-L axiom. Higherdimensional integrals will be defined via iterated integration and Fubini’s theorem.

The corresponding sections in Lavendhomme’s book are 1.1.3,1.1.4, 1.2 and 1.3.

This talk is part of the Synthetic Differential Geometry Seminar series.

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