The Fermat functors
type: Item
instance of: scholarly article
zbMATH Open document ID: 1365.18011
author: Enxin Wu
publication date: 2016-10-11T00:00:00Z
full work available at URL: https://arxiv.org/abs/1603.09266
full work available at URL: 31-28abs.html
Mathematics Subject Classification ID: 18F99
Mathematics Subject Classification ID: 57P99
zbMATH DE Number: 6637044
zbMATH Keywords: Fermat reals
zbMATH Keywords: Fermat spaces
zbMATH Keywords: diffeological spaces
zbMATH Keywords: the adding infinitesimal functor
zbMATH Keywords: the deleting infinitesimal functor
MaRDI profile type: Publication
arXiv ID: 1603.09266
arXiv classification: math.CT
title: The Fermat functors
published in: Theory and Applications of Categories
review text: The theory of Fermat reals [\textit{P. Giordano}, Adv.\ Math.\ 225, No. 4, 2050--2075 (2010; Zbl 1205.26051)] is a theory of infinitesimal numbers in which infinitesimals are nilpotent (as in synthetic differential geometry), but is still compatible with classical logic. The paper under review provides some fundamentals for doing differential geometry using diffeological spaces and infinitesimals.NEWLINENEWLINEThe author defines a Grothendieck site, called the Fermat site, where objects are open subsets of \(\mathbb R^n\) enlarged with Fermat infinitesimals. Then the goal is to construct comparison functors between concrete sheaves on this site, called \textit{Fermat spaces}, and diffeological spaces in the sense of Souriau. These functors are called ``adding infinitesimals'' and ``deleting infinitesimals''. The author establishes the basic properties of these functors and computes some examples, showing in particular that in his approach a \(1\)-dimensional irrational torus yields an interesting Fermat space, solving a deficiency of Giordano's earlier approach.
reviewed by: Augusto Stoffel
description: scientific article; zbMATH DE number 6637044