Functorial Differential Spaces and the Infinitesimal Structure of Space-Time

Leszek Pysiak , Wiesław Sasin , Michael Heller , Tomasz Miller

Abstract

We generalize the differential space concept as a tool for developing differential geometry, and enrich this geometry with infinitesimals that allow us to penetrate into the superfine structure of space. This is achieved by Yoneda embedding a ring of smooth functions into the category of loci. This permits us to define a category of functorial differential spaces. By suitably choosing various algebras as “stages” in this category, one obtains various classes of differential spaces, both known from the literature and many so far unknown. In particular, if one chooses a Weil algebra, infinitesimals are produced.