Abstract

Can we make precise the idea that the geometry of a space is affine, euclidean, or projective at an infinitesimal level?

Yes, we can, in principle. In fact, there is a construction for first-order theories, which we call their infinitesimalisation. The models of the infinitesimalisation may be considered as spaces, which are models of that theory at an infinitesimal level.

What is considered to be at an infinitesmal level for a space is defined by a structure, which we call infinitesimal structure. For a one-sorted first-order theory the construction of infinitesimalisation introduces an infinitesimal structure, replaces every operation by a partial operation with the domains of definition specified by the infinitesimal structure, and every relation is required to factor through the infinitesimal structure. The axioms of the theory are adapted accordingly.

With these notions at hand we can show that every formal manifold in Synthetic Differential Geometry is an infinitesimal model of the algebraic theory of affine combinations, and that for every Lie group the space of points, which are infinitesimal neighbours of the neutral element, yields an infinitesimal model of a group.