Invitation to Functorial Spaces

Abstract

We propose a generalization of the differential space concept that we call functorial differential space. It consists in replacing the standard differential structure C∞(M) of a differential space by the differential structure consisting of functions from an extended space ¯ M(A) to an algebra A, satisfying certain general conditions, but not necessarily even commutative, and ¯ M(A) being a set of mappings C∞(M) → A, interprtetd as new points. In this way, we obtain a whole family of theories of differential spaces (depending on the algebra A). It is this family that is a functorial differential space. An important class of spaces arises if we tensor multiply the differential structure of a functorial differential space by a Grassmann algebra. This leads to the concept of a functorial differential superspace (or supermanifold). In this context we also consider Einstein-Grassmann functorial super algebras.