Monoidal geometries

Abstract

The main idea behind this research proposal is the replacement of the vector space category by another symmetric monoidal category in order to do geometry (i.e. to understand and classify additional geometric structures on smooth manifolds) in the spirit of Alain Connes's noncommutative geometry. The commutative monoids of this category replacing the vector spaces should play the role that the commutative algebras of functions on the manifold play in noncommutative geometry. The first step consists of establishing a Gelfand-Naimark type duality between smooth manifolds and the commutative monoids. The second step is to understand noncommutative monoids in terms of some additional geometric structures present on smooth manifolds. We call geometric structures that arise this way "monoidal geometries". The last step is to classify the geometric structures by classifying their corresponding monoids. An immediate first goal is to show that Poisson geometry is a monoidal geometry using, as a symmetric monoidal category, the microsymplectic category recently constructed by A. Cattaneo, A. Weinstein and the applicant in the recent article "Symplectic microgeometry I: micromorphisms", J. Sympl. Geom. Volume 8, Number 2, 1--19, 2010. (AU)