Toposes of monoid actions and noncommutative geometry.

To each monoid we can naturally associate a topos, consisting of sets with a right action of this monoid. This opens the door to many geometrical invariants associated to the monoid, following the philosophy of toposes as generalized topological spaces. For example, toposes have points, and for the toposes associated to monoids, calculating the points can give surprising results. A simple example is the monoid of nonzero natural numbers under multiplication. Alain Connes and Caterina Consani showed that the points of the associated topos are up to isomorphism given by a double quotient featuring the finite adeles. They then constructed a structure sheaf on the topos, and showed that this combination of topos and structure sheaf, their Arithmetic Site, is related to the noncommutative geometry approach to the Riemann Hypothesis. In this research project, we will systematically study the toposes associated to monoids from a geometric point of view. In certain cases, we will construct structure sheaves on these toposes, leading to generalized Connes-Consani arithmetic sites.

Keywords:F1-GEOMETRY

Disciplines:Algebraic geometry, Associative rings and algebras, Category theory, homological algebra, Number theory, Order, lattices, ordered algebraic structures