Information, Geometry, and Physics Seminar

In joint work with S. Nakamura, I showed that the category of projective geometries and collineations can be fully and faithfully embedded into Connes and Consani's ΓSet model for modules over the field with one element. It follows that one can produce simplicial sets from projective geometries via G. Segal's "delooping" procedure, an extension of the standard classifying space construction for Abelian groups. Dynkin systems are seemingly unrelated structures generalizing σ-algebras in measure theory. Historically, as a result of Dynkin's πσ-Theorem, they have been useful in studying Markov processes. However, they have also been proposed as an alternative to σ-algebras in quantum probability and arise as sets of "precise" events in imprecise probability theory. In this talk, I will describe a recent computation of the delooping of the discrete projective geometry on n points. The resulting simplicial set is the n-fold wedge sum of a simplicial set of (pointed) Dynkin systems containing σ-algebras as a sub-simplicial set. In particular, the delooping of the one-point trivial geometry is the simplicial set of Dynkin systems on finite sets.