Semi-Cosimplicial Hilbert Spaces with Isometric Coface Operators

Semi-cosimplicial objects in the category of Hilbert spaces with isometries which are motivated by non-commutative probability theory, in particular by the distributional symmetry of spreadability, are introduced and systematically developed in various directions: partial shifts, cohomology, Hessenberg form, a related graph, decomposition into labeled subspaces, representation theory of the infinite symmetric and braid groups, classification and extensions for semi-cosimplicial sets with injective coface maps and a toy version of spreadability.