Generalized information structures and their cohomology. (arXiv:1709.07807v2 [cs.IT] UPDATED)

D. Bennequin and P. Baudot introduced a cohomological construction adapted to information theory, called 'information cohomology', that characterizes Shannon entropy through a cocycle condition. This text develops the relation between information cohomology and topos theory. We also introduce several new constructions and results. First, we define generalized information structures, as categories of finite random variables related by a notion of extension or refinement; classical and quantum probability spaces appear as models (representations) for these general structures. Generalized information structures form a category with finite products and coproducts. We prove that information cohomology is invariant under isomorphisms of generalized structures. Secondly, we prove that the relatively-free bar construction gives a projective object for the computation of cohomology. Thirdly, we provide detailed computations of $H^1$ for classical probabilities and describe the degenerate cases. 查看全文>>