2022/12/18 by N. C. Combe, Combe, N. C.
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Computer and information sciences #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Information Theory (cs.IT) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2212.09138
openalex publication_date 2022/12/18 · openalex created_date 2023/01/04 · openalex updated_date 2026/07/28
This article develops, via the perspective of (arithmetic) algebraic geometry and category theory, different aspects of geometry of information. First, we describe in the terms of Eilenberg--Moore algebras over a Giry monad, the collection Capn of all probability distributions on the measurable space (Ωn, A) (where Ω is discrete with n issues) and it turns out that there exists an embedding relation of Segre type among the product of Capn's. We unravel hidden symmetries of these type of embeddings and show that there exists a hypercubic relation. Secondly, we show that the Manin conjecture -- initially defined concerning the diophantine geometry of Fano varieties -- is true in the case of exponential statistical manifolds, defined over a discrete sample space. Thirdly, we introduce a modified version of the parenthesised braids (mPaB), which forms a key tool in code-correction. This modified version mPaB presents all types of mistakes that could occur during a transmission process. We show that the standard parenthesised braids PaB form a full subcategory of mPaB. We discuss the role of the Grothendieck--Teichmüller group in relation to the modified parenthesised braids. Finally, we prove that the motivic Galois group is contained in the automorphism Aut(\widehatmPaB). We conclude by presenting an open question concerning rational points, Commutative Moufang Loops and information geometry.