Variation on sur information theory: categories, cohomology, entropy


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Variation on sur information theory: categories, cohomology, entropy
<abstract language="fre"> L'entropie d'une variable aléatoire discrète, introduit par Shannon et généralisé par Kolmogorov, Sinai et autres, satisfait l'identité 0=H(Y|X)-H(XY)+H(X). On verra que, si on considère une catégorie S de “observables” et un topos de Grothendieck associé à ce catégorie (les préfaisceaux sur S), l'égalité ci-dessus s'interprète au niveau cohomologique comme une condition de cocycle. Sous certaines hypothèses, l'entropie apparaît comme le générateur du premier groupe de “cohomologie de l'information”, introduit par Benenquin-Baudot l'année dernière. On parlera de cette approximation catégorique aux probabilités classiques et quantiques; quelques constructions de Gromov seront aussi mentionnés. </abstract>
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Variations sur l'information : catégories, cohomologie, entropie
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Vigneaux, Juan-Pablo
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Institut des Hautes Études Scientifiques (IHÉS)
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Geometry Spacetime Free group Link (knot theory) Logarithm Multiplication sign 1 (number) Survival analysis Function (mathematics) Content (media) Variable (mathematics) Mathematics Term (mathematics) Homologiegruppe Physical law Selectivity (electronic) Subtraction Area Musical ensemble Series (mathematics) Geometry Distribution (mathematics) Expression Theory Algebraic structure Incidence algebra Measurement Computer animation Function (mathematics) Right angle Cycle (graph theory) Quantum gravity Directed graph
Commutative property Entropy Musical ensemble Group action State of matter Multiplication sign Insertion loss Mereology Variable (mathematics) Variance Connected space Object (grammar) Vector space Physical law Module (mathematics) Partition (number theory) Logical constant Spacetime Theory of relativity Real number Moment (mathematics) Physicalism Variable (mathematics) Measurement Functional (mathematics) Partition (number theory) Category of being Quantum mechanics Simplex algorithm Arrow of time Relation <Mathematik> Abelian category Identical particles Point (geometry) Momentum Set (mathematics) Real number Connectivity (graph theory) Algebraic structure Modulform Functional (mathematics) Number Measurement Wave Frequency Lecture/Conference Energy level Quantum Normal (geometry) Marginal distribution Operator (mathematics) Algebraic structure Mathematical model Set (mathematics) Group action Local Group Algebra Commutator Function (mathematics) Atomic number Physics Momentum Object (grammar) Family
Geometry Commutative property Complex (psychology) Spacetime Topology Real number Modulform Kontraktion <Mathematik> Basis (linear algebra) Infinity Mereology Theory Open set Power (physics) Number Derivation (linguistics) Chain Plane (geometry) Operator (mathematics) Vector space Convex set Physical law Homologiegruppe Continuous function Quantum Condition number Rhombus Rule of inference Spacetime Gradient Complex (psychology) Operator (mathematics) Existence Functional (mathematics) Local Group Sign (mathematics) Computer animation Vector space Function (mathematics) Chain Universe (mathematics) Helmholtz decomposition Theorem Quantum Condition number Right angle Abelian category
Area Topology Image resolution Point (geometry) Complex (psychology) Mathematical model Group action Group theory Grothendieck topology Variable (mathematics) Computer animation Homologiegruppe Abelian category
Spacetime Group action State of matter Set (mathematics) Angle Perspective (visual) Number Vector space Homologiegruppe Selectivity (electronic) Equation Multiplication Spacetime Image resolution Cycle (graph theory) Complex (psychology) Morley's categoricity theorem Mathematical model Set (mathematics) Term (mathematics) Group theory Sequence Functional (mathematics) Local Group Entropy Flow separation Category of being Computer animation Function (mathematics) Simplex algorithm Atomic number Equation Element (mathematics) Condition number Cycle (graph theory) Abelian category Funktionalgleichung Matching (graph theory)
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