@prefix dcat: . @prefix dct: . @prefix foaf: . @prefix vcard: . @prefix xsd: . a dcat:Dataset ; dct:description """ This thesis is devoted to an abstract theory of braided objects and its applications to a study of algebraic and topological structures. Part I presents our general homology theory for braided vector spaces and braided modules, based on the quantum co-shuffle coproduct. The construction of structural braidings characterizing different algebraic structures - self-distributive (SD) structures, associative / Leibniz algebras, their representations - allows then to generalize and unify familiar homologies. Loday's hyper-boundaries and certain homology operations are efficiently treated via our braided tools. We further introduce a concept of braided system and multi-braided module over it. This enables a thorough study of bialgebras, crossed products, bimodules, Yetter-Drinfel'd and Hopf (bi)modules: their braided interpretation, homologies and adjoint actions. A theory of multi-braided tensor products of algebras gives a unifying context for Heisenberg and Drinfel'd doubles, the algebras X of Cibils-Rosso and Y and Z of Panaite. Part III is topology-oriented. We start with a hom-set type categorification of virtual braid groups in terms of braided objects in a symmetric category (SC). This double braiding approach provides a source of representations of V Bn and a new categorical treatment for Manturov's virtual racks and the twisted Burau representation. We then define SD structures in an arbitrary SC and endow them with a braiding. The associativity and Jacobi identities in an SC are interpreted as SD conditions. Hopf algebras enter in the SD framework as well. Braided techniques from part I give a homology theory of categorical SD structures. """ ; dct:identifier "tel-00775857" ; dct:issued "2026-05-15T07:39:11.600881"^^xsd:dateTime ; dct:language "en" ; dct:modified "2026-05-15T07:39:11.600900"^^xsd:dateTime ; dct:publisher ; dct:title "Braided Objects: Unifying Algebraic Structures and Categorifying Virtual Braids" ; dcat:contactPoint [ a vcard:Organization ; vcard:fn "CCSD" ] ; dcat:distribution ; dcat:keyword "algebra-x", "algebraic-homology", "algebre-de-battage-quantique", "algebre-de-leibniz", "algebre-x", "auto-distributivite-categorique", "bimodule-de-hopf", "braided-module", "braided-object", "braided-system", "caractere", "categorical-self-distributivity", "character", "complexe-de-koszul", "crossed-product", "double-de-drinfel-", "double-de-heisenberg", "drinfeld-double", "free-virtual-quandle", "free-virtual-shelf", "groupes-de-tresses-virtuelles", "heisenberg-double", "hochschild-homology", "homologie-algebrique", "homologie-de-hochschild", "homologie-de-quandlerack", "hopf-bimodule", "hyper-bord-de-loday", "infoeu-reposemanticsdoctoralthesis", "koszul-complex", "leibniz-algebra", "lodays-hyper-boundaries", "mathmath-atmathematics-mathalgebraic-topology-mathat", "mathmath-ctmathematics-mathcategory-theory-mathct", "mathmath-ktmathematics-mathk-theory-and-homology-mathkt", "mathmath-qamathematics-mathquantum-algebra-mathqa", "module-de-yetter-drinfel-d", "module-tresse", "multi-braided-tensor-product", "objet-tresse", "produit-croise", "produit-tensoriel-multi-tresse", "quantum-shuffle-algebra", "r-matrice", "r-matrix", "rack-virtuel", "rackquandle-homology", "representation-de-burau-tordue", "structures-auto-distributives-libres", "systeme-tresse", "theses", "twisted-burau-representation", "virtual-braid-groups", "virtual-rack", "yetter-drinfel-d-module" ; dcat:landingPage . a dcat:Distribution ; dct:format "HTML" ; dct:issued "2026-05-15T07:39:11.768044"^^xsd:dateTime ; dct:modified "2026-05-15T07:39:11.417802"^^xsd:dateTime ; dct:title "Braided Objects: Unifying Algebraic Structures and Categorifying Virtual Braids" ; dcat:accessURL . a foaf:Agent ; foaf:name "test_moissonnage_selune" . a foaf:Document .