The cohomology of commutative semigroups : an overview / / Pierre Antoine Grillet

• Autore: Grillet Pierre A (Pierre Antoine), <1941->
• Pubblicazione: Cham, Switzerland : , : Springer, , [2022]; ©2022
• Descrizione fisica: 1 online resource (191 pages)
• Disciplina: 514.23
• Soggetto topico: Cohomology operations; Commutative semigroups; Homologia
• Soggetto genere / forma: Llibres electrònics
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: Intro -- Preface -- Contents -- List of Symbols -- 1 The Beginning -- 1.1 The Congruence H -- 1.1.1 Basics -- 1.1.2 Commutative Group Coextensions -- 1.2 Construction -- 1.2.1 Schreier's Method -- 1.2.2 Split Coextensions -- 1.2.3 Enter Cohomology -- 1.2.4 Finite Semigroups -- 2 Beck Cohomology -- 2.1 General Beck Cohomology -- 2.1.1 Simple Cohomology -- 2.1.2 Abelian Group Objects -- 2.1.3 Objects Over S -- 2.1.4 Beck Cohomology -- 2.1.5 Main Properties -- 2.1.6 Beck Extensions -- 2.2 Commutative Semigroups -- 2.2.1 Commutative Semigroups Over S -- 2.2.2 Abelian Group Objects Over S -- 2.2.3 Beck Extensions of S -- 2.3 Beck Cohomology of Commutative Semigroups -- 2.3.1 The `Free Commutative Semigroup' Adjunction -- 2.3.2 The `Free Commutative Semigroup' Comonad -- 2.3.3 Cochains -- 2.3.4 Cohomology -- 2.3.5 Properties -- 3 Symmetric Cohomology -- 3.1 Definition -- 3.1.1 Cochains -- 3.1.2 Symmetric Cochains -- 3.1.3 Symmetric Cohomology -- 3.1.4 An Example -- 3.2 Comparison with Beck Cohomology -- 3.2.1 Dimension 1 -- 3.2.2 Dimension 2 -- 3.2.3 Dimensions 3 and 4 -- 3.3 Main Properties -- 3.4 Normalization -- 3.4.1 Dimension 2 -- 3.4.2 Dimension 3 -- 4 Calvo-Cegarra Cohomology -- 4.1 Small Categories -- 4.2 Cohomology of Simplicial Sets -- 4.2.1 Definition -- 4.2.2 Cochains -- 4.2.3 The Classifying Simplicial Set -- 4.3 Cohomology of Commutative Semigroups -- 4.3.1 The Double Classifying Simplicial Set -- 4.3.2 Cochains -- 4.4 Extended Cochains -- 4.4.1 Definition -- 4.4.2 Comparison with Symmetric Cohomology -- 4.4.3 An Example -- 4.5 Properties -- 5 The Third Cohomology Group -- 5.1 Groupoids -- 5.1.1 Groupoids -- 5.1.2 Monoidal Groupoids -- 5.1.3 Reduction -- 5.1.4 The Base -- 5.2 Symmetric 3-Cocycles -- 5.2.1 Cocycle Objects -- 5.2.2 Morphisms -- 5.3 Classification -- 5.3.1 Isomorphisms -- 5.3.2 Equivalence -- 5.3.3 Lone Cocycles. ; 5.4 Braided Groupoids -- 5.4.1 Definition -- 5.4.2 Reduction -- 5.4.3 The Base -- 5.4.4 Extended Cocycle Objects -- 5.4.5 Classification -- 6 The Overpath Method -- 6.1 Paths and Overpaths -- 6.1.1 Free Commutative Monoids -- 6.1.2 Congruences -- 6.1.3 Paths -- 6.1.4 Overpaths -- 6.2 Main Result -- 6.2.1 Minimal Cocycles -- 6.2.2 Main Result -- 6.2.3 Examples -- 6.2.4 Semigroups with One Relator -- 6.3 Other Results -- 6.3.1 Branching -- 6.3.2 Relations -- 6.3.3 Partially Free Semigroups -- 6.3.4 Nilmonoids -- 6.3.5 Semigroups with Zero Cohomology -- 7 Symmetric Chains -- 7.1 Symmetric Mappings -- 7.1.1 Symmetry -- 7.1.2 Bases -- 7.2 Chain Groups -- 7.2.1 Definition -- 7.2.2 Properties -- 7.2.3 Symmetric n-chains -- 7.3 Chain Functors -- 7.3.1 Thin Chain Functors -- 7.3.2 General Chain Functors -- 7.4 Semiconstant Functors -- 7.4.1 Definition -- 7.4.2 Chain Groups -- 7.4.3 Properties -- 7.4.4 Homology -- 7.4.5 Cohomology -- 8 Inheritance -- 8.1 The Universal Coboundary -- 8.1.1 Symmetry Properties -- 8.1.2 The Universal Coboundary -- 8.1.3 The Group D -- 8.2 One Equality Between Variables -- 8.3 Results -- 8.3.1 Method -- 8.3.2 Order 5 -- 8.3.3 Other Orders -- 9 Appendixes -- 9.1 Extensions -- 9.1.1 Group Extensions -- 9.1.2 Rédei Extensions -- 9.1.3 The Leech Categories -- 9.1.4 Cosets -- 9.1.5 Group Coextensions -- 9.1.6 Congruences Contained in H -- 9.1.7 Leech Coextensions -- 9.1.8 Leech Cohomology -- 9.2 Monads and Algebras -- 9.2.1 Adjunctions -- 9.2.2 Monads -- 9.2.3 Algebras -- 9.3 Simplicial Objects -- 9.3.1 Simplicial Sets -- 9.3.2 The Simplicial Category -- 9.3.3 The Classifying Simplicial Set -- 9.3.4 Cohomology -- 9.4 Monoidal Categories -- 9.4.1 Strict Monoidal Categories -- 9.4.2 General Monoidal Categories -- 9.4.3 Monoidal Functors -- 9.4.4 Braided Monoidal Categories -- 9.5 Modules -- 9.5.1 S-Modules -- 9.5.2 Quasiconstant Functors. ; 9.5.3 Conclusions -- References -- Index.
• Titolo autorizzato: The Cohomology of Commutative Semigroups
• ISBN: 9783031082122; 9783031082115
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910616380303321
• Lo trovi qui: Univ. Federico II
• Serie: Lecture Notes in Mathematics