05456nam 2200517 450 99649034520331620231110224518.09783031082122(electronic bk.)9783031082115(MiAaPQ)EBC7102127(Au-PeEL)EBL7102127(CKB)24950465600041(PPN)264952618(EXLCZ)992495046560004120230224d2022 uy 0engurcnu||||||||txtrdacontentcrdamediacrrdacarrierThe cohomology of commutative semigroups an overview /Pierre Antoine GrilletCham, Switzerland :Springer,[2022]©20221 online resource (191 pages)Lecture Notes in Mathematics ;v.2307Print version: Grillet, Pierre Antoine The Cohomology of Commutative Semigroups Cham : Springer International Publishing AG,c2022 9783031082115 Includes bibliographical references and index.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.Lecture Notes in Mathematics Cohomology operationsCommutative semigroupsHomologiathubLlibres electrònicsthubCohomology operations.Commutative semigroups.Homologia514.23Grillet Pierre A(Pierre Antoine),1941-54303MiAaPQMiAaPQMiAaPQ996490345203316The Cohomology of Commutative Semigroups2920196UNISA02101nam 2200457z- 450 991055753200332120211118(CKB)5400000000044258(oapen)https://directory.doabooks.org/handle/20.500.12854/73769(oapen)doab73769(EXLCZ)99540000000004425820202111d2020 |y 0engurmn|---annantxtrdacontentcrdamediacrrdacarrierIntegrity of the Autonomic Nervous System in Psychiatric and Neurological DisordersFrontiers Media SA20201 online resource (96 p.)2-88963-627-5 This eBook is a collection of articles from a Frontiers Research Topic. Frontiers Research Topics are very popular trademarks of the Frontiers Journals Series: they are collections of at least ten articles, all centered on a particular subject. With their unique mix of varied contributions from Original Research to Review Articles, Frontiers Research Topics unify the most influential researchers, the latest key findings and historical advances in a hot research area! Find out more on how to host your own Frontiers Research Topic or contribute to one as an author by contacting the Frontiers Editorial Office: frontiersin.org/about/contactMedicine and NursingbicsscNeurology and clinical neurophysiologybicsscassessmentautonomicneurologypsychiatrytreatmentMedicine and NursingNeurology and clinical neurophysiologySiepmann Timoedt1281715Barlinn KristianedtMin-Woo Illigens BenedtSiepmann TimoothBarlinn KristianothMin-Woo Illigens BenothBOOK9910557532003321Integrity of the Autonomic Nervous System in Psychiatric and Neurological Disorders3018587UNINA