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
Summary
This book provides an organized exposition of the current state of the theory of commutative semigroup cohomology, a theory which was originated by the author and has matured in the past few years. The work contains a fundamental scientific study of questions in the theory. The various approaches to commutative semigroup cohomology are compared. The problems arising from definitions in higher dimensions are addressed. Computational methods are reviewed. The main application is the computation of extensions of commutative semigroups and their classification. Previously the components of the theory were scattered among a number of research articles. This work combines all parts conveniently in one volume. It will be a valuable resource for future students of and researchers in commutative semigroup cohomology and related areas
