Cohomologies of crossed modules

I. Categories, Functors and Natural Transformations.- 1. Axioms for Categories.- 2. Categories.- 3. Functors.- 4. Natural Transformations.- 5. Monics, Epis, and Zeros.- 6. Foundations.- 7. Large Categories.- 8. Hom-sets.- II. Constructions on Categories.- 1. Duality.- 2. Contravariance and Opposites.- 3. Products of Categories.- 4. Functor Categories.- 5. The Category of All Categories.- 6. Comma Categories.- 7. Graphs and Free Categories.- 8. Quotient Categories.- III. Universals and Limits.- 1. Universal Arrows.- 2. The Yoneda Lemma.- 3. Coproducts and Colimits.- 4. Products and Limits.- 5. Categories with Finite Products.- 6. Groups in Categories.- IV. Adjoints.- 1. Adjunctions.- 2. Examples of Adjoints.- 3. Reflective Subcategories.- 4. Equivalence of Categories.- 5. Adjoints for Preorders.- 6. Cartesian Closed Categories.- 7. Transformations of Adjoints.- 8. Composition of Adjoints.- V. Limits.- 1. Creation of Limits.- 2. Limits by Products and Equalizers.- 3. Limits with Parameters.- 4. Preservation of Limits.- 5. Adjoints on Limits.- 6. Freyd's Adjoint Functor Theorem.- 7. Subobjects and Generators.- 8. The Special Adjoint Functor Theorem.- 9. Adjoints in Topology.- VI. Monads and Algebras.- 1. Monads in a Category.- 2. Algebras for a Monad.- 3. The Comparison with Algebras.- 4. Words and Free Semigroups.- 5. Free Algebras for a Monad.- 6. Split Coequalizers.- 7. Beck's Theorem.- 8. Algebras are T-algebras.- 9. Compact Hausdorff Spaces.- VII. Monoids.- 1. Monoidal Categories.- 2. Coherence.- 3. Monoids.- 4. Actions.- 5. The Simplicial Category.- 6. Monads and Homology.- 7. Closed Categories.- 8. Compactly Generated Spaces.- 9. Loops and Suspensions.- VIII. Abelian Categories.- 1. Kernels and Cokernels.- 2. Additive Categories.- 3. Abelian Categories.- 4. Diagram Lemmas.- IX. Special Limits.- 1. Filtered Limits.- 2. Interchange of Limits.- 3. Final Functors.- 4. Diagonal Naturality.- 5. Ends.- 6. Coends.- 7. Ends with Parameters.- 8. Iterated Ends and Limits.- X. Kan Extensions.- 1. Adjoints and Limits.- 2. Weak Universality.- 3. The Kan Extension.- 4. Kan Extensions as Coends.- 5. Pointwise Kan Extensions.- 6. Density.- 7. All Concepts are Kan Extensions.- Table of Terminology.

background

Categories for the Working Mathematician

This advanced textbook introduces students to cohomology theory. No knowledge of homological algebra is assumed beyond what is normally taught in a first course in algebraic topology.

Cohomology of Groups

Since it was first published in 1967, Simplicial Objects in Algebraic Topology has been the standard reference for the theory of simplicial sets and their relationship to the homotopy theory of topological spaces. J. Peter May gives a lucid account of the basic homotopy theory of simplicial sets (discrete analogs of topological spaces) which have played a central role in algebraic topology ever since their introduction in the late 1940s. "Simplicial Objects in Algebraic Topology presents much of the elementary material of algebraic topology from the semi-simplicial viewpoint. It should prove very valuable to anyone wishing to learn semi-simplicial topology. [May] has included detailed proofs, and he has succeeded very well in the task of organizing a large body of previously scattered material."--Mathematical Review

Additional excerpts

Simplicial objects in algebraic topology

Braided Tensor Categories

I. Modules.- 1. Modules.- 2. The Group of Homomorphisms.- 3. Sums and Products.- 4. Free and Projective Modules.- 5. Projective Modules over a Principal Ideal Domain.- 6. Dualization, Injective Modules.- 7 Injective Modules over a Principal Ideal Domain.- 8. Cofree Modules.- 9. Essential Extensions.- II. Categories and Functors.- 1. Categories.- 2. Functors.- 3. Duality.- 4. Natural Transformations.- 5. Products and Coproducts Universal Constructions.- 6. Universal Constructions (Continued) Pull-backs and Push-outs.- 7. Adjoint Functors.- 8. Adjoint Functors and Universal Constructions.- 9. Abelian Categories.- 10. Projective, Injective, and Free Objects.- III. Extensions of Modules.- 1. Extensions.- 2. The Functor Ext.- 3. Ext Using Injectives.- 4. Computation of some Ext-Groups.- 5. Two Exact Sequences.- 6. A Theorem of Stein-Serre for Abelian Groups.- 7. The Tensor Product.- 8. The Functor Tor.- IV. Derived Functors.- 1. Complexes.- 2. The Long Exact (Co) Homology Sequence.- 3. Homotopy.- 4. Resolutions.- 5. Derived Functors.- 6. The Two Long Exact Sequences of Derived Functors.- 7. The Functors Extn? Using Projectives.- 8. The Functors % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbWexLMBb50ujbqegm0B % 1jxALjharqqr1ngBPrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY- % Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq % 0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaae % aaeaaakeaadaqdaaqaaGqaaiaa-veacaWF4bGaa8hDaaaadaqhaaWc % baacciGae43MdWeabaGaamOBaaaaaaa!40A3! $$ \overline {Ext} _\Lambda ^n $$ Using Injectives.- 9. Extn and n-Extensions.- 10. Another Characterization of Derived Functors.- 11. The Functor Torn?.- 12. Change of Rings.- V. The Kiinneth Formula.- 1. Double Complexes.- 2. The Kunneth Theorem.- 3. The Dual Kunneth Theorem.- 4. Applications of the Kunneth Formulas.- VI. Cohomology of Groups.- 1. The Group Ring.- 2. Definition of (Co) Homology.- 3. H0, H0.- 4. H1, H1 with Trivial Coefficient Modules.- 5. The Augmentation Ideal, Derivations, and the Semi-Direct Product.- 6. A Short Exact Sequence.- 7. The (Co) Homology of Finite Cyclic Groups.- 8. The 5-Term Exact Sequences.- 9. H2, Hopf's Formula, and the Lower Central Series.- 10. H2 and Extensions.- 11. Relative Projectives and Relative Injectives.- 12. Reduction Theorems.- 13. Resolutions.- 14. The (Co) Homology of a Coproduct.- 15. The Universal Coefficient Theorem and the (Co)Homology of a Product.- 16. Groups and Subgroups.- VII. Cohomology of Lie Algebras.- 1. Lie Algebras and their Universal Enveloping Algebra.- 2. Definition of Cohomology H0, H1.- 3. H2 and Extensions.- 4. A Resolution of the Ground Field K.- 5. Semi-simple Lie Algebras.- 6. The two Whitehead Lemmas.- 7. Appendix : Hubert's Chain-of-Syzygies Theorem.- VIII. Exact Couples and Spectral Sequences.- 1. Exact Couples and Spectral Sequences.- 2. Filtered Differential Objects.- 3. Finite Convergence Conditions for Filtered Chain Complexes.- 4. The Ladder of an Exact Couple.- 5. Limits.- 6. Rees Systems and Filtered Complexes.- 7. The Limit of a Rees System.- 8. Completions of Filtrations.- 9. The Grothendieck Spectral Sequence.- IX. Satellites and Homology.- 1. Projective Classes of Epimorphisms.- 2. ?-Derived Functors.- 3. ?-Satellites.- 4. The Adjoint Theorem and Examples.- 5. Kan Extensions and Homology.- 6. Applications: Homology of Small Categories, Spectral Sequences.- X. Some Applications and Recent Developments.- 1. Homological Algebra and Algebraic Topology.- 2. Nilpotent Groups.- 3. Finiteness Conditions on Groups.- 4. Modular Representation Theory.- 5. Stable and Derived Categories.

Cohomologies of crossed modules

References

Categories for the Working Mathematician

Cohomology of Groups

Simplicial objects in algebraic topology

Braided Tensor Categories

A Course in Homological Algebra