In this book we want to explore aspects of coherence in homological algebra, that already appear in the classical situation of abelian groups or abelian categories. Lattices of subobjects are shown to play an important role in the study of homological systems, from simple chain complexes to all the structures that give rise to spectral sequences. A parallel role is played by semigroups of endorelations. These links rest on the fact that many such systems, but not all of them, live in distributive sublattices of the modular lattices of subobjects of the system. The property of distributivity allows one to work with induced morphisms in an automatically consistent way, as we prove in a 'Coherence Theorem for homological algebra'. (On the contrary, a 'non-distributive' homological structure like the bifiltered chain complex can easily lead to inconsistency, if one explores the interaction of its two spectral sequences farther than it is normally done.) The same property of distributivity also permits representations of homological structures by means of sets and lattices of subsets, yielding a precise foundation for the heuristic tool of Zeeman diagrams as universal models of spectral sequences. We thus establish an effective method of working with spectral sequences, called 'crossword chasing', that can often replace the usual complicated algebraic tools and be of much help to readers that want to apply spectral sequences in any field.

Description-Table Of Contents

Introduction. 0.1. Homological algebra in a non-abelian setting. 0.2. The coherence problem for subquotients. 0.3. The transfer functor. 0.4. Distributivity and coherence. 0.5. Universal models and crossword chasing. 0.6. Outline. 0.7. Further extensions. 0.8. Literature and terminology. 0.9. Acknowledgements -- 1. Coherence and models in homological algebra. 1.1. Some basic notions. 1.2. Coherence and distributive lattices. 1.3. Coherence and crossword diagrams. 1.4. Coherence and representations of spectral sequences. 1.5. Introducing p-exact categories. 1.6. A synopsis of the projective approach. 1.7. Free modular lattices -- 2. Puppe-exact categories. 2.1. Abelian and p-exact categories. 2.2. Subobjects, quotients and the transfer functor. 2.3. Projective p-exact categories and projective spaces. 2.4. Categories with a regular involution. 2.5. Relations for p-exact categories. 2.6. Exact squares, subquotients, induction. 2.7. Coherence, distributivity, orthodoxy. 2.8. Weak induction and the distributive expansion -- 3. Involutive categories. 3.1. RO-categories. 3.2. The 2-category of RO-categories. 3.3. Projection-completion and epi-mono factorisations. 3.4. RE-categories, I. 3.5. RE-categories, II. 3.6. RE-functors and RE-transformations. 3.7. Strict completeness of the 2-category of RE-categories -- 4. Categories of relations as RE-categories. 4.1. Puppe-exact categories and RE-categories. 4.2. Modular relations and transfer functors of RE-categories. 4.3. Complements on subquotients and regular induction. 4.4. Coherence, distributivity and orthodoxy, II. 4.5. Idempotent RE-categories. 4.6. Universal distributive and idempotent RE-categories. 4.7. Distributive joins in inverse categories -- 5. Theories and models. 5.1. Graphs and RE-graphs. 5.2. RE-theories and universal models. 5.3. Properties of RE-theories. 5.4. Universal projective models. 5.5. Criteria for idempotent theories. 5.6. EX-theories and classifying p-exact categories. 5.7. Models in semitopological spaces. 5.8. Crossword models and the Birkhoff theorem -- 6. Homological theories and their universal models. 6.1. The bifiltered object. 6.2. The sequence of relations. 6.3. The bounded filtered chain complex. 6.4. Applications. 6.5. The real filtered chain complex. 6.6. The double complex. 6.7. Eilenberg's exact system. 6.8. The discrete exact system and Massey's exact couple. 6.9. Some non-idempotent theories.