We propose here a study of 'semiexact' and 'homological' categories as a basis for a generalised homological algebra. Our aim is to extend the homological notions to deeply non-abelian situations.
This is a sequel of a book [G20], referred to as Part I, and achieves its goals but can be read independently of the latter.
Part I develops homological algebra in p-exact categories, i.e. exact categories in the sense of Puppe and Mitchell [P2, Mt, HeS, AHS, FS] - a moderate generalisation of abelian categories that is nevertheless crucial for a theory of 'coherence' and 'universal models' of (even abelian) homological algebra. The main motivation of the present, much wider extension is that the exact sequences or spectral sequences produced by unstable homotopy theory cannot be dealt with in the previous framework.
As in Part I, our research follows a sort of 'projective' approach, where lattices (and sublattices) of normal subobjects play an important role. The properties of modularity or distributivity of these lattices are crucial, in contrast with the abelian case where modularity is automatically granted and distributivity is impossible at all.
The role of additivity is more important here than in Part I. Indeed, while the additive (or even semiadditive) p-exact case reduces to the abelian one, and was not developed in Part I, additive homological categories need not be abelian. Homotopies of chain complexes on these categories are considered here, if in a marginal way; in fact, this study is mainly related to such constructions as mapping cylinder and suspension, derived and triangulated categories, that should rather be viewed as a part of homotopical algebra, and need an appropriate investigation.
According to the present definitions, a semiexact category is a category equipped with an ideal of 'null' morphisms and provided with kernels and cokernels with respect to this ideal. A homological category satisfies some further conditions that allow the construction of subquotients and induced morphisms, in particular the homology of a chain complex or the spectral sequence of an exact couple. Both settings are self-dual.
These notions have been introduced in [G10, G11, G13, G14]. Extending abelian categories, and also the p-exact ones, they include the usual domains of homology and homotopy theories, e.g. the category of 'pairs' of topological spaces or groups; they also include their codomains, since the sequences of homotopy 'objects' for a pair of pointed spaces or a fibration can be viewed as exact sequences in a homological category, whose objects are actions of groups on pointed sets.
Our view is thus quite distinct from the more usual affine generalisations of the abelian framework, that are based on finite limits: Barr-exact categories, their extensions and variations. There seems to be a sort of opposition between a projective and an affine approach, that will be further analysed in the Introduction.

Preface vii

Introduction 1

0.1 Categorical settings for homological algebra 1

0.2 Semiexact, homological and generalised exact categories 3

0.3 Subquotients and homology 5

0.4 Satellites 5

0.5 Exact centres, expansions, fractions and relations 6

0.6 Applications 6

0.7 Homological theories and biuniversal models 8

0.8 Modularity and additivity 8

0.9 A list of examples 9

0.10 Terminology and notation 11

0.11 Acknowledgements 12

1 Semiexact categories 14

1.1 Some basic notions 14

1.2 Lattices and Galois connections 21

1.3 The main definitions 29

1.4 Structural examples 36

1.5 Semiexact categories and normal subobjects 43

1.6 Other examples of semiexact and homological categories 53

1.7 Exact functors 61

2 Homological categories 73

2.1 The transfer functor and ex2-categories 73

2.2 Characterisations of homological and g-exact categories 77

2.3 Modular semiexact categories and diagram lemmas 85

2.4 Additivity and monoidal structures 90

2.5 Abstract categories of pairs 95

2.6 The epi-mono completion of a category 103

2.7 From homotopy categories to homological categories 114

2.8 A digression on weak subobjects and smallness 119

3 Subquotients, homology and exact couples 125

3.1 Subquotients in homological categories 125

3.2 Induction, exactness and modularity 131

3.3 Chain complexes and homology 136

3.4 Chain homotopies, semiadditivity and additivity 145

3.5 The exact couple 148

4 Satellites 155

4.1 Connected sequences and satellites 155

4.2 Calculus of satellites 164

4.3 Effacements 171

4.4 Satellites for categories of pairs 179

4.5 Chain homology functors as satellites 185

5 Universal constructions 191

5.1 Coreflectors, exact centres and expansions 191

5.2 Fractions for semiexact categories 199

5.3 Complements and reflectors 206

5.4 Relations for g-exact categories 210

5.5 Fractions for g-exact categories 221

5.6 The homological structure of EX and AB 232

6 Applications to algebraic topology 238

6.1 Homology theories and their transformations 239

6.2 Singular homology for fibrations 248

6.3 Homotopy spectral sequences for path-connected spaces 251

6.4 Actions and homotopy theory 262

6.5 Homotopy spectral sequences and normalised actions 271

7 Homological theories and biuniversal models 278

7.1 Categories for biuniversal models 278

7.2 Homological and g-exact theories 284

7.3 The modular bifiltered object 288

7.4 The modular sequence of morphisms 296

7.5 The modular exact couple 300

Appendix A Some points of category theory 307

A1 Basic notions 307

A2 Limits and colimits 318

A3 Adjoint functors 323

A4 Monoidal categories and two-dimensional categories 325

References 331

Index 337

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