Homological Algebra: In Strongly Non-Abelian Settings

356 Pages·2013·2.56 MB·English

by  Grandis

Homological
Algebra
In Strongly Non-Abelian Settings
8608hc_9789814425919_tp.indd 1 19/11/12 4:28 PM
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
Homological
Algebra
In Strongly Non-Abelian Settings
Marco Grandis
Università di Genova, Italy
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI
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HOMOLOGICAL ALGEBRA
In Strongly Non-Abelian Settings
Copyright © 2013 by World Scientific Publishing Co. Pte. Ltd.
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ISBN 978-981-4425-91-9
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To my granddaughters
Alessandra and Elena
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Preface
We propose here a study of ‘semiexact’ and ‘homological’ categories as
a basis for a generalised homological algebra. Our aim is to extend the
homologicalnotions 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.
PartI develops homologicalalgebra in p-exact categories, i.e. exact cat-
egories 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) homo-
logicalalgebra. The mainmotivationofthe present,muchwider 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(andsublattices)ofnormalsubobjectsplayanimportantrole. 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 abel-
ian one, and was not developed in Part I, additive homological categories
neednotbeabelian. Homotopiesofchaincomplexesonthesecategoriesare
considered here, if in a marginal way; in fact, this study is mainly related
tosuchconstructionsasmappingcylinderandsuspension,derivedandtri-
angulatedcategories,thatshouldratherbeviewedasapartofhomotopical
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
furtherconditions thatallowthe constructionofsubquotientsandinduced
vii
viii Preface
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]. Extend-
ing abelian categories, and also the p-exact ones, they include the usual
domains ofhomologyandhomotopytheories,e.g.the categoryof‘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
canbe viewedas exact sequences in a homologicalcategory,whose objects
are actions of groups on pointed sets.
Our view is thus quite distinct from the more usual affine generalisa-
tions of the abelian framework, that are based on finite limits: Barr-exact
categories,their extensions andvariations. Thereseems to be a sortofop-
position between a projective and an affine approach, that will be further
analysed in the Introduction.
Contents
Preface vii
Introduction 1
0.1 Categorical settings for homological algebra 1
0.2 Semiexact, homologicaland 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
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