* A second book on homological algebra
@ 2013-03-12 20:06 Marco Grandis
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From: Marco Grandis @ 2013-03-12 20:06 UTC (permalink / raw)
To: categories
My second book on homological algebra has appeared
Homological Algebra In Strongly Non-Abelian Settings
World Scientific
http://www.worldscientific.com/worldscibooks/10.1142/8608
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,
where satellites and spectral sequences can still be studied.
This is a sequel of a book on ‘Homological Algebra, The interplay of
homology with distributive lattices and orthodox semigroups’,
published by the same Editor, but can be read independently of the
latter.
The previous book develops homological algebra in p-exact categories,
i.e. exact categories in the sense of Puppe and Mitchell — 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.
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.
Extending abelian categories, and also the p-exact ones, these
notions 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.
______________
The first book, ‘Homological Algebra, The interplay of homology with
distributive lattices and orthodox semigroups’
(World Scientific) can be found at:
http://www.worldscientific.com/worldscibooks/10.1142/8483
______________
With best regards
Marco Grandis
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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