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Homological Algebra: In Strongly Non-abelian Settings (Hardcover)
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Homological Algebra: In Strongly Non-abelian Settings (Hardcover)
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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.
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