A second book on homological algebra

A second book on homological algebra

[Marco Grandis]

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


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