Homological Algebra : In Strongly Non-Abelian Settings.

Homological Algebra : In Strongly Non-Abelian Settings.

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...

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Bibliographic Details
Main Author: Grandis, Marco
Format: eBook
Language:English
Published: Singapore : World Scientific Pub. Co., 2013.
Subjects:
Algebra, Homological.
Homology theory.
MATHEMATICS > Group Theory.
Algebra, Homological
Homology theory
Online Access:Click for online access
Table of Contents:
  • Preface; Contents; Introduction; 0.1 Categorical settings for homological algebra; 0.2 Semiexact, homological and generalised exact categories; 0.3 Subquotients and homology; 0.4 Satellites; 0.5 Exact centres, expansions, fractions and relations; 0.6 Applications; 0.7 Homological theories and biuniversal models; 0.8 Modularity and additivity; 0.9 A list of examples; 0.10 Terminology and notation; 0.11 Acknowledgements; 1 Semiexact categories; 1.1 Some basic notions; 1.1.1 Lattices; 1.1.2 Distributive and modular lattices; 1.1.3 Galois connections; 1.1.4 Contravariant Galois connections.
  • 1.1.5 Isomorphisms, monomorphisms and epimorphisms1.1.6 Pointed categories; 1.1.7 Kernels and cokernels; 1.2 Lattices and Galois connections; 1.2.1 Definition; 1.2.2 Monos and epis; 1.2.3 Kernels and cokernels; 1.2.4 The normal factorisation; 1.2.5 Exact connections; 1.2.6 Normal monos and epis; 1.2.7 The semiadditive structure; 1.2.8 Modular connections; 1.3 The main definitions; 1.3.1 Ideals of null morphisms; 1.3.2 Closed ideals; 1.3.3 Semiexact categories; 1.3.4 Remarks; 1.3.5 Kernel duality and short exact sequences; 1.3.6 Homological and generalised exact categories; 1.3.7 Subcategories.
  • 1.4 Structural examples1.4.1 Lattices and connections; 1.4.2 A basic homological category; 1.4.3 A p-exact category; 1.4.4 Graded objects; 1.4.5 The canonical enriched structure; 1.4.6 Proposition; 1.5 Semiexact categories and normal subobjects; 1.5.1 Semiexact categories and local smallness; 1.5.2 Exact sequences; 1.5.3 Lemma (Annihilation properties); 1.5.4 Theorem (Two criteria for semiexact categories); 1.5.5 Normal factorisations and exact morphisms; 1.5.6 Direct and inverse images; 1.5.7 Lemma (Meets and detection properties); 1.5.8 Theorem and Definition (The transfer functor).
  • 1.5.9 Remarks1.6 Other examples of semiexact and homological categories; 1.6.1 Groups, rings and groupoids; 1.6.2 Abelian monoids, semimodules, preordered abelian groups; 1.6.3 Topological vector spaces; 1.6.4 Pointed sets and spaces; 1.6.5 Categories of partial mappings; 1.6.6 General modules; 1.6.7 Categories of pairs; 1.6.8 Groups as pairs; 1.6.9 Two examples; 1.7 Exact functors; 1.7.0 Basic definitions; 1.7.1 Exact functors and normal subobjects; 1.7.2 Conservative exact functors; 1.7.3 Proposition and Definition (Semiexact subcategories); 1.7.4 Examples.
  • 1.7.5 Left exact functors and right adjoints1.7.6 Categories of maps and kernel functors; 1.7.7 Categories of functors; 1.7.8 Pseudolimits in EX1; 1.7.9 Proposition (Closed ideals and adjunctions); 2 Homological categories; 2.1 The transfer functor and ex2-categories; 2.1.1 Fully normal monos and epis; 2.1.2 Theorem (Exactness properties of the transfer functor); 2.1.3 Ex2-categories; 2.1.4 The associated projective category; 2.2 Characterisations of homological and g-exact categories; 2.2.1 Lemma (The special 3×3-lemma); 2.2.2 Theorem (Homological categories).

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