Homological Algebra: In Strongly Non-abelian Settings

In Strongly Non-Abelian Settings

Marco Grandis

• Publisher: WSPC
• Year: 2013
• Print ISBN: 9789814425919
• eBook ISBN: 9789814425933
• Topic: Biological Sciences
• Subtopic: Science General
• Index: Biological Sciences

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.

Homological Algebra: The Interplay of Homology with Distributive Lattices and Orthodox Semigroups

Contents:

• Introduction
• Semiexact categories
• Homological Categories
• Subquotients, Homology and Exact Couples
• Satellites
• Universal Constructions
• Applications to Algebraic Topology
• Homological Theories and Biuniversal Models
• Appendix A. Some Points of Category Theory

Readership: Graduate students, professors and researchers in pure mathematics, in particular category theory and algebraic topology.

1

Semiexact categories

After a description of the category Ltc of lattices and Galois connections, and an elementary study of its exactness properties (Section 1.2), we introduce in Section 1.3 our main definitions: semiexact, homological and generalised exact categories. All these notions are self-dual.

Various examples of semiexact and homological categories are presented in Sections 1.4 and 1.6. Semiexact categories are studied in Section 1.5, and exact functors between them in Section 1.7; the study of homological categories is deferred to the next chapter.

In every semiexact category E, normal subobjects, with their direct and inverse images, produce a ‘transfer functor’ Nsb: E → Ltc (see Theorem 1.5.8) with values in the aforementioned category of lattices (that is homological).

This transfer functor is a prominent tool of the present analysis; it is also the natural extension to semiexact categories of a tool developed in Part I: the transfer functor of subobjects Sub: E → MIc of a p-exact category (where all subobjects are normal), that takes values in the category of modular lattices and modular connections, a p-exact subcategory of Ltc.

1.1 Some basic notions

This elementary review of basic topics, like lattices, monomorphisms and epimorphisms, is meant to form a common basis for readers with different backgrounds. Most of these things can also be found in Part I [G20], but we want to make the present book essentially independent of the former.

1.1.1 Lattices

Lattices of substructures and lattices of quotients will play an important role. We recall here some basic facts about the theory of lattices; the interested reader is referred to the classical texts by Birkhoff and Grätzer [Bi, Grz].

Classically, a lattice is defined as a (partially) ordered set X such that every pair x, x′ of elements has a join x

x′ = sup{x, x′} (the least upper bound) and a meet x

x′ = inf{x,x′} (the greatest lower bound).

But in this book (as in Part I) it is more convenient to adopt a slightly stronger definition: lattice will always mean an ordered set with finite joins and meets. This is equivalent to requiring the existence of binary joins and meets together with the least element 0 = ∨

(the empty join) and the greatest element 1 = ∧

(the empty meet). These bounds are the unit of the join and meet operations, respectively; they coincide in the one-point lattice, and only there. A lattice is said to be complete if every subset has a supremum, or - equivalently - every subset has an infimum.

In the lattice L(A) = SubA of subgroups of an abelian group A, the meet of two subgroups H, K is their intersection H ∩ K, and the join is their ‘sum’

H + K = {h + k | h ∈ H, k ∈ K}.

But we always use the notation H

K and H

K, for the sake of uniformity with lattices of subobjects in a...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Dedication
• Preface
• Contents
• Introduction
• 1 Semiexact categories
• 2 Homological categories
• 3 Subquotients, homology and exact couples
• 4 Satellites
• 5 Universal constructions
• 6 Applications to algebraic topology
• 7 Homological theories and biuniversal models
• Appendix A Some points of category theory
• References
• Index