In an abelian category , homological algebra is the homotopy theory of chain complexes in up to quasi-isomorphism of chain complexes. Hence it is the study of the (infinity,1)-categorical localization of the category of chain complexes at the class of quasi-isomorphisms, or in other words the derived (infinity,1)-category of .
When considering nonnegatively graded chain complexes, homological algebra may be viewed as a linearized version of the homotopy theory of homotopy types or infinity-groupoids, by the Dold-Kan correspondence. When considering unbounded chain complexes, it may be viewed as a linearized and stabilized version, by the stable Dold-Kan correspondence. Conversely, we may view homotopical algebra as a nonabelian generalization of homological algebra.
Hence homological algebra is
The study of a particularly simple sort of stable (∞,1)-categories, namely those derived from categories of chain complexes. See As a toolbox in stable homotopy theory below and the discussion at cosmic cube.
A simple fragment of, and toolbox for, stable homotopy theory — and hence, by extension, unstable homotopy theory. From this point of view, an archetypical motivating example is the chain complex of singular chains in a topological space , whose chain homology is the singular homology of , which is a linear approximation to the homotopy groups of . Accordingly, itself serves as a linearized approximation to the homotopy type of .
With homological algebra being a topic in stabilized homotopy theory, it is really the study of stable (∞,1)-categories of chain complexes – and thus, by the stable Dold-Kan correspondence, of Eilenberg-MacLane module spectra.
Historically this modern perspective has developed only in stages out of more “concrete” (more 1-categorical) notions, which now form the body of homological algebra, in the form of a box of tools for computing linearized problems in homotopy theory. The following list indicates how these traditional notions serve to present constructions in stable homotopy theory.
The notion of quasi-isomorphism between chain complexes – chain maps which induce isomorphisms on homology groups – is the stable version of weak homotopy equivalences of topological spaces. The derived category of chain complexes obtained by localizing at these weak equivalences is the corresponding homotopy category, the context where all chain maps are identified up to chain homotopy between good representatives of these objects. (On the other hand, in more general situations this correspondence is less immediate, and the notion of quasi-isomorphism may not be the best choice; see at Whitehead theorem.)
By the discussion at localization the morphisms in are zig-zags of chain maps that involve resolutions by non-isomorphic but quasi-isomorphic chain complexes. By the various model structures on chain complexes these resolutions can concretely be constructed as injective resolutions, projective resolutions and/or more general sorts of resolutions (such as flat resolutions, soft, flabby, etc.) of chain complexes, and much of the theory revolves around handling these.
Notably, functors between categories of chain complexes may extend to functors on these derived categories by evaluating them on suitable resolutions – accordingly called derived functors. (In homological algebra, the phrase “derived functor” is traditionally applied to the homology groups of what abstract homotopy theory calls the “derived functor”, these being the invariants that one can compute.) Much of the theory revolves around computing and characterizing derived functors, for instance in the definition of abelian sheaf cohomology and hence there are powerful tools for these computations, notably spectral sequences.
However, the derived category is still a rather coarse approximation to the full stable (∞,1)-category of chain complexes in . There is a series of extra property and structures added to it which gives better approximations, and large parts of modern homological algebra study these:
First of all the derived category is automatically a triangulated category, which is a means of remembering the operation of suspension and de-suspension (looping) of chain complexes. Further structure added to these goes by names such as enhanced triangulated category. A stable derivator is a strong enhancement which encodes basically all the requisite structure for internal computations. Finally, the further promotion of these to stable model categories or pretriangulated dg-categories/linear A-∞ categories of chain complexes makes them capture the full information present in the stable (∞,1)-category.
Algebra in stable homotopy theory is higher algebra over E-∞ rings, and homological algebra provides approximations to that: by the stable Dold-Kan correspondence chain complexes of -modules are a presentation for HR-module spectra. Moreover, A-infinity algebras in -module spectra are presented by dg-algebras, hence by ordinary associative algebras in chain complexes. Similarly E-infinity algebras are presented by commutative dg-algebras, hence by commutative algebras internal to chain complexes. By variation of this theme a multitude of notions in higher algebra finds their representation in homological algebra, for instance L-∞ algebras in terms of dg-Lie algebras: Lie algebras internal to chain complexes.
There are variants of the tools of homological algebra that can also be applied to more non-linear phenomena, see for instance at Dold-Kan correspondence the section non-abelian case. These include non-Abelian (co)homology and crossed and quadratic versions that use a small degree of non-linearity in the models. These latter theories make extensive use of techniques from homotopical algebra in the wide sense of that term and simplicial homotopy theory to avoid the crushing of homotopical information that can occur when passing to chain complexes.
The following lists references on homological algebra:
Classical historical accounts include
D. A. Buchsbaum, Exact categories and duality, Transactions of the American Mathematical Society Vol. 80, No. 1 (1955), pp. 1-34 (JSTOR)
Saunders Mac Lane, Homology (1975) reprinted as Classics in Mathematics. Springer-Verlag, Berlin, 1995. x+422 pp. ISBN 3-540-58662-8
A standard modern textbook introduction is
and a more systematic modern development of the theory is in sections 8 and 12-18 of
Non-abelian variants of homological algebra are disussed for instance in
The foundations of the formulation in the broader context of stable (∞,1)-category theory is in
Other textbooks include
I. Bucur, A. Deleanu, Introduction to the theory of categories and functors, 1968
S. I . Gelfand, Yu. I. Manin, Methods of homological algebra
Julia Collins, Homological algebra (2006) (pdf)
Discussion of homological algebra in constructive mathematics is in
Discussion of a formalization in type theory is in
From the introduction of (Collins):
The word “homology” was first used in a topological context by Poincaré in 1895, who used it to think about manifolds which were the boundaries of higher-dimensional manifolds. It was Emmy Noether in the 1920s who began thinking of homology in terms of groups, and who developed algebraic techniques such as the idea of modules over a ring. These are both absolutely crucial ingredients in the modern theory of homological algebra, yet for the next twenty years homology theory was to remain confined to the realm of topology.
In 1942 came the first move forward towards homological algebra as we know it today, with the arrival of a paper by Samuel Eilenberg and Saunders MacLane. In it we find Hom and Ext defined for the very first time, and along with it the notions of a functor and natural isomorphism. These were needed to provide a precise language for talking about the properties of ; in particular the fact that it varies naturally, contravariantly in and covariantly in .
Only three years later this language was expanded to include category and natural equivalence. However, this terminology was not widely accepted by the mathematical community until the appearance of Cartan and Eilenberg’s book in 1956. Cartan and Eilenberg’s book was truly a revolution in the subject, and in fact it was here that the term “Homological Algebra” was first coined. The book used derived functors in a systematic way which united all the previous homology theories, which in the past ten years had arisen in group theory, Lie algebras and algebraic geometry. The sheer list of terms that were first defined in this book may give the reader an idea of how much of this project is due to the existence of that one book! They defined what it means for an object to be projective or injective, and defined the notions of projective and injective resolutions. It is here that we find the first mention of being left exact and the first occurrence of as the right derived functors of .