nLab homological algebra

Idea

In an abelian category $\mathcal{A}$, homological algebra is the homotopy theory of chain complexes in $\mathcal{A}$ 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 $\mathcal{A}$.

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.

• The study of properties of modules over rings of various types, by the use of methods adapted from topological homology theory.

• 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 $C_\bullet(X)$ of singular chains in a topological space $X$, whose chain homology is the singular homology $H_\bullet(X)$ of $X$, which is a linear approximation to the homotopy groups of $X$. Accordingly, $C_\bullet(X)$ itself serves as a linearized approximation to the homotopy type of $X$.

As a toolbox in stable homotopy theory

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.

1. 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 $D(\mathcal{A})$ obtained by localizing $Ch_\bullet(\mathcal{A})$ 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.)

2. By the discussion at localization the morphisms in $D(\mathcal{A})$ 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.

3. 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.

4. However, the derived category $D(\mathcal{A})$ is still a rather coarse approximation to the full stable (∞,1)-category of chain complexes in $\mathcal{A}$. 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.

5. 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 $R$-modules are a presentation for HR-module spectra. Moreover, A-infinity algebras in $HR$-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.

Non-abelian variants

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.

Entries in homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

• additive and abelian categories

  • Ab-enriched category

  • pre-additive category

  • additive category

  • pre-abelian category

  • abelian category

  • Grothendieck category

• abelian sheaves

• semi-abelian category

Basic definitions

• kernel, cokernel

• complex

  • differential

  • homology

• category of chain complexes

  • chain complex

  • chain map

  • chain homotopy

• chain homology and cohomology

  • quasi-isomorphism

  • homological resolution

  • simplicial homology

  • generalized homology

• exact sequence,

  • short exact sequence, long exact sequence, split exact sequence

• injective object, projective object

  • injective resolution, projective resolution

  • flat resolution

Stable homotopy theory notions

• derived category

• triangulated category, enhanced triangulated category

• stable (∞,1)-category

  • stable model category

  • pretriangulated dg-category

  • A-∞-category

• (∞,1)-category of chain complexes

• derived functor, derived functor in homological algebra

  • Tor, Ext

  • homotopy limit, homotopy colimit

  • lim^1 and Milnor sequences

  • fr-code

• abelian sheaf cohomology

Constructions

• double complex

• Koszul-Tate resolution, BRST-BV complex

• spectral sequence

  • spectral sequence of a filtered complex

  • spectral sequence of a double complex

  • Grothendieck spectral sequence

    • Leray spectral sequence

    • Serre spectral sequence

    • Hochschild-Serre spectral sequence

Lemmas

diagram chasing

• 3x3 lemma

• four lemma, five lemma

• snake lemma, connecting homomorphism

• horseshoe lemma

• Baer's criterion

Schanuel's lemma

Homology theories

• singular homology

• cyclic homology

Theorems

• Dold-Kan correspondence / monoidal, operadic

  • Moore complex, Alexander-Whitney map, Eilenberg-Zilber map

• Eilenberg-Zilber theorem

  • cochain on a simplicial set

• universal coefficient theorem

• Künneth theorem

References

The following lists references on homological algebra:

• General

• Lecture notes and course notes

• History

General

Classical accounts:

• Samuel Eilenberg, Saunders MacLane, Group Extensions and Homology, Annals of Mathematics 43 4 (1942) 757-831 [doi:10.2307/1968966, jstor:1968966]

• D. A. Buchsbaum, Exact categories and duality, Transactions of the American Mathematical Society Vol. 80, No. 1 (1955), pp. 1-34 (JSTOR)

• Henri Cartan, Samuel Eilenberg, Homological Algebra, Princeton Univ. Press (1956), Princeton Mathematical Series 19 (1999) [ISBN:9780691049915, doi:10.1515/9781400883844, pdf]

• A. Grothendieck, Sur quelques points d'algèbre homologique (1957) (part1, part2)

• Roger Godement, Topologie algébrique et theorie des faisceaux, Actualités Sci. Ind. 1252, Hermann, Paris (1958) [webpage, pdf]

• Saunders MacLane, Homology, Grundlehren der Mathematischen Wissenschaften 114, Springer (1963, 1974), reprinted as: Classics in Mathematics, Springer (1995) [pdf, doi:10.1007/978-3-642-62029-4]

• Peter Hilton, Urs Stammbach, A course in homological algebra, Springer-Verlag, New York, 1971, Graduate Texts in Mathematics, Vol. 4 (doi:10.1007/978-1-4419-8566-8, pdf)

• Saunders Mac Lane, Homology (1975) reprinted as Classics in Mathematics. Springer-Verlag, Berlin, 1995. x+422 pp. ISBN 3-540-58662-8 (doi:10.1007/978-3-642-62029-4)

A standard modern textbook introduction is

• Charles Weibel, An introduction to homological algebra, Cambridge Studies in Adv. Math. 38, Cambridge University Press (1994) [doi:10.1017/CBO9781139644136, pdf]

and a more systematic modern development of the theory is in sections 8 and 12-18 of

• Masaki Kashiwara, Pierre Schapira, Categories and Sheaves, Grundlehren der Mathematischen Wissenschaften 332, Springer (2006)

Non-abelian variants of homological algebra are disussed for instance in

• Francis Borceux, Dominique Bourn, Mal'cev, protomodular, homological and semi-abelian categories, Mathematics and Its Applications 566, Kluwer (2004)

The foundations of the formulation in the broader context of stable (∞,1)-category theory is in

• Jacob Lurie, Stable Infinity-Categories, (math.CT/0608228)

Other textbooks include

• I. Bucur, A. Deleanu, Introduction to the theory of categories and functors, 1968

• Sergei Gelfand, Yuri Manin, Methods of homological algebra, transl. from the 1988 Russian (Nauka Publ.) original, Springer (1996, 2002) [doi:10.1007/978-3-662-12492-5]

See also

• Wikipedia, Homological algebra

• Springer Online Encyclopeadia of Mathematics: homological algebra

Lecture notes and course notes

• Alexander Beilinson, Introduction to homological algebra (handwritten notes, summer 2007, pdf) lec1, lec2, lec3, lec4

• Julia Collins, Homological algebra (2006) [pdf]

• Rick Jardine, Homological algebra, course notes, 2009 (index)

• Peter May, Notes on Tor and Ext (pdf)

• Pierre Schapira, Categories and homological algebra, lecture notes (2011) (pdf)

• Urs Schreiber, Introduction to Homological Algebra

In constructive mathematics

Discussion of homological algebra in constructive mathematics:

• Thierry Coquand, Arnaud Spiwack, Towards constructive homological algebra in type theory, in: Towards Mechanized Mathematical Assistants. MKM Calculemus 2007, Lecture Notes in Computer Science 4573 Springer (2007) [doi:10.1007/978-3-540-73086-6_4, pdf]

  (via type theory)

• Julio Rubio, Francis Sergeraert, Constructive homological algebra and applications [arXiv:1208.3816]

History

• Charles Weibel, A history of homological algebra, dvi

From the introduction of Collins 2006:

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 $Hom(A,B)$; in particular the fact that it varies naturally, contravariantly in $A$ and covariantly in $B$.

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. 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 $Hom$ being left exact and the first occurrence of $Ext^n$ as the right derived functors of $Hom$.