Contents

Idea

From a modern perspective, homological algebra is the study of algebraic objects, (such as groups, rings or Lie algebras, or sheaves of such objects), by ‘resolving them’, replacing them by more stable objects whose homotopy category is the derived category of an abelian category.

Functors between the original algebraic objects lead to derived functors between the homotopy categories, and much effort is devoted to the study of those derived functors, and the interpretation of their properties in terms of structure of the original algebraic objects.

There are variants of the above idea that handle more non-linear phenomena. 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 methods to avoid the crushing of homotopical information that can occur when passing to chain complexes.

Homological algebra thus studies, in particular, the homology of chain complexes in abelian categories – therefore the name.

One of the most refined 1-categorical approximations to the context in which such chain complexes live is a derived category with the structure of a triangulated category. Perhaps better contexts are frameworks in which higher coherences are better taken care of, like $A_{\mathrm{\infty }}$-categories, DG-categories and stable ∞-categories. There are enriched versions of derived categories which have any of the 3 mentioned higher structures. For the corresponding stable ∞-category, the derived category is just the homotopy category. It may be possibly enriched by derived Kan extensions as in the theory of dérivateurs.

Entries on concepts in homological algebra

• additive and abelian categories, chain complex, derived category, triangulated category, A-infinity category, differential graded category, stable infinity-category, projective object, derived functor, injective object, five lemma, snake lemma, nonabelian homological algebra, satellite, Tor, Ext

References

• Charles Weibel, An introduction to homological algebra, Cambridge Studies in Adv. Math. 38, CUP 1994

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

• H. Cartan, S. Eilenberg, Homological algebra, Princeton Univ. Press 1956.

• M. Kashiwara and P. Schapira, Categories and Sheaves, Springer (2000)

• S. I . Gelfand, Yu. I. Manin, Methods of homological algebra

• Jacob Lurie, Derived algebraic geometry I: Stable Infinity-Categories, math.CT/0608228

• Springer Online Encyclopeadia of Mathematics: homological algebra

• A. Grothendieck, Sur quelques points d’algèbre homologique, Tohoku, part1, part2

• Charles Weibel, A history of homological algebra, dvi

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