(also nonabelian homological algebra)
Context
Basic definitions
Stable homotopy theory notions
Constructions
Lemmas
Homology theories
Theorems
∞-ary regular and exact categories
arity class: unary, finitary, infinitary
regularity
regular category = unary regular
coherent category = finitary regular
geometric category = infinitary regular
exactness
exact category = unary exact
A category $C$ is called homological if it is
Many of the standard results of classical homological algebra in abelian categories extend to homological categories:
the five lemma
the nine lemma
the snake lemma
the Noether isomorphism theorem
a version of the Jordan-Hölder theorem.
A homological category which is Barr-exact and has finite coproducts is semiabelian.
The category Grp of all groups (including non-abelian groups) is homological. Namely it is
regular, by this example,
pointed protomodular by this example.
Francis Borceux, Dominique Bourn, Chapter IV in: Mal'cev, protomodular, homological and semi-abelian categories, Mathematics and Its Applications 566, Kluwer, 2004 (doi:10.1007/978-1-4020-1962-3)
Francis Borceux, Marco Grandis, Jordan-Hölder, modularity and distributivity in non-commutative algebra, J. Pure Appl. Algebra 208 (2007), no. 2, 665–689. (doi, MR2007k:18021)
Tamar Janelidze, Foundations of relative non-abelian homological algebra, 2009 (pdf, pdf, hdl:11427/4891)
Last revised on January 15, 2024 at 03:16:13. See the history of this page for a list of all contributions to it.