nLab quasi-abelian category

Definition

A quasi-abelian category is an additive category admitting the kernels and the cokernels which satisfies the following conditions:

(i) the strict epimorphisms are stable by base changes,

(ii) the strict monomorphisms are stable by co-base changes

This is definition 2.1.1 in (Kashiwara).

Examples

• The category of bornological vector spaces over the complex numbers is quasi-abelian (Prosmans-Schneiders 00)

• The category of bornological abelian groups is quasi-abelian (Bambozzi 14).

Related concepts

• abelian category

• pseudo-abelian category

• semi-abelian category

References

• Jean-Pierre Schneiders, Quasi-abelian categories and sheaves., Mémoires de la Société Mathématique de France 76 (1999), 144 pp. [doi:10.24033/msmf.389]

• Fabienne Prosmans, Jean-Pierre Schneiders, A homological study of bornological spaces, Prepublications Mathematiques de l’Universite Paris 13 46 (2000) [pdf]

• Wolfgang Rump, Almost abelian categories, Cahiers de topologie et géométrie différentielle catégoriques, tome 42, no 3 (2001) 163–225. [numdam:CTGDC_2001__42_3_163_0]

• Federico Bambozzi, section 1 of: On a generalization of affinoid varieties [arXiv:1401.5702]

• Masaki Kashiwara, Equivariant derived category and representation of real semisimple Lie groups, Lecture Notes in Mathematics 1931 (2008) 137–234 [doi:10.1007/978-3-540-76892-0, pdf]

• Rhiannon Savage, Koszul monoids in quasi-abelian categories [arXiv:2203.08789]