nLab Puppe exact category

Idea

Loosely speaking the notion of Puppe exact categories is that of abelian categories without the requirement of additivity. The remaining exactness properties still permit a considerable amount of homological algebra.

Definition

A well-powered category $\mathcal{C}$ is called Puppe exact (or p-exact for short) iff

• $\mathcal{C}$ has a zero object.

• $\mathcal{C}$ has kernels and cokernels.

• every mono is a kernel and every epi is a cokernel.

• every morphism has an epi-mono factorization.

Related entries

• abelian category, semi-abelian category

• exact category

• homological category

History

The notion of Puppe exact categories was introduced by Puppe (1962) as a weakening of the notion of abelian categories. Their homological algebra was studied by Mitchell (1965). They were extensively used by Marco Grandis in his work on distributive homological algebra in the 1980s.

Grandis worked on further generalizations that he called semi-exact or homological. This family differs from the concepts around semi-abelian categories studied in particular in the 2004 monograph by Francis Borceux and Dominique Bourn with respect to the existence of finite limits in the latter. Grandis (2010) proposed viewing the difference as akin to the difference between projective and affine spaces.

References

The original articles:

• Dieter Puppe, Korrespondenzen in abelschen Kategorien , Math. Ann. 148 (1962) pp.1-30. (gdz)

• Barry Mitchell, Theory of categories, Pure and Applied Mathematics 17, Academic Press, 1965.

See also:

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

• Francis Borceux , Marco Grandis, Jordan-Hölder, modularity and distributivity in non-commutative algebra, JPAA 208 (2007), pp.665-689. (doi)

• Aurelio Carboni, Marco Grandis, Categories of projective spaces, JPAA 110 (1996), pp.241-258. (web)

• Peter Freyd, A. Scedrov, Categories, Allegories, North-Holland Amsterdam 1990.

• Marco Grandis, Transfer functors and projective spaces, Math. Nachr. 118 (1984) pp.147-165. (doi)

• Marco Grandis, Homotopy spectral sequences, J. Homotopy Relat. Struct. 5 (2010), pp.213-252. (arXiv:1007.0632, pdf)