# nLab Puppe exact category

### Context

#### Regular and exact categories

[[!include regular and exact categories - contents]]

#### Category Theory

[[!include category theory - contents]]

## Idea

Loosely speaking a Puppe exact category is an abelian category without additivity. The subsisting exactness properties permit a considerable amount of homology theory.

Introduced by D. Puppe in the early 1960s as a weakening of abelian categories, their homology theory was studied by B. Mitchell in his textbook on category theory. They were extensively used by M. Grandis in his work on distributive homological algebra in the 1980s.

## Remark

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 monograph by Bourn and Borceux with respect to the existence of finite limits in the latter. Grandis (2010) proposes to view the difference as akin to the difference between projective and affine spaces.

## 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.

## References

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

• A. Carboni, M. Grandis, Categories of projective spaces , JPAA 110 (1996), pp.241-258.

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

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

• M. Grandis, Homotopy spectral sequences , J. Homotopy Relat. Struct. 5 (2010), pp.213-252. (preprint)

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

Last revised on October 19, 2014 at 21:50:16. See the history of this page for a list of all contributions to it.