Geodesic
Homological algebra in pre-Abelian categories
Zapiski Nauchnykh Seminarov POMI, Rings and modules. Part 2, Tome 94 (1979), pp. 131-141 Cet article a éte moissonné depuis la source Math-Net.Ru

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We construct derived functors in additive categories in which each morphism has a kernel, co-kernel, image, and coimage, but the image and coimage are not necessarily isomorphic. We prove that these derived functors possess the usual properties. The main difficulty is that the $3\times3$-lemma does not necessarily hold in the categories under consideration. 
@article{ZNSL_1979_94_a14,
     author = {A. V. Yakovlev},
     title = {Homological algebra in {pre-Abelian} categories},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {131--141},
     year = {1979},
     volume = {94},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1979_94_a14/}
}
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A. V. Yakovlev. Homological algebra in pre-Abelian categories. Zapiski Nauchnykh Seminarov POMI, Rings and modules. Part 2, Tome 94 (1979), pp. 131-141. http://geodesic.mathdoc.fr/item/ZNSL_1979_94_a14/