These categories are called Puppe-exact or p-exact categories. See “Jordan-Hölder, modularity and distributivity in non-commutative algebra”, paragraph 1.1, by Francis Borceux and Marco Grandis (JPAA 208 (2007), 665-689 ; available here: http://www.dima.unige.it/~grandis/BGwe.Abs.html), for non-abelian examples. And see the papers of Marco Grandis (e.g. this one: http://www.numdam.org:80/numdam-bin/item?id=CTGDC_1992__33_2_135_0) and Mitchell's book “Theory of categories” for homological results in this context (as a general rule, all homological lemmas non involving direct products hold).
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These categories are called Puppe-exact or p-exact categories. See “Jordan-Hölder, modularity and distributivity in non-commutative algebra”, paragraph 1.1, by Francis Borceux and Marco Grandis (JPAA 208 (2007), 665-689 ; available here: http://www.dima.unige.it/~grandis/BGwe.Abs.html), for non-abelian examples. And see the papers of Marco Grandis (e.g. this one: http://www.numdam.org:80/numdam-bin/item?id=CTGDC_1992__33_2_135_0) for homological results in this context (as a general rule, all homological lemmas non involving direct products hold). |
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