Lemmas on exactness in exact categories and in generalized Abelian categories

Abstract

Weakly idempotent complete exact categories and generalized Abelian categories are two important generalizations of Abelian categories. Snake lemma is one of the most fundamental lemmas in homological algebra. In this paper, we point out that the three forms of Snake lemma, the fundamental theorem of homological algebra, and the$3\\times~3$ lemma, are true and equivalent, in the weakly idempotent complete exact categories; and they imply other lemmas on exactness. In generalized Abelian categories, the three forms of Snake lemma are true and equivalent; and they imply the fundamental theorem of homological algebra and other lemmas on exactness.