Abstract
In this thesis, we study different abstract approaches to classical lemmas about exact
sequences in the sense of homological algebra. The first main result is that a diagram lemma
of a 3×3-shape holds for all cyclic groups if and only if it holds in the abstract setting of a
noetherian form, and hence, if and only if it holds for all groups. The second main result is
that a theory of subfactors in the context of a noetherian form can fully capture Isbell’s work
on subfactors of groups in relation to the Butterfly lemma. Further work on noetherian forms
presented in the thesis includes: a relational approach to the Butterfly lemma and Goursat’s
Theorem, and an extension of Bergman’s approach to proving homological diagram lemmas.
The third main result of the thesis is a characterization of Yoneda quasi-abelian categories
(better known in the literature as Quillen exact categories) by using forms over an additive
category. We then show that by replacing additivity in our characterization with an new
additional form-theoretic property, it is possible to fully recover in this more general nonadditive
context the homological diagram lemmas that are known to hold in such categories.
Such a non-additive context is an alternative context to that of noetherian forms, which
includes all homological categories in the sense of Borceux and Bourn.