  A noetherian form is an abstract self-dual framework suitable for
establishing homomorphism theorems (such as the isomorphism theorems and
homological diagram lemmas) for non-abelian, as well as abelian, group-like
structures. It can be seen as a unification of semi-abelian categories and
Grandis exact categories (two separate generalisations of abelian categories)
that retains the self-dual character of Grandis exact categories. In this paper
we axiomatise a noetherian form over the category of sets, which, in a suitable
sense, is an optimal noetherian form. We then show that every topos has such
noetherian form as well, while in the context of a pointed category with finite
products and sums, the dual of such noetherian form is exactly the form of
subobjects of a semi-abelian category.
