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The search, of almost a century long, for a unified axiomatic framework forestablishing homomorphism theorems of classical algebra (such as Noetherisomorphism theorems and homological diagram lemmas) has led to the notion of a`noetherian form', which is a generalization of an abelian category suitable toencompass categories of non-abelian algebraic structures (such as non-abeliangroups, or rings with identity, or cocommutative Hopf algebras over any field,and many others). In this paper, we show that, surprisingly, even the categoryof sets, and more generally, any topos, fits under the framework of anoetherian form. Moreover, we give an intrinsic characterization of suchnoetherian form and show that it is very closely related to the knownnoetherian form of a semi-abelian category. In fact, we show that for a pointedcategory having finite products and sums, the existence of the type ofnoetherian form that any topos possesses is equivalent to the category beingsemi-abelian (this result is unexpected since only trivial toposes can besemi-abelian). We also show that these noetherian forms are optimal, in asuitable sense.

Every topos has an optimal noetherian form