# Biproducts and commutators for noetherian forms

## Francois Koch Van Niekerk

Noetherian forms provide an abstract self-dual context in which one can establish homomorphism theorems (Noether isomorphism theorems and homological diagram lemmas) for groups, rings, modules and other group-like structures. In fact, any semi-abelian category in the sense of G. Janelidze, L. M\arki and W. Tholen, as well as any exact category in the sense of M. Grandis (and hence, in particular, any abelian category), can be seen as an example of a noetherian form. In this paper we generalize the notion of a biproduct of objects in an abelian category to a noetherian form and apply it do develop commutator theory in noetherian forms. In the case of semi-abelian categories, biproducts give usual products of objects and our commutators coincide with the so-called Huq commutators (which in the case of groups are the usual commutators of subgroups). This paper thus shows that the structure of a noetherian form allows for a self-dual approach to products and commutators in semi-abelian categories, similarly as has been known for homomorphism theorems.

Keywords: Forms, product, commutators, Semi-abelian categories

2010 MSC: 18D99, 18A30

Theory and Applications of Categories, Vol. 34, 2019, No. 30, pp 961-992.

Published 2019-10-03.

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