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Biproducts and commutators for noetherian forms

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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.

http://www.tac.mta.ca/tac/volumes/34/30/34-30.pdf

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