Further remarks on the “Smith is Huq” condition

Martins-Ferreira, Nelson;Van der Linden, Tim
(2015) Applied Categorical Structures — Vol. 23, n° 4, p. 527-541 (2015)

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Abstract
We compare the Smith is Huq condition (SH) with three com- mutator conditions in semi-abelian categories: first an apparently weaker con- dition which arose in joint work with Bourn and turns out to be equivalent with (SH), then an apparently equivalent condition which takes commutation of non-normal subobjects into account and turns out to be stronger than (SH). This leads to the even stronger condition that weighted commutators in the sense of Gran, Janelidze and Ursini are independent of the chosen weight, which is known to be false for groups but turns out to be true in any two-nilpotent semi-abelian category.
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Martins-Ferreira, N., & Van der Linden, T. (2015). Further remarks on the “Smith is Huq” condition. Applied Categorical Structures, 23(4), 527-541. https://doi.org/10.1007/s10485-014-9368-5 (Original work published 2015)