2
$\begingroup$

Call a subobject normal if it's an equivalence class of some internal equivalence relation on the codomain. In a finitely complete category, normality is closed under finite intersection and finite product and is also pullback stable. This can be proved using a metatheorem which reduces to verification in the category of sets.

In the category of groups, normality is also closed under joins in the subobject poset (see this MSE question). Is the finite analogue of this state true in some nice protomodular categories, e.g homological ones?

$\endgroup$
1
$\begingroup$

This paper of Tomas Everaert and Tim Van der Linden shows that it is true for semi-abelian categories (see Proposition 2.7). Exactness implies that every normal monomorphism is a kernel, and they need the existence of coproduct to define joins of subobjects; so I'm not sure that this would still hold in a category that is only homological.

$\endgroup$
  • $\begingroup$ Brilliant, thank you! $\endgroup$ – Arrow Jun 6 '17 at 9:36

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.