Normalizers, Centralizers and Action Representability in Semi-Abelian Categories

Abstract

We introduce the notion of normalizer as motivated by the classical notion in the category of groups. We show for a semi-abelian category ℂ that the following conditions are equivalent:

• (a) ℂ is action representable and normalizers exist in ℂ;

• (b) the category Mono(ℂ) of monomorphisms in ℂ is action representable;

• (c) the category ℂ² of morphisms in ℂ is action representable;

• (d) for each category $\mathbb {D}$ with a finite number of morphisms the category ${\mathbb {C}} ^{\mathbb {D}}$ is action representable.

Moreover, when in addition ℂ is locally well-presentable, we show that these conditions are further equivalent to:

• (e) ℂ satisfies the amalgamation property for protosplit normal monomorphism and ℂ satisfies the axiom of normality of unions;

• (f) for each small category $\mathbb {D}$, the category ${\mathbb {C}} ^{\mathbb {D}}$ is action representable.

We also show that if ℂ is homological, action accessible, and normalizers exist in ℂ, then ℂ is fiberwise algebraically cartesian closed.