James Richard Andrew Gray

Algebraic Exponentiation in General Categories

We study a categorical-algebraic concept of exponentiation, namely, right adjoints for the pullback functors between D. Bourn’s categories of points. We introduce and study them in the situations where the ordinary pullback functors between bundles do not admit right adjoints—in particular, for semi-abelian, protomodular, (weakly) Mal’tsev, (weakly) unital, and more general categories. We present a number of examples and counter examples for the existence of such right adjoints.

PERI-ABELIAN CATEGORIES AND THE UNIVERSAL CENTRAL EXTENSION CONDITION

We study the relation between Bourns notion of peri-abelian category and conditions involving the coincidence of the Smith, Huq and Higgins commutators. In particular, we show that a semi-abelian category is peri-abelian if and only if for each normal subobject K◁XK◁X, the Higgins commutator of K with itself coincides with the normalisation of the Smith commutator of the denormalisation of K with itself. We show that if a category is peri-abelian, then the condition (UCE), which was introduced and studied by Casas and the second author, holds for that category. In addition, we show, using amongst other things a result by Cigoli, that all categories of interest in the sense of Orzech are peri-abelian and therefore satisfy the condition (UCE).

ON THE NORMALITY OF HIGGINS COMMUTATORS

In a semi-abelian context, we study the condition (NH) asking that Higgins commutators of normal subobjects are normal subobjects. We provide examples of categories that do or do not satisfy this property. We focus on the relationship with the Smith is Huq condition (SH) and characterise those semi-abelian categories in which both (NH) and (SH) hold in terms of reflection and preservation properties of the change of base functors of the fibration of points.

Normalizers, Centralizers and Action Representability in Semi-Abelian Categories

AbstractWe 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 D

On Algebraic and More General Categories Whose Split Epimorphisms Have Underlying Product Projections

\mathbb {D}

Normalizers and Split Extensions

with a finite number of morphisms the category ℂD

A Note on the Distributivity of the Huq Commutator Over Finite Joins

{\mathbb {C}} ^{\mathbb {D}}

Aspects of algebraic exponentiation

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 D

Algebraically coherent categories

\mathbb {D}

Algebraic exponentiation for categories of Lie algebras

, the category ℂD