arXiv subjects

James Richard Andrew Gray

Publications and source records attributed to James Richard Andrew Gray.

9 recordsLinked to original sources

arXiv2026

More on action representability and normalizers

We show that under mild assumptions a functor is a prefibration if and only if it has terminal objects in its fibers and admits precartesian liftings of monomorphisms. We use this to show that the following conditions on a pointed category $\mathbb{C}$ are equivalent: (a) The category of morphisms of $\mathbb{C}$ admits generic split extensions; (b) Each functor category of $\mathbb{C}$ with finite domain category admits generic split extensions; (c) $\mathbb{C}$ admits normalizers and generic split extensions; (d) The kernel functor from the category split extensions in $\mathbb{C}$ to $\mathbb{C}$ is a prefibration. In addition we show that the category of morphisms of a pointed protomodular $\mathbb{C}$ admits generic split extensions if and only if for each morphism $f:X\to Z$ the functor sending each object $B$ to the isomorphism class of split extensions in $\mathbb{C}^{\mathbf{2}}$ of $(B,B,1_B)$ with kernel $(X,Z,f)$, is representable.

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arXiv2026

Nice exact categories are coexact

Several important types of categories have been shown to be both exact and coexact (in the sense of Barr). The first type consists of abelian categories, which due to their self-dual definition, can be seen to be both exact and coexact by Tierney's characterization of them as additive exact categories. The next type consists of elementary toposes which are well-known to be exact, but have also been shown to be coexact and coprotomodular by Bourn. In this paper we study a condition weaker than extensivity and equivalent to additivity for pointed categories. We show that for a finitely cocomplete category this condition together with exactness implies coexactness and coprotomodularity. As a special case we obtain that a finitely cocomplete pretopos is coexact.

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arXiv2022

A note on the relationship between action accessible and weakly action representable categories

The main purpose of this paper is to show that the converse of the known implication weakly action representable implies action accessible is false. In particular we show that both action accessibility, as well as the (at least formally stronger) condition requiring the existence of all normalizers do not imply weakly-action-representability even for varieties. In addition we show that in contrast to both action accessibility and the condition requiring the existence of all normalizers, weakly-action representability is not necessarily inherited by Birkoff subcategories.

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arXiv2022

A note on the Huq-commutativity of normal monomorphisms

We give an alternative criteria for when a pair of Bourn-normal monomorphisms Huq-commute in a unital category. We use this to prove that in a unital category, in which a morphism is a monomorphism if and only if its kernel is zero morphism, a pair of Bourn-normal monomorphisms with the same codomain Huq-commute as soon as they have trivial pullback. As corollaries we show that several facts known only in the protomodular context are in fact true in more general contexts.

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arXiv2021

Complete objects in categories

We introduce the notions of proto-complete, complete, complete* and strong-complete objects in pointed categories. We show under mild conditions on a pointed exact protomodular category that every proto-complete (respectively complete) object is the product of an abelian proto-complete (respectively complete) object and a strong-complete object. This together with the observation that the trivial group is the only abelian complete group recovers a theorem of Baer classifying complete groups. In addition we generalize several theorems about groups (subgroups) with trivial center (respectively, centralizer), and provide a categorical explanation behind why the derivation algebra of a perfect Lie algebra with trivial center and the automorphism group of a non-abelian (characteristically) simple group are strong-complete.

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arXiv2020

Action representability of the category of internal groupoids

When $\mathbb C$ is a semi-abelian category, it is well known that the category $\mathsf{Grpd}(\mathbb C)$ of internal groupoids in $\mathbb C$ is again semi-abelian. The problem of determining whether the same kind of phenomenon occurs when the property of being semi-abelian is replaced by the one of being action representable (in the sense of Borceux, Janelidze and Kelly) turns out to be rather subtle. In the present article we give a sufficient condition for this to be true: in fact we prove that the category $\mathsf{Grpd}(\mathbb C)$ is a semi-abelian action representable algebraically coherent category with normalizers if and only if $\mathbb C$ is a semi-abelian action representable algebraically coherent category with normalizers. This result applies in particular to the categories of internal groupoids in the categories of groups, Lie algebras and cocommutative Hopf algebras, for instance.

math.CT↗

arXiv2020

A note on images of cover relations

For a category $\mathbb{C}$, a small category $\mathbb{I}$, and a pre-cover relation $\sqsubset$ on $\mathbb C$ we prove, under certain completeness assumptions on $\mathbb C$, that a morphism $g: B\to C$ in the functor category $\mathbb {C}^{\mathbb I}$ admits an image with respect to the pre-cover relation on $\mathbb C^{\mathbb I}$ induced by $\sqsubset$ as soon as each component of $g$ admits an image with respect to $\sqsubset$. We then apply this to show that if a pointed category $\mathbb{C}$ is: (i) algebraically cartesian closed; (ii) exact protomodular and action accessible; or (iii) admits normalizers, then the same is true of each functor category $\mathbb{C}^{\mathbb I}$ with $\mathbb{I}$ finite. In addition, our results give explicit constructions of images in functor categories using limits and images in the underlying category. In particular, they can be used to give explicit constructions of both centralizers and normalizers in functor categories using limits and centralizers or normalizers (respectively) in the underlying category.

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arXiv2019

Hall's criterion for nilpotence in semi-abelian categories

A well-known theorem of P. Hall, usually called Hall's criterion for nilpotence, states: a group G is nilpotent whenever it has a normal subgroup N such that G/[N,N] and N are nilpotent. We widely generalize this result, replacing groups with objects in an abstract semi-abelian category satisfying suitable conditions. In particular, these conditions are satisfied in any algebraically coherent semi-abelian category, and hence in many categories of classical algebraic structures (including a few where Hall's theorem is already known). Note that the categories of crossed modules, crossed Lie algebras, n-cat groups, and cocommutative Hopf algebras over fields are also algebraically coherent. We give, however, a counter-example showing that Hall's theorem does not hold in the (semi-abelian) variety of non-associative rings.

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arXiv2013

Normalizers and split extensions

We make explicit a larger structural phenomenon hidden behind the existence of normalizers in terms of existence of certain cartesian maps related to the kernel functor.

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