arXiv subjects

Marino Gran

Publications and source records attributed to Marino Gran.

At least 19 recordsLinked to original sources

arXiv2026

A new characterization of arithmetical categories and Pixley's theorem

We characterize the arithmetical categories - the exact Mal'tsev categories whose lattices of congruences are distributive - by a simple condition on some suitable pushouts and a finite limit: a regular category is exact arithmetical if and only if, for every triple of regular epimorphisms with common domain, the pairwise pushouts exist and the comparison morphism to the limit of the diagram they form is a regular epimorphism. For a regular category, the validity of this condition for pairs and for triples is equivalent to its validity for $n$-tuples for all $n \ge 2$, and characterizes arithmetical categories among regular ones. For pairs alone the condition reduces to the characterization of exact Mal'tsev categories among regular ones, due to Carboni, Kelly and Pedicchio. For $n$-tuples with $n \ge 4$ nothing further is obtained: arithmeticity is the third and last rung of that ladder. The proof rests on a categorical form of the Chinese Remainder Theorem; in the exact Mal'tsev context this form of the theorem lies close to results of Hoefnagel on majority categories. As a direct application, we give a new proof of Pixley's characterization of arithmetical varieties using a suitable diagram of free algebras.

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arXiv2026

Torsion and exactness

We give an equivalent definition of a non-pointed torsion theory, also called a pretorsion theory in the literature. This uses a new, more general, notion of a near-torsion theory, and does not use non-pointed exactness. Furthermore, a general method of associating the `largest' (non-pointed) torsion theory to any given near-torsion theory is described. We also describe a near-torsion theory formed by the class of cokernel diagrams and the class of kernel diagrams in a pointed category with kernels and cokernels, and its associated torsion theory.

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arXiv2026

Central series of cocommutative Hopf braces

By extending some classical results known for groups and skew braces, we define and investigate central series of cocommutative Hopf braces. Both left and right central series are defined using a $\star$-product that measures the difference between the two algebra operations, and naturally leads to introducing the notions of socle and of annihilator of a cocommutative Hopf brace. We characterize the central extensions relative to the subcategories of cocommutative Hopf algebras and of commutative and cocommutative Hopf algebras, respectively. Since the category of cocommutative Hopf braces is semi-abelian and it has enough projectives with respect to the class of cleft extensions, one can then establish suitable Hopf formulae for their homology. These are expressed in terms of the corresponding notions of relative commutators of cocommutative Hopf braces. In particular, the one relative to the subcategory of commutative and cocommutative Hopf algebras turns out to be the Huq commutator.

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arXiv2025

Hopf formulae for cocommutative Hopf algebras

The adjunction between coalgebras and Hopf algebras, first described by Takeuchi, allows one to prove that the semi-abelian category of cocommutative Hopf algebras has enough $\mathcal E$-projective objects with respect to the class $\mathcal{E}$ of cleft extensions. One then proves that, for any cocommutative Hopf algebra, there exists a weak $\mathcal{E}$-universal normal (=central) extension. This fact allows one to apply the methods of categorical Galois theory to classify normal $\mathcal{E}$-extensions and to provide an explicit description of the fundamental group of a cocommutative Hopf algebra in terms of a generalized Hopf formula. Moreover, with any cleft extension, we associate a 5-term exact sequence in homology that can be seen as a Hopf-theoretic analogue of the classical Stallings-Stammbach exact sequence in group theory.

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arXiv2025

Double groupoids and $2$-groupoids in regular Mal'tsev categories

We prove that the category 2-$ \mathrm{Grpd}(\mathscr{C}) $ of internal $2$-groupoids is a Birkhoff subcategory of the category $ \mathrm{Grpd}^2(\mathscr{C}) $ of double groupoids in a regular Mal'tsev category $\mathscr{C}$ with finite colimits. In particular, when $\mathscr{C}$ is a Mal'tsev variety of universal algebras, the category 2-$ \mathrm{Grpd}(\mathscr{C}) $ is also a Mal'tsev variety, of which we describe the corresponding algebraic theory. When $\mathscr{C}$ is a naturally Mal'tsev category, the reflector from $ \mathrm{Grpd}^2(\mathscr{C}) $ to 2-$ \mathrm{Grpd}(\mathscr{C}) $ has an additional property related to the commutator of equivalence relations. We prove that the category 2-$ \mathrm{Grpd}(\mathscr{C}) $ is semi-abelian when $\mathscr{C}$ is semi-abelian, and then provide sufficient conditions for 2-$ \mathrm{Grpd}(\mathscr{C}) $ to be action representable.

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arXiv2025

Hopf braces and semi-abelian categories

Hopf braces have been introduced as a Hopf-theoretic generalization of skew braces. Under the assumption of cocommutativity, these algebraic structures are equivalent to matched pairs of actions on Hopf algebras, that can be used to produce solutions of the quantum Yang-Baxter equation. We prove that the category of cocommutative Hopf braces is semi-abelian and strongly protomodular. In particular, this implies that the main homological lemmas known for groups, Lie algebras and other classical algebraic structures also hold for cocommutative Hopf braces. Abelian objects are commutative and cocommutative Hopf algebras, that form an abelian Birkhoff subcategory of the category of cocommutative Hopf braces. Moreover, we show that the full subcategories of "primitive Hopf braces" and of "skew braces" form an hereditary torsion theory in the category of cocommutative Hopf braces, and that "skew braces" are also a Birkhoff subcategory and a localization of the latter category. Finally, we describe central extensions and commutators for cocommutative Hopf braces.

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arXiv2023

Ideals and Congruences in L-algebras and Pre-L-algebras

We link the recent theory of $L$-algebras to previous notions of Universal Algebra and Categorical Algebra concerning subtractive varieties, commutators, multiplicative lattices, and their spectra. We show that the category of $L$-algebras is subtractive and normal in the sense of Zurab Janelidze, but neither the category of $L$-algebras nor that of pre-$L$-algebras are Mal'tsev categories, hence in particular they are not semi-abelian. Therefore $L$-algebras are a rather peculiar example of an algebraic structure.

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arXiv2023

Groupoids and skeletal categories form a pretorsion theory in $\mathsf{Cat}$

We describe a pretorsion theory in the category $Cat$ of small categories: the torsion objects are the groupoids, while the torsion-free objects are the skeletal categories, i.e., those categories in which every isomorphism is an automorphism. We infer these results from two unexpected properties of coequalizers in $Cat$ that identify pairs of objects: they are faithful and reflect isomorphisms.

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arXiv2023

Conditional flatness, fiberwise localizations, and admissible reflections

We extend the group-theoretic notion of conditional flatness for a localization functor to any pointed category, and investigate it in the context of homological categories and of semi-abelian categories. In the presence of functorial fiberwise localization analogous results to those obtained in the category of groups hold, and we provide existence theorems for certain localization functors in specific semi-abelian categories. We prove that a Birkhoff subcategory of an ideal determined category yields a conditionally flat localization, and explain how conditional flatness corresponds to the property of admissibility of an adjunction from the point of view of categorical Galois theory. Under the assumption of fiberwise localization we give a simple criterion to determine when a (normal epi)-reflection is a torsion-free reflection. This is shown to apply in particular to nullification functors in any semi-abelian variety of universal algebras. We also relate semi-left-exactness for a localization functor $L$ with what is called right properness for the $L$-local model structure.

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arXiv2023

Central extensions of preordered groups

We prove that the category of preordered groups contains two full reflective subcategories that give rise to some interesting Galois theories. The first one is the category of the so-called commutative objects, which are precisely the preordered groups whose group law is commutative. The second one is the category of abelian objects, that turns out to be the category of monomorphisms in the category of abelian groups. We give a precise description of the reflector to this subcategory, and we prove that it induces an admissible Galois structure and then a natural notion of categorical central extension. We then characterize the central extensions of preordered groups in purely algebraic terms: these are shown to be the central extensions of groups having the additional property that their restriction to the positive cones is a special homogeneous surjection of monoids.

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arXiv2023

Pretorsion theories in lextensive categories

We propose a construction of a stable category for any pretorsion theory in a lextensive category. We prove the universal property of the stable category, that extends previous results obtained for the stable category of internal preorders in a pretopos. Some examples are provided in the categories of topological spaces and of (small) categories.

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arXiv2022

The stable category of preorders in a pretopos II: the universal property

We prove that the stable category associated with the category $\mathsf{PreOrd}(\mathbb C)$ of internal preorders in a pretopos $\mathbb C$ satisfies a universal property. The canonical functor from $\mathsf{PreOrd}(\mathbb C)$ to the stable category $\mathsf{Stab}(\mathbb C)$ universally transforms a pretorsion theory in $\mathsf{PreOrd}(\mathbb C)$ into a classical torsion theory in the pointed category $\mathsf{Stab}(\mathbb C)$. This also gives a categorical insight into the construction of the stable category first considered by Facchini and Finocchiaro in the special case when $\mathbb C$ is the category of sets.

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arXiv2022

The stable category of preorders in a pretopos I: general theory

In a recent article Facchini and Finocchiaro considered a natural pretorsion theory in the category of preordered sets inducing a corresponding stable category. In the present work we propose an alternative construction of the stable category of the category $\mathsf{PreOrd} (\mathbb C)$ of internal preorders in any coherent category $\mathbb C$, that enlightens the categorical nature of this notion. When $\mathbb C$ is a pretopos we prove that the quotient functor from the category of internal preorders to the associated stable category preserves finite coproducts. Furthermore, we identify a wide class of pretoposes, including all $σ$-pretoposes and all elementary toposes, with the property that this functor sends any short $\mathcal Z$-exact sequences in $\mathsf{PreOrd} (\mathbb C)$ (where $\mathcal Z$ is a suitable ideal of trivial morphisms) to a short exact sequence in the stable category. These properties will play a fundamental role in proving the universal property of the stable category, that will be the subject of a second article on this topic.

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arXiv2021

Some remarks on protolocalizations and protoadditive reflections

We investigate additional properties of protolocalizations, introduced and studied by F. Borceux, M. M. Clementino, M. Gran, and L. Sousa, and of protoadditive reflections, introduced and studied by T. Everaert and M. Gran. Among other things we show that there are no non-trivial (protolocalizations and) protoadditive reflections of the category of groups, and establish a connection between protolocalizations and Kurosh--Amitsur radicals of groups with multiple operators whose semisimple classes form subvarieties.

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arXiv2021

Non-associative algebras

A non-associative algebra over a field $\mathbb{K}$ is a $\mathbb{K}$-vector space $A$ equipped with a bilinear operation \[ {A\times A\to A\colon\; (x,y)\mapsto x\cdot y=xy}. \] The collection of all non-associative algebras over $\mathbb{K}$, together with the product-preserving linear maps between them, forms a variety of algebras: the category $\mathsf{Alg}_\mathbb{K}$. The multiplication need not satisfy any additional properties, such as associativity or the existence of a unit. Familiar categories such as the varieties of associative algebras, Lie algebras, etc. may be found as subvarieties of $\mathsf{Alg}_\mathbb{K}$ by imposing equations, here $x(yz)=(xy)z$ (associativity) or $xy =- yx$ and $x(yz)+z(xy)+ y(zx)=0$ (anti-commutativity and the Jacobi identity), respectively. The aim of these lectures is to explain some basic notions of categorical algebra from the point of view of non-associative algebras, and vice versa. As a rule, the presence of the vector space structure makes things easier to understand here than in other, less richly structured categories. We explore concepts like normal subobjects and quotients, coproducts and protomodularity. On the other hand, we discuss the role of (non-associative) polynomials, homogeneous equations, and how additional equations lead to reflective subcategories.

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arXiv2020

Torsion theories and coverings of preordered groups

In this article we explore a non-abelian torsion theory in the category of preordered groups: the objects of its torsion-free subcategory are the partially ordered groups, whereas the objects of the torsion subcategory are groups (with the total order). The reflector from the category of preordered groups to this torsion-free subcategory has stable units, and we prove that it induces a monotone-light factorization system. We describe the coverings relative to the Galois structure naturally associated with this reflector, and explain how these coverings can be classified as internal actions of a Galois groupoid. Finally, we prove that in the category of preordered groups there is also a pretorsion theory, whose torsion subcategory can be identified with a category of internal groups. This latter is precisely the subcategory of protomodular objects in the category of preordered groups, as recently discovered by Clementino, Martins-Ferreira, and Montoli.

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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.

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arXiv2020

Pretorsion theories in general categories

We present a setting for the study of torsion theories in general categories. The idea is to associate, with any pair ($\mathcal T$, $\mathcal F$) of full replete subcategories in a category $\mathcal C$, the corresponding full subcategory $\mathcal Z = \mathcal T \cap \mathcal F$ of \emph{trivial objects} in $\mathcal C$. The morphisms which factor through $\mathcal Z$ are called $\mathcal Z$-trivial, and these form an ideal of morphisms, with respect to which one can define $\mathcal Z$-prekernels, $\mathcal Z$-precokernels, and short $\mathcal Z$-preexact sequences. This naturally leads to the notion of pretorsion theory, which is the object of study of this article, and includes the classical one in the abelian context when $\mathcal Z$ is reduced to the $0$-object of $\mathcal C$. We study the basic properties of pretorsion theories, and examine some new examples in the category of all endomappings of finite sets and in the category of preordered sets.

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