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Alan S. Cigoli

Publications and source records attributed to Alan S. Cigoli.

14 recordsLinked to original sources
arXiv2025

On the representability of actions of Leibniz algebras and Poisson algebras

In a recent paper, motivated by the study of central extensions of associative algebras, G. Janelidze introduces the notion of weakly action representable category. In this paper, we show that the category of Leibniz algebras is weakly action representable and we characterize the class of acting morphisms. Moreover, we study the representability of actions of the category of Poisson algebras by describing explicitly a universal strict general actor.

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arXiv2023

The third cohomology 2-group

In this paper we show that a finite product preserving opfibration can be factorized through an opfibration with the same property, but with groupoidal fibres. If moreover the codomain is additive, one can endow each fibre of the new opfibration with a canonical symmetric 2-group structure. We then apply such factorization to the opfibration that sends a crossed extension of a group C to its corresponding C-module. The symmetric 2-group structure so obtained on the fibres, defines the third cohomology 2-group of C, with coefficients in a C-module. We show that the usual third and second cohomology groups are recovered as its homotopy invariants. Furthermore, even if all results are presented in the category of groups, their proofs are valid in any strongly protomodular semi-abelian category, once one adopts the corresponding internal notions.

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arXiv2022

On pseudofunctors sending groups to 2-groups

For a category B with finite products, we first characterize pseudofunctors from B to Cat whose corresponding opfibration is cartesian monoidal. Among those, we then characterize the ones which extend to pseudofunctors from internal groups to 2-groups. If B is additive, this is the case precisely when the corresponding opfibration has groupoidal fibres.

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arXiv2021

Fibred categorical theory of obstruction and classification of morphisms

We set up a fibred categorical theory of obstruction and classification of morphisms that specializes to the one of monoidal functors between categorical groups and also to the Schreier-Mac Lane theory of group extensions. Further applications are provided, as for example a classification of unital associative algebra extensions with non-abelian kernel in terms of Hochschild cohomology.

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arXiv2020

Discrete and conservative reflections of fibrations

We focus on two factorization systems for opfibrations in the 2-category Fib(B) of fibrations over a fixed base category B. The first one is the internal version of the so called comprehensive factorization, where the right orthogonal class is given by internal discrete opfibrations. The second one has as its right orthogonal class internal opfibrations in groupoids, i.e. with groupoidal fibres. These factorizations can be obtained by means of a single step 2-colimit. Namely, their left orthogonal parts are nothing but suitable coidentifiers and coinverters respectively. We will show how these results follow from their analogues in Cat. To this end, we first provide suitable conditions on a 2-category C, allowing the transfer of the construction of coinverters and coidentifiers from C to Fib(B).

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arXiv2019

Galois theory and the categorical Peiffer commutator

We show that the Peiffer commutator previously defined by Cigoli, Mantovani and Metere can be used to characterize central extensions of precrossed modules with respect to the subcategory of crossed modules in any semi-abelian category satisfying an additional property. We prove that this commutator also characterizes double central extensions, obtaining then some Hopf formulas for the second and third homology objects of internal precrossed modules.

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arXiv2018

Fibered aspects of Yoneda's regular span

In this paper we start by pointing out that Yoneda's notion of a regular span $S \colon \mathcal{X} \to \mathcal{A} \times \mathcal{B}$ can be interpreted as a special kind of morphism, that we call fiberwise opfibration, in the 2-category $\mathsf{Fib}(\mathcal{A})$. We study the relationship between these notions and those of internal opfibration and two-sided fibration. This fibrational point of view makes it possible to interpret Yoneda's Classification Theorem given in his 1960 paper as the result of a canonical factorization, and to extend it to a non-symmetric situation, where the fibration given by the product projection $Pr_0 \colon \mathcal{A} \times \mathcal{B} \to \mathcal{A}$ is replaced by any split fibration over $\mathcal{A}$. This new setting allows us to transfer Yoneda's theory of extensions to the non-additive analog given by crossed extensions for the cases of groups and other algebraic structures.

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arXiv2018

A relative monotone-light factorisation system for internal groupoids

Given an exact category $\mathcal{C}$, it is well known that the connected component reflector $π_0\colon\mathsf{Gpd}(\mathcal{C})\to\mathcal{C}$ from the category $\mathsf{Gpd}(\mathcal{C})$ of internal groupoids in $\mathcal{C}$ to the base category $\mathcal{C}$ is semi-left-exact. In this article we investigate the existence of a monotone-light factorisation system associated with this reflector. We show that, in general, there is no monotone-light factorisation system $(\mathcal{E}',\mathcal{M}^*)$ in $\mathsf{Gpd}(\mathcal{C})$, where $\mathcal{M}^*$ is the class of coverings in the sense of the corresponding Galois theory. However, when restricting to the case where $\mathcal{C}$ is an exact Mal'tsev category, we show that the so-called comprehensive factorization of regular epimorphisms in $\mathsf{Gpd}(\mathcal{C})$ is the relative monotone-light factorisation system (in the sense of Chikhladze) in the category $\mathsf{Gpd}(\mathcal{C})$ corresponding to the connected component reflector, where $\mathcal{E}'$ is the class of final functors and $\mathcal{M}^*$ the class of regular epimorphic discrete fibrations.

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arXiv2015

Algebraically coherent categories

We call a finitely complete category algebraically coherent when the change-of-base functors of its fibration of points are coherent, which means that they preserve finite limits and jointly strongly epimorphic pairs of arrows. We give examples of categories satisfying this condition; for instance, coherent categories, categories of interest in the sense of Orzech, and (compact) Hausdorff algebras over a semi-abelian algebraically coherent theory. We study equivalent conditions in the context of semi-abelian categories, as well as some of its consequences: including amongst others, strong protomodularity, and normality of Higgins commutators for normal subobjects, and in the varietal case, fibre-wise algebraic cartesian closedness.

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arXiv2015

Peiffer product and Peiffer commutator for internal pre-crossed modules

In this work we introduce the notions of Peiffer product and Peiffer commutator of internal pre-crossed modules over a fixed object B, extending the corresponding classical notions to any semi-abelian category C. We prove that, under mild additional assumptions on C, crossed modules are characterized as those pre-crossed modules X whose Peiffer commutator is trivial. Furthermore we provide suitable conditions on C (fulfilled by a large class of algebraic varietes, including among others groups, associative algebras, Lie and Leibniz algebras) under which the Peiffer product realizes the coproduct in the category of crossed modules over B.

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arXiv2014

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.

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arXiv2013

Characteristic subobjects in semi-abelian categories

We extend to semi-abelian categories the notion of characteristic subobject, which is widely used in group theory and in the theory of Lie algebras. Moreover, we show that many of the classical properties of characteristic subgroups of a group hold in the general semi-abelian context, or in stronger ones.

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