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Federico Campanini

Publications and source records attributed to Federico Campanini.

16 recordsLinked to original sources
arXiv2026

Spaces of subobjects as spectral spaces

We study natural topologies on spaces of subobjects of a fixed object in a suitable category, with the aim of determining when these spaces are spectral. A central step in our approach, which is also of independent interest, is the comparison between the categorical notion of a $λ$-generated object and the order-theoretic notion of a $λ$-compact element in a lattice of subobjects. We prove that these notions coincide under some natural and mild assumptions . In the finitary case, this allows us to describe the finitely generated subobjects purely in order-theoretic terms and to construct, inside each interval of a subobject lattice, a canonical algebraic core. We study this construction abstractly for complete lattices and characterize it by a universal property. We then introduce the categorical Zariski topology on spaces of subobjects, relate it to the Scott topology, and obtain spectrality criteria for the whole subobject space and for its algebraic core.

math.CT↗
arXiv2026

Categorical Algebra of Atomic Monoids: Presentability, Regularity, and Pretorsion Theories

We study the category $\mathsf{AtoMon}$ of atomic monoids and atom-preserving homomorphisms. We prove that $\mathsf{AtoMon}$ is locally finitely presentable by exhibiting a strong generator consisting of compact objects. We show that $\mathsf{AtoMon}$ admits (regular epi, mono)-factorizations but that it is not a regular category: we construct a regular epimorphism which is not pullback-stable. We also establish adjunctions for the group of units and explicitly construct the ``cofree atomic monoid'' over an arbitrary monoid. Finally, we exhibit a way to lift torsion theories of $\mathsf{Grp}$ to pretorsion theories of $\mathsf{AtoMon}$ and extend this construction to a more general setting.

math.CT↗
arXiv2025

Lattices of pretorsion classes

Since their introduction, torsion theories have played a key role in the study of abelian and pointed categories. In representation theory, torsion theories and lattices of torsion classes of mod$ A$, for $A$ a finite-dimensional algebra, have been widely studied. The more recent definition of pretorsion theories, that can be given for any category, has expanded the theory, giving many more instances of ``non-pointed torsion theories'' in unexpected settings. In this work, we introduce and study the lattice $\mathcal{L}_t(A)$ of pretorsion classes of mod$ A$. These lattices are in close connection with the lattices tors$ A$ of torsion classes of mod$ A$. We fully describe the completely join-irreducible elements of $\mathcal{L}_t(A)$. Moreover, we characterise and give a full classification of when $\mathcal{L}_t(A)$ is distributive and further describe when it can be identified with the \emph{distributive closure} of tors$ A$. Finally, we show how the lattices of pretorsion classes, together with their duals, can be used to build pretorsion theories in mod$ A$.

math.RT↗
arXiv2025

Homomorphisms with semilocal endomorphism rings between modules

We study the category $\operatorname{Morph}(\operatorname{Mod} R)$ whose objects are all morphisms between two right $R$-modules. The behavior of objects of $\operatorname{Morph}(\operatorname{Mod} R)$ whose endomorphism ring in $\operatorname{Morph}(\operatorname{Mod} R)$ is semilocal is very similar to the behavior of modules with a semilocal endomorphism ring. For instance, direct-sum decompositions of a direct sum $\oplus_{i=1}^nM_i$, that is, block-diagonal decompositions, where each object $M_i$ of $\operatorname{Morph}(\operatorname{Mod} R)$ denotes a morphism $μ_{M_i}\colon M_{0,i}\to M_{1,i}$ and where all the modules $M_{j,i}$ have a local endomorphism ring $\operatorname{End}(M_{j,i})$, depend on two invariants. This behavior is very similar to that of direct-sum decompositions of serial modules of finite Goldie dimension, which also depend on two invariants (monogeny class and epigeny class). When all the modules $M_{j,i}$ are uniserial modules, the direct-sum decompositions (block-diagonal decompositions) of a direct-sum $\oplus_{i=1}^nM_i$ depend on four invariants.

math.RA↗
arXiv2025

Factorizations of polynomials with integral non-negative coefficients

We study the structure of the commutative multiplicative monoid $\mathbb N_0[x]^*$ of all the non-zero polynomials in $\mathbb Z[x]$ with non-negative coefficients. We show that $\mathbb N_0[x]^*$ is not a half-factorial monoid and is not a Krull monoid, but has a structure very similar to that of Krull monoids, replacing valuations into $\mathbb N_0$ with derivations into $\mathbb N_0$. We study ideals, chain of ideals, prime ideals and prime elements of $\mathbb N_0[x]^*$. Our monoid $\mathbb N_0[x]^*$ is a submonoid of the multiplicative monoid of the ring $\mathbb Z[x]$, which is a left module over the Weyl algebra $A_1(\mathbb Z)$.

math.AC↗
arXiv2025

Exactness of cochain complexes via additive functors

We investigate the relation between the notion of $e$-exactness, recently introduced by Akray and Zebary, and some functors naturally related to it, such as the functor $P\colon\operatorname{Mod} R\to \operatorname{Spec}(\operatorname{Mod} R)$, where $\operatorname{Spec}(\operatorname{Mod} R)$ denotes the spectral category of $\operatorname{Mod} R$, and the localization functor with respect to the singular torsion theory.

math.RA↗
arXiv2025

The Category of Atomic Monoids: Universal Constructions and Arithmetic Properties

We introduce and investigate the category $\mathsf{AtoMon}$ of atomic monoids and atom-preserving monoid homomorphisms, which is a (non-full) subcategory of the usual category of monoids. In particular, we compute all limits and colimits, showing that $\mathsf{AtoMon}$ is a complete and cocomplete category. We also address certain arithmetic properties of products and coproducts, providing explicit formulas for some fundamental invariants associated with factorization lengths in atomic monoids.

math.RA↗
arXiv2024

Building pretorsion theories from torsion theories

Torsion theories play an important role in abelian categories and they have been widely studied in the last sixty years. In recent years, with the introduction of pretorsion theories, the definition has been extended to general (non-pointed) categories. Many examples have been investigated in several different contexts, such as topological spaces and topological groups, internal preorders, preordered groups, toposes, V-groups, crossed modules, etc. In this paper, we show that pretorsion theories naturally appear also in the "classical" framework, namely in abelian categories. We propose two ways of obtaining pretorsion theories starting from torsion theories. The first one uses "comparable" torsion theories, while the second one extends a torsion theory with a Serre subcategory. We also give a universal way of obtaining a torsion theory from a given pretorsion theory in additive categories. We conclude by providing several applications in module categories, internal groupoids, recollements and representation theory.

math.CT↗
arXiv2024

Some remarks on Prüfer rings with zero-divisors

Let $A$ be the fiber product $R\times_TB$, where $B\to T$ is a surjective ring homomorphism with regular kernel and $R\subseteq T$ is a ring extension where $T$ is an overring of $R$. In this paper we provide a characterization of when $A$ has distinguished Prüfer-like properties and new constructions of Prüfer rings with zero-divisors. Furthermore we give examples of homomorphic images of Prüfer rings that are Prüfer without assuming that the kernel of the surjection is regular. Finally we provide some remarks on the ideal theory of pre-Prüfer rings.

math.AC↗
arXiv2024

On bi-amalgamated constructions

Let $f:A\longrightarrow B, g:A\longrightarrow C$ be ring homomorphisms and let $\mathfrak{b}$ (resp., $\mathfrak{c}$) be an ideal of $B$ (resp., $C$) satisfying $f^{-1}(\mathfrak{b})=g^{-1}(\mathfrak{c})$. Recently Kabbaj, Louartiti and Tamekkante defined and studied the following subring $$A\bowtie^{f,g}(\mathfrak{b},\mathfrak{c}) :=\{(f(a)+b, g(a)+c)\mid a\in A, b\in\mathfrak{b}, c\in \mathfrak{c} \}$$ of $B\times C$, called the bi-amalgamation of $A$ with $(B,C)$ along $(\mathfrak{b}, \mathfrak{c})$, with respect to $(f,g)$. This ring construction is a natural generalization of the amalgamated algebras, introduced and studied by D'Anna, Finocchiaro and Fontana. The aim of this paper is to continue the investigation started by Kabbaj, Louartiti and Tamekkante, by providing a deeper insigt on the ideal-theoretic structure of bi-amalgamations.

math.AC↗
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.

math.CT↗
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.

math.CT↗
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.

math.CT↗
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.

math.CT↗