arXiv subjects

Lucrezia Bottegoni

Publications and source records attributed to Lucrezia Bottegoni.

7 recordsLinked to original sources

arXiv2026

Frobenius functors and one-sided Hopf algebras in braided monoidal categories

Evidence suggests a tight connection between the existence of antipode-like maps on bialgebra-like structures, and the Frobenius property for the associated free Hopf module functor. In this paper, we prove that in any braided monoidal category satisfying few mild assumptions, a bimonoid is a one-sided Hopf monoid and the antipode is a bimonoid anti-homomorphism, if and only if the free Hopf module functor is Frobenius. Our interest in these structures stems from having found genuine examples of one-sided Hopf monoids in contexts where the braiding is non-trivial. In fact, we provide a construction of the free one-sided Hopf monoid over a comonoid in any symmetric monoidal category with some non-restrictive additional assumptions. We present several examples of this construction, and describe the resulting one-sided (sometimes two-sided) Hopf monoids.

math.QA↗

arXiv2026

Protomodularity of cocommutative Hopf monoids in duoidal categories and quasitriangular Hopf algebras

In this work, we extend the protomodularity of the category of cocommutative Hopf algebras to the quasitriangular setting. Every quasitriangular Hopf algebra admits a minimal quasitriangular Hopf subalgebra and, as we show, can be regarded as a cocommutative bimonoid in the tensor-braided duoidal category of bimodules over it. This leads us to investigate protomodularity in the broader context of Hopf monoids in duoidal categories. To this end, we adopt a slight modification of Böhm's notion of antipode associated with a reversion, further refining an earlier one due to Böhm-Lack. This framework allows us to study Hopf monoids in this setting, Galois and co-Galois maps, and the factorization of Hopf monoids. Using these tools, we prove a factorization of points, the Split Short Five Lemma, and the existence of pullbacks of split epimorphisms along arbitrary morphisms in the category of cocommutative Hopf monoids with monic unit in any tensor-braided duoidal category with a reversion; hence this category is protomodular. As applications, we recover the protomodularity of cocommutative Hopf algebras in symmetric monoidal categories under mild assumptions, and we obtain that of the coslice category of quasitriangular (resp. triangular) Hopf algebras under a fixed subobject; in the triangular case, this can be traced back to a category of generalized internal groups, introduced in the present work. When the fixed subobject is minimal, we infer the protomodularity of the category of quasitriangular Hopf algebras whose minimal quasitriangular Hopf subalgebra is isomorphic to the fixed subobject, which we interpret as the protomodularity of an essential fibre of a functor. As a byproduct, our results extend the double cross product of cocommutative Hopf algebras to the quasitriangular setting.

math.CT↗

arXiv2026

Semiseparability of induction functors in a monoidal category

For any algebra morphism in a monoidal category, we provide sufficient conditions (which are also necessary if the unit is a left tensor generator) for the attached induction functor being semiseparable. Under mild assumptions, we prove that the semiseparability of the induction functor is preserved if one applies a lax monoidal functor. Similar results are shown for the coinduction functors attached to coalgebra morphisms in a monoidal category. As an application, we study the semiseparability of combinations of (co)induction functors in the context of duoidal categories.

math.CT↗

arXiv2025

Infinitesimal $\mathcal{R}$-matrices for some families of Hopf algebras

Given a bialgebra $H$ such that the associated trivial topological bialgebra $H[[\hbar]]$ admits a quasitriangular structure $\tilde{\mathcal{R}}=\mathcal{R}(1\otimes 1+\hbarχ+\mathcal{O}(\hbar^2))$, one gets a distinguished element $χ\in H \otimes H$ which is an infinitesimal $\mathcal{R}$-matrix, according to the definition given in [1]. In this paper we classify infinitesimal $\mathcal{R}$-matrices for some families of well-known Hopf algebras, among which are the generalized Kac-Paljutkin Hopf algebras $H_{2n^2}$, the Radford Hopf algebras $H_{(r,n,q)}$, and the Hopf algebras $E(n)$.

math.QA↗

arXiv2023

On (naturally) semifull and (semi)separable semifunctors

The notion of semifunctor between categories, due to S. Hayashi (1985), is defined as a functor that does not necessarily preserve identities. In this paper we study how several properties of functors, such as fullness, full faithfulness, separability, natural fullness, can be formulated for semifunctors. Since a full semifunctor is actually a functor, we are led to introduce a notion of semifullness (and then semifull faithfulness) for semifunctors. In order to show that these conditions can be derived from requirements on the hom-set components associated with a semifunctor, we look at "semisplitting properties" for seminatural transformations and we investigate the corresponding properties for morphisms whose source or target is the image of a semifunctor. We define the notion of naturally semifull semifunctor and we characterize natural semifullness for semifunctors that are part of a semiadjunction in terms of semisplitting conditions for the unit and counit attached to the semiadjunction. We study the behavior of semifunctors with respect to (semi)separability and we prove Rafael-type Theorems for (semi)separable semifunctors and a Maschke-type Theorem for separable semifunctors. We provide examples of semifunctors on which we test the properties considered so far.

math.CT↗

arXiv2023

Semiseparable functors and conditions up to retracts

In a previous paper we introduced the concept of semiseparable functor. Here we continue our study of these functors in connection with idempotent (Cauchy) completion. To this aim, we introduce and investigate the notions of (co)reflection and bireflection up to retracts. We show that the (co)comparison functor attached to an adjunction whose associated (co)monad is separable is a coreflection (reflection) up to retracts. This fact allows us to prove that a right (left) adjoint functor is semiseparable if and only if the associated (co)monad is separable and the (co)comparison functor is a bireflection up to retracts, extending a characterization pursued by X.-W. Chen in the separable case. Finally, we provide a semi-analogue of a result obtained by P. Balmer in the framework of pre-triangulated categories.

math.CT↗

arXiv2022

Semiseparable functors

In this paper we introduce and investigate the notion of semiseparable functor. One of its first features is that it allows a novel description of separable and naturally full functors in terms of faithful and full functors, respectively. To any semiseparable functor we attach an invariant, given by an idempotent natural transformation, which controls when the functor is separable and yields a characterization of separable functors in terms of (dual) Maschke and conservative functors. We prove that any semiseparable functor admits a canonical factorization as a naturally full functor followed by a separable functor. Here the main tool is the construction of the coidentifier category attached to the associated idempotent natural transformation. Then we move our attention to the semiseparability of functors that have an adjoint. First we obtain a Rafael-type Theorem. Next we characterize the semiseparability of adjoint functors in terms of the (co)separability of the associated (co)monads and the natural fullness of the corresponding (co)comparison functor. We also focus on functors that are part of an adjoint triple. In particular, we describe bireflections as semiseparable (co)reflections, or equivalently, as either Frobenius or naturally full (co)reflections. As an application of our results, we study the semiseparability of functors traditionally attached to ring homomorphisms, coalgebra maps, corings and bimodules, introducing the notions of semicosplit coring and semiseparability relative to a bimodule which extend those of cosplit coring and Sugano's separability relative to a bimodule, respectively.

math.CT↗