arXiv subjects

Tony Zorman

Publications and source records attributed to Tony Zorman.

8 recordsLinked to original sources

arXiv2026

Gabi-Monads

We study gabi-monads on skew-closed categories, extending the gabi-algebras of Berger, the second author, and Vercruysse beyond the linear case. Our main reconstruction theorem identifies gabi-monad structures on a monad with skew-closed structures on its Eilenberg--Moore category for which the canonical forgetful functor is strict closed. We compare this notion with closed monads in the sense of Kock, showing that in representation-theoretic cases these notions are quite different. On closed monoidal categories, every left Hopf monad is a normal gabi-monad, but the converse fails in general. We characterise when a gabi-monad is Hopf by the invertibility of the corresponding parametric mates, which recovers the ring-theoretic result that normal gabi-algebras over a commutative base ring are Hopf algebras. The theory of gabi-monads admits several natural examples, such as torsion-free modules, reflexive digraphs, and simplicial complexes, that we will explore in detail; we also study pointed sets as a quasi-example.

math.CT↗

arXiv2026

Categorical Reconstruction Theory

We generalise classical reconstruction results in algebra, using the language of monads, monoidal categories, module categories, as well as various notions of duality, such as closedness, Grothendieck--Verdier duality (also known as *-autonomy), and rigidity.

math.CT↗

arXiv2025

Duoidal R-Matrices

In this note, we define an analogue of R-matrices for bialgebras in the setting of a monad that is opmonoidal over two tensor products. Analogous to the classical case, such structures bijectively correspond to duoidal structures on the Eilenberg--Moore category of the monad. Further, we investigate how a cocommutative version of this lifts the linearly distributive structure of a normal duoidal category.

math.CT↗

arXiv2025

Simple algebras and exact module categories

We verify a conjecture of Etingof and Ostrik, stating that an algebra object in a finite tensor category is exact if and only if it is a finite direct product of simple algebras. Towards that end, we introduce an analogue of the Jacobson radical of an algebra object, similar to the Jacobson radical of a finite-dimensional algebra. We give applications of our main results in the context of incompressible finite symmetric tensor categories.

math.RT↗

arXiv2024

Reconstruction of module categories in the infinite and non-rigid settings

By building on the notions of internal projective and injective objects in a module category introduced by Douglas, Schommer-Pries, and Snyder, we extend the reconstruction theory for module categories of Etingof and Ostrik. More explicitly, instead of algebra objects in finite tensor categories, we consider quasi-finite coalgebra objects in locally finite tensor categories. Moreover, we show that module categories over non-rigid monoidal categories can be reconstructed via lax module monads, which generalize algebra objects. For the monoidal category of finite-dimensional comodules over a (non-Hopf) bialgebra, we give this result a more concrete form, realizing module categories as categories of contramodules over Hopf trimodule algebras -- this specializes to our tensor-categorical results in the Hopf case. In this context, we also give a precise Morita theorem, as well as an analogue of the Eilenberg--Watts theorem for lax module monads and, as a consequence, for Hopf trimodule algebras. Using lax module functors we give a categorical proof of the variant of the fundamental theorem of Hopf modules which applies to Hopf trimodules. We also give a characterization of fusion operators for a Hopf monad as coherence cells for a module functor structure, using which we similarly reinterpret and reprove the Hopf-monadic fundamental theorem of Hopf modules due to Brugui\`eres, Lack, and Virelizier.

math.QA↗

arXiv2023

Diagrammatics for Comodule Monads

We extend Willerton's graphical calculus for bimonads to comodule monads, a monadic interpretation of module categories over a monoidal category. As an application, we prove a version of Tannaka--Krein duality for these structures.

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arXiv2023

Duality in Monoidal Categories

We compare closed and rigid monoidal categories. Closedness is defined by the tensor product having a right adjoint: the internal hom functor. Rigidity, on the other hand, generalises the duality of finite-dimensional vector spaces. In the latter, the internal hom functor is implemented by tensoring with the respective duals. This raises the question: can one decide whether a closed monoidal category is rigid, simply by verifying that the internal hom is tensor-representable? We provide a counterexample in terms of the category of sl2-crystals. As a byproduct, we obtain characterisations of the Grothendieck-Verdier duality and rigidity of functor categories endowed with Day convolution as their tensor product. This has various applications, three of which we study in detail: generalisations of quasi-Frobenius algebras, called QF-2 algebras; Mackey functors, where we prove that, as expected due to work of Bouc, an object being rigidly dualisable is equivalent to it being finitely-generated projective; and crossed modules of finite groups, where we associate to each of these objects a Grothendieck-Verdier category of group-graded representations.

math.CT↗

arXiv2022

Pivotality, twisted centres and the anti-double of a Hopf monad

Finite-dimensional Hopf algebras admit a correspondence between so-called pairs in involution, one-dimensional anti-Yetter--Drinfeld modules and algebra isomorphisms between the Drinfeld and anti-Drinfeld double. We extend it to general rigid monoidal categories and provide a monadic interpretation under the assumption that certain coends exist. Hereto we construct and study the anti-Drinfeld double of a Hopf monad. As an application the connection with the pivotality of Drinfeld centres and their underlying categories is discussed.

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