Research

Contents

• Papers
  • Published
  • Preprints
• Talks
• Posters
• Seminars
• Theses

⊕ My research interests lie in the intersection of category theory, Hopf algebras, and representation theory. In particular, I like anything involving dualities, like rigid monoidal or *-autonomous categories. Amongst other things, this involves studying how algebraic gadgets lift into more categorical frameworks, and how much theory still works in that context. When no one’s looking, I also like to study categories for their own sake.

Papers§

All of my papers—published or not—are readily available on the arXiv.⊕

Published§

• Duality in Monoidal Categories [HZ26]
  Joint work with Sebastian Halbig.
  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 finitely-generated projective objects in an abelian k-linear category. A byproduct of our work is that we obtain characterisations of the Grothendieck–Verdier duality, also called *-autonomy, and rigidity of functor categories endowed with Day convolution as their tensor product. Applied to Mackey functors, this yields a proof of a sketched argument by Bouc linking rigidity of an object to it being finitely-generated projective.

• Diagrammatics for Comodule Monads [HZ24a]
  Joint work with Sebastian Halbig.
  We extend Willerton’s [Wil08] 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.

• Pivotality, twisted centres, and the anti-double of a Hopf monad [HZ24b]
  Joint work with Sebastian Halbig.
  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.

Preprints§

• Gabi-Monads [HSZ26]
  
  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.

• Duoidal R-Matrices [Zor25]
  
  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.

• Simple algebras and exact module categories [CSZ25]
  Joint work with Kevin Coulembier and Mateusz Stroiński.
  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.

• Reconstruction of module categories in the infinite and non-rigid settings [SZ24]
  Joint work with Mateusz Stroiński.
  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 generalise algebra objects. For the category of finite-dimensional comodules over a bialgebra, we give this result a more concrete form, realising module categories as categories of contramodules over Hopf trimodule algebras—this specialises to our tensor-categorical results in the Hopf case. 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 characterisation 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 Bruguières, Lack, and Virelizier.

Talks§

• Lifting Adjunctions to Categories of Algebras
  2026-08-06, COSUMS, Dresden.

• Closed reconstruction
  2026-05-12, Seminar gmm, Dresden.
  2026-05-26, Seminar on Quantum groups, Hopf algebras and monoidal categories, Brussels.
  2026-06-17, European Quantum Algebra Lectures, Online; slides (handout).

• The Reflection Equation and Braided Module Categories
  2026-02-19, Seminario di Algebra e Teoria dei Numeri, Turin.

• Category Theory and All That
  2025-11-06, ZMP Seminar, Hamburg; slides.

• Categorical Reconstruction Theory
  2025-10-21, AMP Seminar, Hamburg; slides.

• Categorical Reconstruction Theory
  2025-07-04, Dresden; slides.
  My PhD thesis defence; see here for more information.

• Reconstruction for Lax Module Monads
  2025-07-17, CT2025, Brno; slides and accompanying website.
  2025-04-25, Hopf25, Brussels; slides and accompanying website.
  2025-05-19, Higher Structures for Hopf Algebras, Dresden.
  A classical result by Moerdijk and McCrudden is that Tannaka–Krein reconstruction for bialgebras may be lifted to bimonads: there is a bijection between bimonad structures on a given monad, and monoidal structures on its Eilenberg–Moore category that are compatible with the forgetful functor. This theorem may even be generalised to comodules over a bimonad.
  In contrast to these kinds of reconstruction results, we study reconstruction results that do not require a forgetful functor. This comes at the cost of not recovering the algebraic object of interest on-the-nose, but only up to Morita equivalence. This talk generalises a result of Ostrik about Hopf algebras on finite tensor categories to the general case of characterising lax module monads on a nice module category over a general abelian monoidal category with enough projectives. Crucially, the proof does not need any rigidity assumptions on the underlying category. As an application, we give conceptual proofs of the fundamental theorem of Hopf modules, and the fact that a bimonad is Hopf if and only if it is strong as a module monad over its base category.
  The talk is based on joint work with Matti Stroiński [SZ24].

• 2-Categorical Centre Constructions
  2024-11-01, Seminar “Factorisation homology”, Kleinwalsertal.
  Based on [FH23] and [Str04].

• Locally Finitely Presentable Categories and Ind-Completions
  2024-06-13, Seminar “Factorisation homology”, Bonn.

• The Kelly–Deligne Tensor Product
  2023-11-25, Seminar “Factorisation homology”, Dresden.
  Based on Tensor products of finitely cocomplete and abelian categories by López Franco [Lóp13].

• Duality in Monoidal Categories
  2023-01-16, Seminar gmm, Dresden.
  2023-05-23, hatc23, Marburg.
  2023-07-26, Uppsala Algebra Seminar, Uppsala.
  Dualities are an important tool in the study of monoidal categories and their applications. For example, underlying the construction of Tor and Ext functors is the tensor–hom adjunction in the category of bimodules over a unital ring—this is referred to as a closed monoidal structure. A stronger concept, rigidity, models the behaviour of finite-dimensional vector spaces; that is, the existence of evaluation and coevaluation morphisms, implementing a notion of dual basis. Under delooping, this corresponds to the concept of an adjunction in a bicategory, with coevaluation as unit and evaluation as counit. Grothendieck–Verdier duality, also called *-autonomy, lies between the strict confinements of rigidity, and the generality of monoidal closedness. It is closely linked to linearly distributive categories with negation.
  An immediate consequence of rigidity is that the internal-hom functor is tensor representable. That is, a dualising functor sending any object to its dual exists, and tensoring with the object is left adjoint to tensoring with its dual.
  This raises a naive question:

  Is a monoidal category with tensor representable internal-hom automatically rigid?

  While it is expected that this is not true in general, constructing counterexamples is non-trivial; we will provide one. Additionally, a weaker version of the above statement is true: every monoidal category with tensor representable internal-hom is Grothendieck–Verdier.
  This talk is based on joint work with Sebastian Halbig [HZ26].

• Abstract Schur Functors
  2023-07-21, “Operads” seminar, Bonn; notes
  Based on the paper Schur Functors and Categorified Plethysm by Baez, Moeller, and Trimble [BMT21].

• Abstract Mackey Functors
  2023-07-15, Mackey functors seminar, Dresden; see Section 6 of the script.

• Operads as Functors
  2022-12-15, “Operads” seminar, Bonn.

• Pivotality, twisted centres and the anti-double of a Hopf monad
  2022-05-12, Seminar of the Czech Academy of Sciences, Prague; slides.
  2022-05-15, pssl 106, Brno; slides.
  2022-05-30, qgs: Quantum Group Seminar, Online.
  
  Pairs in involution are an algebraic structure whose systematic study is motivated by their applications in knot theory, representation theory and cyclic homology theories. We will explore a categorical version of these objects from the perspective of representation theory of monoidal categories. A focus will lie on illustrating how their existence is linked to a particular well-behaved notion of duality, called pivotality. As a central point, we show how the language of monads allows us to combine the algebraic and categorical perspective on such pairs.
  Based on joint work with Sebastian Halbig [HZ24b].

• Optics in functional programming—a categorical perspective
  2022-01-10, Seminar gmm, Dresden; slides.
  A talk about the categorical aspects of (profunctor) optics, as done by Riley [Ril18] and Clark et al [CEGLMPR20], as well as connections to earlier mathematical work by Pastro and Street [PS08].

• Visual Category Theory
  2021-07-26, Seminar gmm, Dresden.
  The defense of my master’s thesis, concentrating on a higher-dimensional graphical calculus, as first introduced by Willerton [Wil08] and extended in the thesis. The “basic” slides are available—the talk was given on a Wacom tablet and thus contained many live drawings to illustrate the concepts. These, however, are lost to time.

Posters§

• Lax module functors, reconstruction, and Hopf algebras
  2024-09-02, Hopf Algebras and Monoidal Categories, Ferrara;
  pdf and accompanying website;
  Based on joint work with Mateusz Stroiński, [SZ24].

• Pivotality, Twisted Centres, and the Anti-Double of a Hopf Monad
  2024-06-27, CT24, Santiago de Compostela; pdf and accompanying website;
  Based on joint work with Sebastian Halbig—[HZ24b] and [HZ24a]—as well as Mateusz Stroiński, [SZ24].

• Duality in Monoidal Categories
  2023-07-07, ct23, Louvain-la-Neuve; in portrait and landscape format.
  Based on a paper with Sebastian Halbig of the same name [HZ26].

Seminars§

• Higher Structures for Hopf Algebras
  2025-05-19–2025-05-20, Dresden; website here

• Mackey Functors
  2023-07-14–2023-07-15, Dresden; website here.

Theses§

• Categorical Reconstruction Theory
  I completed my dissertation at TU Dresden on the 4th of July 2025, under the supervision of Ulrich Krähmer. Here is everything one could possibly want to know about this:
  • The slides for my thesis defence and the full source code.
  • My dissertation as I handed it in (source code). Additionally, I have made minor improvements as suggested by the referees, but due to some bylaws of the TU Dresden I’m not allowed to modify the version I originally handed in. The pdf of the updated version can be found here. The source code stays the same, one merely has to switch to the default branch. Note that the numbering is still consistent with the original version, I merely corrected some spelling errors and mathematical typos.

• Comodules for Categories
  Master’s thesis; with the help of a higher-dimensional graphical calculus, the Hopf algebraic anti-Drinfeld centre is lifted into the language of comodule monads. A result relating the (anti-)Drinfeld centre and (anti-)Yetter–Drinfeld modules is proven in this monadic setting.

• From Knot Theory to Algebra
  Bachelor’s thesis with a focus on keis; objects arising naturally when trying to generalise the number of 3-colourings of a knot.

References§

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