Research

Contents

• Preprints
• Talks
• Theses

⊕ My research interests can broadly be characterised under the term “category theory”—in particular, I like anything involving dualities, like rigid monoidal or *-autonomous categories. Further, I enjoy 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. Some more keywords include Hopf monads, graphical calculi, duoidal and linearly distributive categories, as well as operads.

Preprints¶

All of my preprints are readily available on the arXiv.⊕

• Duality in Monoidal Categories
  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 concept of duals in the sense of finite-dimensional vector spaces. A consequence of these axioms is that 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? At the Research School on Bicategories, Categorification and Quantum Theory, Heunen suggested that this is not the case. In this note, we will prove his claim by constructing an explicit counterexample.

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

Talks¶

• Duality in Monoidal Categories
  2023-01-16, Seminar gmm, Dresden.
  There are at least three categorical gadgets that capture various notions of mathematical duality: closed monoidal, -autonomous, and rigid monoidal categories. These concepts are all interlinked—rigid monoidal categories are always -autonomous, and they in turn are necessarily closed monoidal. In fact, the resulting internal-hom is very well-behaved. We will explore connections in the other direction: does the shape of the internal-hom of a closed monoidal category already characterise rigidity or *-autonomy?
  Based on joint work with Sebastian Halbig.

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

• 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 and Clark et.al., as well as connections to earlier mathematical work by Pastro and Street.

• 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 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.

Theses¶

• 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. An interesting equivalence relating the (anti-)Drinfeld centre and (anti-)Yetter–Drinfeld modules is proven in this monadically enriched setting.

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