# Research

Most of my work is in “applied” category theory. For example, I like to study how algebraic objects lift into more categorical frameworks and how much of the theory people have developed still works in that context. When no one’s looking, I also like to study categories for their own sake.

## Preprints

As a general note: one can download the source code for every paper on the arXiv! A simple click on “Other formats” on the relevant article will guide you through that.

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

Joint work with Sebastian Halbig; arXiv link.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

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

Based on joint work with Sebastian Halbig.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.

Version of this talk were given at:

*Optics in functional programming—a categorical perspective*

TU Dresden, 2022-01-10; 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*

TU Dresden, 2021-07-26.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.