Research

Contents

• Preprints
• Talks
• Posters
• Seminars
• Theses

⊕ My research interests can broadly be characterised under the term “monoidal 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§

• The Kelly–Deligne Tensor Product
  2023-11-25, Seminar “Factorisation homology”, Dresden.

• Duality in Monoidal Categories
  2023-01-16, Seminar gmm, Dresden.
  2023-05-23, hatc23, Marburg; slides (handout).
  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 in the form of the category of Mackey functors. 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.

• Abstract Schur Functors
  2023-07-21, “Operads” seminar, Bonn; notes
  Based on this paper by Baez, Moeller, and Trimble.

• 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.

• 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.

Posters§

• 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.

Seminars§

• Mackey Functors
  2023-07-14–2023-07-15, Dresden; website here.
  A seminar on Mackey functors and its applications.

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.