Nathanael Arkor

narkor.co

Postdoc researching Category Theory & Type Theory at Tallinn University of Technology

Google ScholarORCID

About me

Hi – I am a postdoctoral researcher in the Logic and Semantics Group at Tallinn University of Technology, working under Tarmo Uustalu. My present research interests fall broadly within pure category theory and type theory, and include more specifically: formal category theory, relative monads, double category theory, 2-category theory, abstract syntax & algebraic type theory, and categorical algebra & logic.

Previously, I was a postdoctoral researcher in the Algebra Group at Masaryk University, working under John Bourke. I completed my PhD in the Programming, Logic, and Semantics Group at the University of Cambridge with Marcelo Fiore.

I am happy to supervise bachelor's and master's projects in any of the areas mentioned above – feel free to email me to discuss potential projects.

Preprints

  1. Idempotence for relative monads (with Andrew Slattery), , arXiv
  2. Magmal characterisations of cocartesian categories, , arXiv
  3. Exponentiable virtual double categories and presheaves for double categories, , arXiv
  4. Bicategories of algebras for relative pseudomonads (with Philip Saville and Andrew Slattery), , arXiv
  5. Enhanced 2-categorical structures, two-dimensional limit sketches and the symmetry of internalisation (with John Bourke and Joanna Ko), , arXiv
  6. Higher-order algebraic theories (with Dylan McDermott), , PDF
    See Chapter 4 of my thesis for an revised account.

Papers

  1. The nerve theorem for relative monads (with Dylan McDermott), Theory and Applications of Categories, , TAC, arXiv
  2. Relative monadicity (with Dylan McDermott), Journal of Algebra, , DOI, arXiv
  3. Adjoint functor theorems for lax-idempotent pseudomonads (with Ivan Di Liberti and Fosco Loregian), Theory and Applications of Categories, , TAC, arXiv
  4. The formal theory of relative monads (with Dylan McDermott), Journal of Pure and Applied Algebra, , DOI, arXiv
  5. Abstract clones for abstract syntax (with Dylan McDermott), FSCD 2021, DOI, arXiv
  6. Algebraic models of simple type theories: a polynomial approach (with Marcelo Fiore), LICS 2020, DOI, arXiv

Thesis

  1. Monadic and Higher-Order Structure, University of Cambridge (2022), PDF, DOI

Software

  1. quiver: a modern commutative diagram editor for the web.
  2. tangle: a modern string diagram editor for the web.

Talks

  1. The three-dimensional structures formed by monoidal categories, bicategories, double categories, etc., Category Theory 2025, Slides
  2. Virtual double categories: Past, present, and future, Séminaire Itinérant de Catégories, Slides
  3. Relative monadicity, PSSL 110, Slides
  4. A (virtual) double category theorist's perspective on polynomials, The Topos Institute Colloquium, Recording, Slides
  5. A recipe for enriched categories, TSEM Seminar, Slides
  6. The pullback theorem for relative monads (with Dylan McDermott), Category Theory 2024, Slides
  7. The pullback theorem for relative monads (with Dylan McDermott), TSEM Seminar, Recording, Slides
  8. Relative monads and distributors (with Dylan McDermott), #LoVe Seminar, Slides
  9. The nature of adjoint functor theorems (with Ivan Di Liberti and Fosco Loregian), PSSL 108, Slides
  10. The formal theory of relative monads (with Dylan McDermott), Category Theory 2023, Recording, Slides
  11. The theory of relative (co)monads (with Dylan McDermott), Comonads Seminar, Slides
  12. The formal theory of relative monads (with Dylan McDermott), PSSL 107, Slides
  13. A 2-dimensional perspective on polymorphism, ICE-TCS Seminar, Slides
  14. Relative monads and their many guises (with Dylan McDermott), Octoberfest 2022, Slides
  15. What is a proof?, DIMEA and FORMELA Seminar, Slides
  16. The (relative) monad–theory correspondence (with Dylan McDermott), Masaryk University Algebra Seminar, Recording, Slides
  17. The (relative) monad–theory correspondence (with Dylan McDermott), TallCat, Slides
  18. The formal theory of theories (first half with Dylan McDermott), CT 20→21, Recording, Slides
  19. Abstract clones for abstract syntax (with Dylan McDermott), FSCD 2021, Recording, Slides
  20. Higher-order algebraic theories and relative monads (with Dylan McDermott), Categories and Companions Symposium 2021, Recording, Slides
  21. Higher-order algebraic theories and relative monads (with Dylan McDermott), Masaryk University Algebra Seminar, Recording, Slides
  22. Algebraic models of simple type theories: a polynomial approach (with Marcelo Fiore), LICS 2020, Recording, Slides
  23. Algebraic simple type theory: a polynomial approach (with Marcelo Fiore), Category Theory 2019, Slides