Hi! I am Christoph, from Berlin, Mathematician, currently based in Oxford.

I am a mathematician working in higher category theory. Higher categories are a generalisation of spaces, adding a notion of “direction” to the latter: while in a space any path can be travelled along in two directions, in higher categories this need not be the case. This is relevant for the description of irreversible processes. In short, my work focuses on the interplay of algebraic and geometric models of higher categories, with computable foundations to “geometric higher category theory” being a central goal. In my PhD thesis I build a combinatorial model of higher categories based on the observation that higher categories can be locally described by certain framed stratified manifolds. Following these ideas, I am presently working (joint with Christopher Douglas) on a foundational exposition of a more general theory of “framed combinatorial topology”.

Written Work

In preparation

  • Framed combinatorial topology (joint with Christopher Douglas)

This book lays foundations of a theory of framed combinatorial topology. The theory synthesizes ideas from classical combinatorial topology with a combinatorial notion of framing. The synthesis has interesting consequences, including the following. Firstly, the presence of combinatorial framings allows to overcome fundamental obstruction to computability questions in classical combinatorial topology. Secondly, the framed setting provides a faithful bridge of combinatorics and topology allowing, for instance, for a proof of a `framed local Hauptvermutung'. The scope of the book is in some sense limited: eventually, in a yet bigger picture, framed combinatorial topology aims to provide a combinatorial, and, importantly, computable, foundation to structured algebraic topology and geometric higher category theory; this, for instance, merges ideas from manifold singularity theory, stratified Morse theory, and combinatorial higher category theory into a single unified theory.

PhD thesis

  • Associative n-categories (Submitted: Dec 2018l. Final: May 2019. ORA record, pdf)

While higher groupoids have a natural model in spaces, higher categories have no such well-accepted model. This makes the question of correctness of a given definition of higher categories difficult to answer. We argue that the question has a simple answer “locally”, namely, categories are locally modelled on so-called manifold diagrams. The idea of locally modelling higher categories by manifold diagrams (most prominently in the 3-dimensional case of Gray-categories) is not new and has been proposed by multiple authors. However, the niceness of this manifold-based perspective on higher categories has in the past been somewhat obfuscated by the complexity of manifold geometry in higher dimensions. My thesis discusses a fully algebraic (and - in some sense - canonical) formulation of manifold diagrams, and thereby resolves the long-standing problem of finding a generalization of string diagrams to higher dimensions. Based on this notion of manifold diagrams, we further define Associative n-categories”, which generalize Gray-categories to higher dimensions.

Some remarks: The first (roughly) 100 pages are a concise summary of all ideas, and should be an easy and self-contained read for most readers familier with basic ordinary category theory. The remaining 400 pages are mostly rather detailed and concrete computations, which explore the combinatorial properties of the theory.


Unpublished and expository writing

  • A datastructure for higher sesqui-categories (Sep 2016). Draft (pdf)

The precursor combinatorics to my thesis, modelling higher “sesqui-categories” and not full higher categories.

  • Basic concepts in homotopy theory (Oct 2015). Expository notes (pdf)

Expository notes, attempting to give a more modern treatment of basic Algebraic Topology, and then discuss its abstraction in model categories and quasicategories.

  • Enriched Category Theory (May 2014). Part III essay (pdf)

Very basic enriched category theory, essentially adding details and computations to Kelly’s book.


Talks

  • Talk series on framed combinatorial topology (Apr - June 2021) @ Oxford

  • Transversal Stratifications (Mar 2020), Higher Categories and Categorification @ MSRI

  • -Excision theorem for factorization homology (Nov 2019), Advanced Topology Seminar, Oxford

  • Higher stratified Morse Theory (Oct 2019), Topology Seminar, Oxford

  • On how to continue the sequence (categories, functors, natural transformations, modifications …) (Mar 2019). K-theory seminar, CUNY, New York

  • Higher categories from higher-dimensional manifolds (Feb 2019). Category theory seminar, Johns Hopkins Univesity, Baltimore

  • Manifolds and higher categories (Nov 2018). McGill Category Theory and Logic Seminar, Montreal

  • Combinatorial Cobordism (Oct 2018). Category Theory Octoberfest 2018, New York

  • Associative n-categories (Sep 2018). New York Category Theory Seminar, CUNY, New York

  • Higher-dimensional Programming (May 2018). Applied Category Theory Seminar, MIT, Cambridge MA

  • Associative n-categories (Apr 2018). 103rd Peripatetic Seminar on Sheaves and Logic, Brno

  • TQFTs and string diagrams (Oct 2017). Junior Algebra and Representation Theory Seminar, Oxford


email : dorn [at] maths.ox.ac.uk