Simon Henry

University of Ottawa

Research Interests

I am assistant professor at the university of Ottawa (Ontario) since 2019. My work revolve around several aspect of category theory including:

• Higher category theory
• Categorical homotopy theory (Model categories, Brown categories and similar notions)
• Categorical logic, (homotopy) type theory
• Topos theory (and its relation to non-commutative geometry)
• Point free topology, constructive mathematics
• General category theory

If you would be interested in doing a PhD or a MSc project at the university of Ottawa under my supervision, you can email me.

Here is a very brief presentation of my research areas. A list of publications can be found below. More details can be also be found in one of these research statements:

• 7 pages in english from January 2018 (link).
• 18 pages in french from January 2018 (link).

which also includes (not totally up to date) informations about my current work and projects.

I've started my research with a phd thesis under the supervision of Alain Connes on the relation between topos theory and non-commutative geometry between 2010 and 2014.

Roughly, topos theory and C*-algebra theory are two generalisations of ordinary topology. My goal was to understand if and how they are related. I studied more precisely continuous fields of Hilbert spaces over toposes, i.e. Hilbert spaces defined in the internal logique of a topos. The algebra of endomorphisms of such a field of Hilbert spaces is a C* algebra closely related to the geometry of the topos. The study of the properties of such continuous fields allow to understand several relations between topos theory and operator algebras.
The introduction of my thesis (linked above) contains a more detailled explanation of theses ideas.

In addition of topos theory and C*-algebra theory, this first project lead me to constructive mathematics (in order to handle Hilbert spaces of the internal logic) and Pointfree topology arised naturally in my works as an extremely powerfull to do constructive analysis.

Since 2015, I have been moving to higher category theory, and more precisely on the construction and comparison of models for higher structures. I was initally interested in higher category theory because of the developement of higher topos theory and homotopy type theory. But, because of the importance of constructive mathematics to topos theory and category theory, I was worried by the non-constructivity of the methods of higher category theory. I hence started to work on the developement on new models and methods for higher category theory and higher structures. And this is at the moment my main area of research. I'm especially interested in the Simpson conjecture, of which I proved a special case , as well as more generally the question of obtaining general strictification theorems. I have also worked on Grothendieck homotopy hypothesis on which I obtained partial results.

I am also active on mathoverflow.

Publications

• Towards a constructive simplicial model of Univalent Foundations (with N. Gambino) - Arxiv
• A constructive account of the Kan-Quillen model structure and of Kan's Ex^∞ functor - Arxiv
• On the homotopy hypothesis in dimension 3 (with E. Lanari) - Arxiv
• An abstract elementary class non-axiomatizable in L_(∞,κ) - Journal of Symbolic Logic - Arxiv
• Weak model categories in classical and constructive mathematics - Arxiv
• Regular polygraphs and the Simpson conjecture - Arxiv
• Non-unital polygraphs form a presheaf category - Higher structures - Arxiv
• The localic istropy group of a topos - Theory and Applications of Categories (TAC) - Arxiv
• The convolution algebra of an absolutely locally compact topos - Arxiv
• Algebraic models of homotopy types and the homotopy hypothesis - Arxiv
• Complete C^*-categories and a topos theoretic Green-Julg theorem - Arxiv
• On toposes generated by cardinal finite objects - Mathematical Proceedings of the Cambridge Philosophical Society - Arxiv
• A Geometric Bohr topos - Arxiv
• Toward a non-commutative Gelfand duality: Boolean locally separated toposes and Monoidal monotone complete C^∗-categories - Arxiv
• Constructive Gelfand duality for non-unital commutative C^*-algebras - Arxiv
• Measure theory over boolean toposes - Mathematical Proceedings of the Cambridge Philosophical Society - Arxiv
• Localic Metric spaces and the localic Gelfand duality Advanced in Mathematics - Arxiv
• Symmetrization of monoïds as hypergroups - Arxiv

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University of Ottawa