 Simon HENRY

uOttawa
STEM Building
Office 625Courriel: shenry2 at uottawa dot ca
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 noncommutative 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:
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 noncommutative 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 nonconstructivity 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 KanQuillen model structure and of Kan's Ex^{∞} functor  Arxiv
 On the homotopy hypothesis in dimension 3 (with E. Lanari)  Arxiv
 An abstract elementary class nonaxiomatizable in L_{(∞,κ)}  Journal of Symbolic Logic  Arxiv
 Weak model categories in classical and constructive mathematics  Arxiv
 Regular polygraphs and the Simpson conjecture  Arxiv
 Nonunital 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 GreenJulg theorem  Arxiv
 On toposes generated by cardinal finite objects  Mathematical Proceedings of the Cambridge Philosophical Society  Arxiv
 A Geometric Bohr topos  Arxiv
 Toward a noncommutative Gelfand duality: Boolean locally separated toposes and Monoidal monotone complete C^{∗}categories  Arxiv
 Constructive Gelfand duality for nonunital 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