I am a researcher in mathematics. I defended my PhD thesis in category theory on 30 June 2008 at the Université catholique de Louvain, in Louvain-la-Neuve (Belgium).
Mathematical interests
My principal interests are in category theory, logic and the foundations of mathematics. In my PhD thesis, I defined abelian and 2-abelian groupoid-enriched categories, which are 2-dimensional versions of abelian categories. In them we can develop 2-dimensional homology theory. At the same time, I've been interested in the question of the foundations and the formalisation of category theory (and, more generally, of mathematics).
Mathematical texts
(All files are in .pdf format.)
Theses
- Systèmes de factorisation dans les 2-catégories (2001; “mémoire de licence”; supervisor : Enrico Vitale).
Diffent notions of proper factorisation system on a 2-category are defined and, for each of these notions, the free 2-category with a proper factorisation system on a given category is constructed.
- Catégories enrichies dans une base munie d'un système de factorisation (2002; “mémoire de D.E.A.”; supervisor : Enrico Vitale).
This work is based on the preprint “Factorizations in bicategories” by Renato Betti, Dietmar Schumacher and Ross Street. With respect to a collection of functors, they define notions of monomorphism, regular epimorphisme and regular factorisation in a 2-category, and of regular and exact 2-category. I do the same in an enriched context, defining these notions with respect to a factorisation system on the enrichment basis and adding appropriate dual notions (which require the introduction of the notion of coupled factorisation systems). There is an error in subsection 3.8.1: everywhere D(ℤ) must be replaced by the free symmetric cat-group on one generator. A cleaner version is in preparation.
- Abelian categories in dimension 2 (2008; PhD thesis; supervisor : Enrico Vitale) {here is the original French version, which is also available on the website of the UCL; the English version is also available on the arXiv}.
The goal of my thesis was to find a notion of 2-dimensional abelian category, for which symmetric 2-groups would play the role of abelian groups. I give two such notions, abelian and 2-abelian groupoid enriched categories. I show that we can develop in them 2-dimensional homology, including the existence of the homology long exact sequence from a short exact sequence of chain complexes. The examples are, in addition to symmetric 2-groups, the 2-modules on a 2-ring and, if we accept the axiom of choice, the Baez-Crans 2-vector spaces.
Paper
- Proper factorization systems in 2-categories (2003; published in the Journal of Pure and Applied Algebra, volume 179, pages 65-86).
This paper is derived from my “mémoire de licence” (see above). Abstract. Starting from known examples of factorization systems
in 2-categories, we discuss possible definitions of proper factorization
system in a 2-category. We focus our attention on the construction
of the free proper factorization system on a given 2-category.
Notes
- Categories without a set of objects, n-orders, constructivism (2008; this was intended to be a warning at the beginning of my PhD thesis, entitled “Warning concerning the framework of this work”, but I removed it at the last minute).
- The two meanings of the word “set” in mathematics (2009; this was the only entry on my blog). Currently unavailable; a new version called The three meanings of the word “set” in mathematics is in preparation.