# Mathieu Dupont

• Main page
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.
Last update : `2013-12-18`. Contact : (this picture is made of letters which appear in the film Zorns lemma by Hollis Frampton; this film should be called “Axiom of choice”, but this name itself is not suitable, because there is only a finite set of finite sets; disclaimer: the fact that I chose pictures from this film doesn't mean that I accept the axiom of choice).