Le séminaire des doctorants se propose de fournir aux doctorants une occasion de s'ouvrir aux autres domaines des mathématiques que le leur. A chaque séance, un intervenant réalise un exposé sur un fait standard de leur domaine d'étude, de niveau adapté à l'ensemble des doctorants.

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In this talk, I will begin by following a note by Tom Leinster entitled "Rethinking Set Theory", which gradually leads to the notions of categories and elementary toposes.

I will then digress on the internal logic of a category and explain how a logical theory can be recovered from a category. In particular, the internal logic of an elementary topos is a higher-order constructive logic. The aim of this digression is simply to share my rough understanding of this link between logic and category.

The underlying goal of this talk is to highlight that the categorical approach provides a programming-friendly way to implement mathematical structures. This approach is used by the GAP package CAP to implement various categories, such as the category of finite sets. I will discuss both the implementation of the category of finite sets and the implementation of the category of finite G-sets, in which I am currently involved. We will see that the latter can be constructed as a "tower" of categorical constructors.

The PhD students seminar aims to provide PhD students with an opportunity to explore other areas of mathematics beyond their own. At each session, a speaker gives a presentation on a standard topic in their field of study, at a level suitable for all doctoral students.