**Title :** Graded (co-)Differential Categories

**Abstract :** Co-differential categories are additive symmetric monoidal categories where you can differentiate certain kind of morphisms with the help of a combination of a monad, monoids and a deriving transformation.

Recently, J.-S. Pacaud Lemay proved that the category of finite-dimensional vector spaces cannot be endowed with a nontrivial structure of co-differential category. And this is very unlikely that it will work better with some familiar categories such as the category of finite sets and relations or the category of all Hilbert spaces.

In this talk, I will introduce a generalisation of co-differential categories: graded co-differential categories, which basically consist in putting the word “graded” everywhere in the axioms of a co-differential category. I will explain how the preceding categories can be endowed with the structure of a graded co-differential category as example of a general categorical construction which use symmetric powers.

This is joint work in progress with Jean-Simon Pacaud-Lemay.