# Slice and comma categories in French

I have to give a presentation in French on category theory, and most of the literature I use is in English. This is not too much of a problem since most of the terms translate fairly easily. However, I am not sure how to translate "comma category" or "(co)-slice category", as I have not found any mention of these in French. Does any French-speaking mathematician know the correct translation?

The case I am particularly interested in is the comma category $e\downarrow K$, where $e$ is an object in a category $\mathcal{E}$ and $K$ a functor $\mathcal{C}\to \mathcal{E}$ (and the dual version $K\downarrow e$ as well, though I guess the two must be quite similar). It seems to me that "catégorie des objets $K$-en-dessous de $e$" should be good, but I couldn't find any reference to this sort of things when the functor $K$ is not the identity functor.

In Au-dessous de $\mathrm{Spec}~\mathbb{Z}$ they use the exact phrasing that you suggest:
$\ldots$ la catégorie $A/\mathrm{Comm}(\mathcal{C})$, des objets de $\mathrm{Comm}(\mathcal{C})$ en dessous de $A$.
• Thank you for the reference. Do you know of anything for the more general case where you have a non-identity functor to $A$? Apr 7, 2016 at 13:58