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.


1 Answer 1


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 $A$.

  • $\begingroup$ Thank you for the reference. Do you know of anything for the more general case where you have a non-identity functor to $A$? $\endgroup$
    – Arnaud D.
    Apr 7, 2016 at 13:58
  • $\begingroup$ @ArnaudD. sorry, I just read your question again and saw that you were looking for this more general case. Nothing springs to mind (I only knew this reference because I've been working on translating it recently), but I'll definitely keep my eye out for something -- this is interesting to me as well. $\endgroup$
    – Tim
    Apr 7, 2016 at 15:18
  • $\begingroup$ I've had a skim through EGA and SGA hoping to find something, but can't see anything (thought admittedly I only spent 10 minutes or so looking). $\endgroup$
    – Tim
    Apr 7, 2016 at 15:51
  • $\begingroup$ For what it's worth, I don't really need to know that anymore. I'm still curious to know, but you surely don't have to get out of your way for this. $\endgroup$
    – Arnaud D.
    Apr 8, 2016 at 10:16

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