I have problem understanding the following sentence in Verdier's paper.

Ob($\mathcal{G}$) est un ensemble réduit à un élément choisi une fois pour toutes.

What is the meaning of this sentence? I think this means a set consisting of two elements so that it reduces to an element once we choose one.

Here is the context of the sentence. enter image description here


2 Answers 2


"A set of a single element". For emphasis, he writes that the element is "chosen once and for all". He basically means $\mathrm{Ob}(\mathcal G)=\{*\}$ for some (fixed) element $*$.


Essentially, what is going on here is we're looking at a category with a group-like structure.

However, the group-like structure does not lie in the objects of the category, but in its arrows.

The category itself will have a single object. We don't care about what that object is, for it doesn't matter; instead we look at the arrows, and on them have a group structure (wherein the operation is composition).

Each element of the original group is given a corresponding arrow to and from that object, and composition of arrows corresponds to group multiplication, and the resulting product also has an arrow in the associated category.


You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .