0
$\begingroup$

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

$\endgroup$

2 Answers 2

7
$\begingroup$

"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 $*$.

$\endgroup$
1
$\begingroup$

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.

$\endgroup$

You must log in to answer this question.

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