2
$\begingroup$

Let $\mathcal F$ be a sheaf (say on a topological space $X$) valued in some category $\mathcal C$.

What do we call $\mathcal F(X)$?

When $\mathcal C$ is some vaguely linear category (e.g. the category of abelian groups) then it is common to call $\mathcal F(X)$ the global sections of $\mathcal F$, or slightly more precisely, the group (or space) of global sections of $\mathcal F$. However when $\mathcal C$ is something more exotic, such as the category of categories, it becomes grammatically awkward to say something like:

The global sections $\mathcal F(X)$ is the object $Y\in\mathcal C$ constructed previously.

since $\mathcal F(X)$ is singular but "global sections" is plural.

In such situations, what to call $\mathcal F(X)$?

(Motivation: I once wrote "the global sections is" in a paper and the referee complained. Luckily, I was in a situation where it could be corrected to "the space of global sections is" and the sentence would still make sense. What would I have written if the sheaf in question were valued in some more arbitrary category $\mathcal C$?)

I hope to avoid writing something awkward (and potentially confusing) such as "global sections object".

Improve this question
$\endgroup$
4
  • 2
    $\begingroup$ "Object of global sections" sounds better to me than "global sections object". I don't see that either of these is potentially confusing, but if it is then one could presumably remove the confusion by giving, with the first use of the terminology, a detailed explanation of what it is intended to mean. $\endgroup$
    – Andreas Blass
    Commented Apr 1, 2017 at 2:20
  • 3
    $\begingroup$ I don't think $\mathcal C$ needs to be 'vaguely linear' to be called the [something] of global sections. If its objects are in some sense sets with extra structure then that's good enough. More generally if there is a noun N such that $\mathcal C$ is called the category of Ns, then I think you can call it the N of global sections -- for example, the category of global sections, the scheme of global sections, ... And of course by the same token 'object of global sections' or '$\mathcal C$-object of global sections'. $\endgroup$
    – Tom Goodwillie
    Commented Apr 1, 2017 at 2:30
  • 3
    $\begingroup$ I would not object to 'global sections category'. $\endgroup$
    – Tom Goodwillie
    Commented Apr 1, 2017 at 2:32
  • 4
    $\begingroup$ I would probably say the "category of global sections". $\endgroup$
    – Mike Shulman
    Commented Apr 1, 2017 at 9:00

0

Reset to default

You must log in to answer this question.