3
$\begingroup$

There is a (seemingly simple) statement in the literature on sheaf theory, namely,

If $E$ is the site of opens of a topological space $X$, the notion of sheaf over $X$ coincides with that of sheaf of $E$.

It is often said that Grothendieck topologies are not a generalization of topological structures on sets but of coverings (see coverage).

What tells you the statement on the nature of sites as spaces?

My intuition tells me that sites are not spaces, but what is minimally needed to define sheaves, they axiomatize not a (classical) topological structure but a process of "localizing" with families or "decomposing" with sieves, but the statement goes in other direction. I would be tempted to say that sheaves include spaces, but this only happens with sober spaces.

When/why is more convenient to have a definition over spaces than over sites (or the other way)?

Improve this question
$\endgroup$
5
  • 3
    $\begingroup$ (This is a bit tangentially related, but I think a good argument that Grothendieck topologies (though not pretopologies) are a generalisation of topological spaces is that a sheaf topos is a a left exact reflective subcategory of $\mathsf{PSh}(\mathcal{C})$, while a topology on a set $X$ is a left exact reflective subcategory of the powerset $\mathcal{P}(X)$ of $X$, viewed as a poset.) $\endgroup$
    – Emily
    Commented Dec 24, 2022 at 15:50
  • 2
    $\begingroup$ (Here the powerset $\mathcal{P}(X)$ of $X$ (=functions $X\to\{\mathrm{true},\mathrm{false}\}$) should be viewed as the set-theoretic analogue of categories of presheaves (=functors $\mathcal{C}^\mathsf{op}\to\mathsf{Set}$). See here for a discussion of this in more detail) $\endgroup$
    – Emily
    Commented Dec 24, 2022 at 15:52
  • 1
    $\begingroup$ @Emily Maybe I am confused, but are locales more precise analogues (in fact, 0-topoi)? $\endgroup$
    – Z. M
    Commented Dec 24, 2022 at 19:10
  • $\begingroup$ There are some subtle differences between locales and topological spaces (for instance, open morphisms might differ). And, yes, things coincide only for sober spaces. For stuff like Hausdorff spaces, for instance, no differences will be present. I also see sites as a minimal setting for defining sheaves. Topological spaces, on the other side, are more oriented towards the notion of convergence and are intended to generalise distance and proximity (at least, that's how I think). $\endgroup$
    – user40276
    Commented Dec 24, 2022 at 20:35
  • $\begingroup$ @Z.M I'm not really sure. A few months back when I was wondering about these things I noticed there was a certain notion that could be called a "2-locale" distinct from a Grothendieck topos, but in the end I couldn't find much about it written anywhere. I've now asked about it here. $\endgroup$
    – Emily
    Commented Dec 24, 2022 at 22:35

0

Reset to default

You must log in to answer this question.