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I'm trying to read parts of McLarty's Grothendieck on Simplicity and Generality. In the article, I read Grothendieck thought of sheaves over some topological space as meter sticks measuring it.

What did Grothendieck mean? What property of a topological space do these meter sticks measure? Why was a meter stick the chosen metaphor and in what sense is it appropriate? Is the fact meter sticks only "measure one dimension" relevant in any way?

Just putting this out there: when I think of a sheaf over a topological space, I think of a big floating cloud containing information comprised of small clouds - one for each open set (look at connected stuff). Then the pasting axiom just says you can look at a proper bunch of smaller clouds as one big cloud. This is probably very naïve, but might help you help me.

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    $\begingroup$ I recommend to ignore such vague metaphors, which are no use whatsoever. Having a good understanding of examples of interesting sheaves (especially the diversity of interesting ones on a given space) and how their use clarifies classical problems and theorems (such the role of topology in complex analysis) is a far more instructive and illuminating thing to do. $\endgroup$
    – grghxy
    Commented Aug 9, 2015 at 22:02
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    $\begingroup$ @grghxy Why should one come at the expense of the other? Also, I don't think trying to understand vague metaphors by someone like Grothendieck is a waste of time. $\endgroup$
    – Arrow
    Commented Aug 9, 2015 at 22:47
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    $\begingroup$ I always think of line bundles on $\mathbb{P}^1$ as types of graph paper, i.e. $\mathcal{O}_{\mathbb{P}^1}(n)$ is the graph paper you would use if you wanted to graph a degree $n$ homogeneous polynomial in two variables. I imagine Grothendieck would think of this graph paper as a family of meter sticks indexed by $\mathbb{P}^1$, but since he does everything in families, he would drop the (implied) word "family" and just call it a meter stick. $\endgroup$
    – zeb
    Commented Aug 9, 2015 at 22:53
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    $\begingroup$ I speak from the experience of understanding EGA, SGA1, SGA2, SGA3, etc. like the back of my hand. One gets intuition not from vague metaphors (which in my experience are useless), but from examples which provide motivation and allow one to understand interesting older results in a new light. Serre often provided the "spark" leading to Grothenieck's grand ideas by deeply studying good examples. A single good example is more valuable than any number of vague metaphors. The fact that you are led to wonder about "one dimension" for a meter stick shows that the metaphor is leading you astray. $\endgroup$
    – grghxy
    Commented Aug 10, 2015 at 0:48
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    $\begingroup$ Looks like an attempt at an allusion to Einstein's relativity, where the observed length of a meter stick may help you think about what happens to distances in different reference frames. No idea whether such a metaphor could be useful here, though sure the one-dimensionality of the stick isn't the point. $\endgroup$
    – Noam D. Elkies
    Commented Aug 10, 2015 at 2:33

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Here are two (related) interpretation of this quote I can think of:

A first interpretation is just that Grothendieck was attach to the idea that you can study a 'space' (whatever this mean) by studing certain family of objects indexed by it, for example sheaves over a topological space, vector bundle over a manifold, Vector bundle or coherent sheave over a scheme or an algebraic stacks etc... From this perspective it might seem natural to think that each sheaves is just "something that will bring you a piece of information about the space" (for example through cohomology, taking globale section etc...) so a "meter stick" seem to be a correct name for this.

The second interpretation is based on another image which is very important for topos theory, and I think which makes this a lot clearer. The idea is that sheaves are just "generalized open sets".

I think if one replace sheaves by open sets in the quote you will understand what is the idea behind ? So I just need to explain in what sense sheaves generalise open sets.

There is two way to think about a sheaf over a topological space $X$:

  • as a set of locale section which yields this image you are talking about of "cloud of thing that you can patch together".

  • as an etale space over $X$, i.e. a topological space with a map $p:Y \rightarrow X$ which is a local homomorphism.

This second way is where we see the "generalised open sets" aspect appears: clearly, any $U \subset X$ defines a sheaf $U \rightarrow X$ and morphism of such sheaves are just inclusion of open sets. Moreover a general sheaf over $X$, as it is locally homeomorphic to $X$, is just a gluing of these sheaves corresponding to open set.

So sheaves are a bit like open set, but more flexible you can add sheaves (disjoint union), a sheaf can "self intersect" (for example $(0,2) \rightarrow \mathbb{R}/\mathbb{Z}$ gives you such an example), etc...

This analogy can be pushed very far, for exemple topos theory can be explain as completely replacing open set by sheaves in the definition of a topological space: instead of specifying a base of open sets to define a topology you will specify a base of sheaves to define a (Grothendieck) topos and this is exactly what a site is.

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