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 naive, but might help you help me.

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.