Robert's Mistakes

What's a sheaf?

Nov 10, 2013

A sheaf is an especially nice presheaf . Then what's a presheaf ? Read on and i'll tell you.

The guy in the picture is Jean Leray , the inventor of sheaves. Modern algebraic geometry has been discussed in terms of category theory ever since Alexander Grothendieck defined the abelian category and presheaf in the 1955's. It'd be nice to do a full intro to category theory here, but also a bit excessive so we skip that today. Look here for a little historical overview on sheaves:

Besides other celebrities i recognized Hassler Whitney who is given as the founder of the modern definition of cohomology , a related notion i'm not going to talk about today. I knew Whitney for his contributions to graph theory and the four color theorem , also no topics for today. Let's get to the meat.

First let's define some symbols. What types of objects are we going to use? X is a set , C a category and U is a subset of X. Lowercase letters shall refer to elements instead of whole sets. Let's define what a presheaf F on X looks like:

For each U we have F(U): U -> C. Down to earth we can say F(U) is a function , so F(U)(u) = c. You see: F is a function , mapping an U to another function F(U). F(U) then is a function mapping u's to c's. F is a higher function. And that deserves it's own name, F is called a functor .

An example, please? Say, we have a measure M: X -> C, mapping any point x in X to it's distance to the origin M(x) = c in C. So, on planet X every point x has a distance M(x) (to the north pole). It's conceivable that we don't want to talk about the whole planet X but only the country U. The previous paragraph asks us to consider having different distance functions for differing countries. But why?

Say, our planet X is not just a sphere but a more complicated manifold . When you read the word manifold it means that altough we can give a coordinate system starting at any point, we can't extend it to cover all of X. It may degrade when we go too far. That's bad. But can't we walk a bit, then start anew, walk a bit further, etc...?

And that is indeed a reasonable thing to do, if our coordinate systems sufficiently fit together. If they do, we have a sheaf . But i still haven't stated the conditions to get a presheaf . Again, some symbols first:

Let's consider some smaller coordinate systems, one for a state V, and one for W-town in V (W is a subset of V is a subset of U). Since we're systematically constructing coordinate systems we can call our method to get an F(V) from F(U) by the name res_V,U. Then res_W,V is the name for what we do to further restrict F(V) to F(W). Until here I have only given names to objects. Here's the condition:

If we apply res_W,V after res_V,U then we get the same result as if we took the direct route res_W,U instead. See the triangle in the picture below. It's called a commutative diagram stating the law of associativity . It's the fourth item in the section titled presheaves here:

I forgot to mention one condition wikpedia's article states early: res_U,U has to be the identity , that is, we get the same data everytime we return to our starting point. That, together with the triangle, is enough to make res a category ! And that is cool because it means we have all we need to keep track of our attached objects along.

To summarize, a presheaf is an axiomatic construction stating what conditions must be met so we can nicely keep track of some data c associated every point in X. Nice means that moving in a circle gets us the same data, and taking a detour also shouldn't change anything.

It somehow generalizes the concept of a neighbourhood. A sheaf will further say something about covering U with a bunch of subsets U_i, and the consistency conditions one wants to hold there. Presehaves are easy to get, a sheaf is what one really likes to have. But that's for next time...

As i said, modern algebraic geometry builds on category theory , so there's a category version here:

The nLab has more on that view here:

Some more from the pedia:

The diagram below is based on one from here: