Topos Theory (Part 3)

Azimuth 2020-01-13

Last time I described two viewpoints on sheaves. In the first, a sheaf on a topological space $X$ is a special sort of presheaf

$F \colon \mathcal{O}(X)^{\mathrm{op}} \to \mathsf{Set}$

Namely, it’s one obeying the ‘sheaf condition’.

I explained this condition in Part 1, but here’s a slicker way to say it. Suppose $U \subseteq X$ is an open set covered by a collection of open sets $U_i \subseteq U.$ Then we get this diagram:

$\displaystyle{ FU \rightarrow \prod_i FU_i \rightrightarrows \prod_{i,j} F(U_i \cap U_j) }$

The first arrow comes from restricting elements of $FU$ to the smaller sets $U_i.$ The other two arrows come from this: we can either restrict from $FU_i$ to $F(U_i \cap U_j),$ or restrict from $FU_j$ to $F(U_i \cap U_j).$

The sheaf condition says that this diagram is an equalizer! This is just another way of saying that a family of $s_i \in FU_i$ are the restrictions of a unique $s \in FU$ iff their restrictions to the overlaps $U_i \cap U_j$ are equal.

In the second viewpoint, a sheaf is a bundle over $X$

$p \colon Y \to X$

with the special property of being ‘etale’. Remember, this means that every point in $Y$ has an open neighborhood that’s mapped homeomorphically onto an open neighborhood in $X.$

Last time I showed you how to change viewpoints. We got a functor that turns presheaves into bundles

$\Lambda \colon \widehat{\mathcal{O}(X)} \to \mathsf{Top}/X$

and a functor that turns bundles into presheaves:

$\Gamma \colon \mathsf{Top}/X \to \widehat{\mathcal{O}(X)}$

Moreover, I claimed $\Lambda$ actually turns presheaves into etale spaces, and $\Gamma$ actually turns bundles into sheaves. And I claimed that these functors restrict to an equivalence between the category of sheaves and the category of etale spaces:

$\mathsf{Sh}(X) \simeq \mathsf{Etale}(X)$

What can we do with these ideas? Right away we can do three things:

• We can describe ‘sheafification’: the process of improving a presheaf to get a sheaf.

• We can can show $\Lambda$ is the left adjoint of $\Gamma.$

• We can see how to push forward and pull back sheaves along a continuous map between spaces.

I’ll do the first two now and the third next time. I’m finding it pleasant to break up these notes into small bite-sized pieces, shorter than my actual lectures.

Sheafification

To turn a presheaf into a sheaf, we just hit it with $\Lambda$ and then with $\Gamma.$ In other words, we turn our presheaf into a bundle and then turn it back into a presheaf. It turns out the result is a sheaf!

Why? The reason is this:

Theorem. If we apply the functor

$\Gamma \colon \mathsf{Top}/X \to \widehat{\mathcal{O}(X)}$

to any object, the result is a sheaf on $X.$

(The objects of $\mathsf{Top}/X$ are, of course, the bundles over $X.$ )

Proving this theorem was a puzzle last time; let me outline the solution. Remember that if we take a bundle

$p \colon Y \to X$

and hit it with $\Gamma,$ we get a presheaf called $\Gamma_p$ where $\Gamma_p U$ is the set of sections of $Y$ over $X,$ and we restrict sections in the usual way, by restricting functions. But you can check that if we have an open set $U$ covered by a bunch of open subsets $U_i,$ and a bunch of sections $s_i$ on the $U_i$ that agree on the overlaps $U_i \cap U_j,$ these sections piece together to define a unique section on all of $U$ that restricts to each of the $s_i.$ So, $\Gamma_p$ is a sheaf!

It follows that $\Gamma \Lambda$ sends presheaves to sheaves. Since sheaves are a full subcategory of presheaves, any $\Gamma \Lambda$ automatically sends any morphism of presheaves to a morphism of sheaves, and we get the sheafification functor

$\Gamma \Lambda \colon \widehat{\mathcal{O}(X)} \to \mathsf{Sh}(X)$

To fully understand this, it’s good to actually take a presheaf and sheafify it. So take a presheaf:

$F \colon \mathcal{O}(X)^{\mathrm{op}} \to \mathsf{Set}$

When we hit this with $\Lambda,$ we get a bundle

$p \colon \Lambda F \to X$

Remember: any section of $F$ on any open neighborhood of $x$ gives a point over $x$ in $\Lambda F,$ all points over $x$ show up this way, and two such sections determine the same point iff they become equal when we restrict them to some sufficiently small open neighborhood of $x.$

When we hit this bundle with $\Gamma,$ we get a sheaf

$\Gamma \Lambda F \colon \mathcal{O}(X)^{\mathrm{op}} \to \mathsf{Set}$

where $(\Gamma \Lambda F)U$ is the set of sections of $p$ over $U.$ This is the sheafification of $F.$

So, if you think about it, you’ll see this: to define a section of the sheafification of $F$ over an open set $U,$ you can just take a bunch of sections of $F$ over open sets covering $U$ that agree when restricted to the overlaps.

Puzzle. Prove the above claim. Give a procedure for constructing a section of $\Gamma \Lambda F$ over $U$ given open sets $U_i \subseteq U$ covering $U$ and sections $s_i$ of $F$ over the $U_i$ that obey

$\displaystyle{ s_i|_{U_i \cap U_j} = s_j|_{U_i \cap U_j} }$

The adjunction between presheaves and bundles

Here’s one nice consequence of the last puzzle. We can always use the trivial cover of $U$ by $U$ itself! Thus, any section of $F$ over $U$ gives a section of $\Gamma \Lambda F$ over $U.$ This is the key to the following puzzle:

Puzzle. Show that for any presheaf $F$ there is morphism of presheaves

$\eta_F \colon F \to \Gamma \Lambda F$

Show that these morphisms are natural in $F,$ so they define a natural transformation $\eta \colon 1 \Rightarrow \Lambda \Gamma.$

Now, this is just the sort of thing we’d expect if $\Lambda$ were the left adjoint of $\Gamma.$ Remember, when you have a left adjoint $L \colon C \to D$ and a right adjoint $R \colon D \to C,$ you always have a ‘unit’

$\epsilon \colon 1 \Rightarrow R L$

and a ‘counit’

$\eta \colon L R \Rightarrow 1$

where the double arrows stand for natural transformations.

And indeed, in Part 2 I claimed that $\Lambda$ is the left adjoint of $\Gamma.$ But I didn’t prove it. What we’re doing now could be part of the proof: in fact Mac Lane and Moerdijk prove it this way in Theorem 2 of Section II.6.

Let’s see if we can construct the counit

$\eta \colon \Lambda \Gamma \Rightarrow 1$

For this I hand you a bundle

$p \colon Y \to X$

You form its sheaf of sections $\Gamma_p,$ and then you form the etale space $\Lambda \Gamma_p$ of that. Then you want to construct a morphism of bundles $\eta_p$ from your etale space $\Lambda \Gamma_p$ to my original bundle.

Mac Lane and Moerdijk call the construction ‘inevitable’. Here’s how it works. We get points in $\Lambda \Gamma_p$ over $x \in X$ from sections of $p \colon Y \to X$ over open sets containing $x.$ But you can just take one of these sections and evaluate it at $x$ and get a point in $Y.$

Puzzle. Show that this procedure gives a well-defined continuous map

$\eta_p \colon \Lambda \Gamma_p \to Y$

and that this is actually a morphism of bundles over $X.$ Show that these morphisms define a natural transformation $\eta \colon \Lambda \Gamma \Rightarrow 1.$

Now that we have the unit and counit, if you’re feeling ambitious you can show they obey the two equations required to get a pair of adjoint functors, thus solving the following puzzle:

Puzzle. Show that

$\Lambda \colon \widehat{\mathcal{O}(X)} \to \mathsf{Top}/X$

is left adjoint to

$\Gamma \colon \mathsf{Top}/X \to \widehat{\mathcal{O}(X)}$

If you’re not feeling so ambitious, just look at Mac Lane and Moerdijk’s proof of Theorem 2 in Section II.6!