# The Unapologetic Mathematician

## Presheaves

Strictly speaking, sheaves are not just about differential topology. And it’s also possible to get away without talking about them explicitly in differential topology. But it’s in differential topology that they really start to make their presence felt.

The best description of a sheaf I ever got doesn’t really translate to text, unfortunately. I forget the originator, as told to me by the professor explaining it — and if he remembers and can fill in the gap this can be recorded for posterity — but it really does capture the essence. A sheaf is a thing where the topology goes this way (moving one’s open hand around in circles at about chest level) and the algebra goes this way (moving one’s hand up and down across chest level). To be a bit more explicit, sheaves are about taking algebraic structures and localizing them to open sets in a topological space. I’ll start with the simplest versions, which are sheaves of sets. Further, I’ll actually start with something simpler: presheaves.

So, a presheaf $\mathcal{F}$ of sets on a topological space $X$ is a choice of a set $\mathcal{F}(U)$ for each open set $U\subseteq X$. We call the members of $\mathcal{F}(U)$ the elements — or the “sections” — of the presheaf “over $U$“. But there’s an important condition on this choice: if $V\subseteq U$ is a smaller open subset of $X$, then we should be able to “restrict” our element $f\in\mathcal{F}(U)$ to an element $f\vert^U_V\in\mathcal{F}(V)$. Thus, if we have an inclusion $V\subseteq U$, then we have a restriction map $\cdot\vert^U_V:\mathcal{F}(U)\to\mathcal{F}(V)$.

These restriction maps are subject to a couple conditions. First of all, if we restrict from $U$ to itself, then we shouldn’t change anything. That is, the restriction map $\cdot\vert^U_U$ is always the identity map. Secondly, if we restrict from $U$ to $V$, and then from $V$ to $W$, we should get the same result as if we just restricted directly from $U$ to $W$. That is, we have the equation $\cdot\vert^V_W\circ\cdot\vert^U_V=\cdot\vert^U_W$. Because of this, we’ll often just write the restriction map as $\cdot\vert_V$, since which subset it came from doesn’t really matter.

We can express this definition more succinctly if we remember that containment of open subsets of a topological space constitutes a partial order, and thus defines a category $\mathrm{Subset}(X)$. The objects are the open sets themselves, and there is a unique arrow from $V$ to $U$ if $V\subseteq U$. If we look closely, we’ll find that what we’ve defined as a presheaf is actually a contravariant functor from this category to the category of sets! For every arrow $V\mapsto U$ we have an arrow $\cdot\vert_U:\mathcal{F}(U)\to\mathcal{F}(V)$ — in the “opposite direction”, since the functor is contravariant. The conditions we impose on the restriction maps just say that they preserve identity arrows and compositions.

Now, there’s nothing inherently special about sets here. We can set up exactly the same construction with any target category to define, say, a presheaf of rings to be a contravariant functor $\mathcal{F}:\mathrm{Subset}(X)\to\mathbf{Ring}$. This assigns a ring $\mathcal{F}(U)$ to every open set $U$, and the restriction maps have to be ring homomorphisms. In the same way we get presheaves of groups, of abelian groups, or of vector spaces over a given field.

The one possibly confusing case is when we talk about a presheaf of modules over a presheaf of rings. In this case, say we have a presheaf $\mathcal{F}$ of rings on a topological space $X$. A presheaf of modules $\mathcal{M}$ over $\mathcal{F}$ assigns an abelian group $\mathcal{M}(U)$ to every open set $U\subseteq X$ — it’s a presheaf of abelian groups — in such a way that $\mathcal{M}(U)$ is a module over $\mathcal{F}(U)$. The restriction map has to work with the restriction map of $\mathcal{F}$, so we have $f(m)\vert_V=f\vert_V\left(m\vert_V\right)$.

The canonical example to keep in mind is continuous, real-valued functions on a topological space. This is a sheaf of real algebras that associates to the open set $U\subseteq X$ the algebra $\mathcal{F}(U)$ of real-valued functions that are defined and continuous there. Clearly we can restrict such a function $f$ to whatever open subset $V$ we want — and, in fact, we have. The nice thing is that this gives us a way of talking about and dealing with functions on our space that may not be defined or continuous everywhere. Just work within a suitable open set where the function does play nice! If you need to work with two functions defined over different open sets, just restrict them both to their common intersection and work there. Many structures we run into in differential geometry will be naturally expressible in terms of presheaves, just like this.

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Posted by John Armstrong | Topology

## 18 Comments	»

1. I’m having a bit of trouble figuring out what $f\in\mathcal F$ is in paragraph 3. A member of the set that is the value of the presheaf at $U$? Or something different?

Comment by Avery Andrews | March 17, 2011 | Reply

2. I mean $f\in\mathcal F(U)$.

Comment by Avery Andrews | March 17, 2011 | Reply

3. Exactly. If $\mathcal{F}(U)$ is a set, then $f\in\mathcal{F}(U)$ is an element of that set. I chose $f$ because of the example listed in the last paragraph. Maybe it would help to read that example and then go back through the definition to see how it fits.

Comment by | March 17, 2011 | Reply

4. [...] the moment we will be more concerned with presheaves, but we may as well go ahead and define sheaves. These embody the way that not only can we restrict [...]

Pingback by | March 17, 2011 | Reply

5. Got it. I wouldn’t have had a problem if “these are” was replaced by “the members of $\mathcal{F}(U)$ are”.

Comment by Avery D Andrews | March 17, 2011 | Reply

6. In the 4th paragraph, I do not understand why the equation is not written the other way around, with restriction from U to V written on the right of restriction from V to W

Comment by Francis | March 18, 2011 | Reply

• You’re right, it probably should be written the other way. Thanks.

Comment by | March 18, 2011 | Reply

7. John, I heard the “this way – that way” description of sheaves from Bill Massey. You might have been there at the time, or you might have gotten it from me later, since I certainly worked that description into my repertoire. I still use it whenever I talk about sheaves to a broad audience.

Comment by Greg Friedman | March 19, 2011 | Reply

8. By the way, in the penultimate paragraph, it might be less confusing to write \$(fm)|_V=f|_V m|_V\$, both because on the left hand side you’re restricting all of \$fm\$ and also because \$f(m)\$ looks like you’re evaluating some function at \$m\$.

Comment by Greg Friedman | March 19, 2011 | Reply

9. In the last paragraph, shouldn’t it be ‘presheaf’ not ‘sheaf’ (a sheaf seems to need an extra restriction).

Comment by Avery D Andrews | March 19, 2011 | Reply

10. Avery, it turns out that the continuous real-valued functions do form a sheaf. But since a sheaf is a presheaf, it’s an example of the latter as well.

Comment by | March 19, 2011 | Reply

11. [...] Between Presheaves As ever, we want our objects of study to be objects in some category, and presheaves (and sheaves) are no exception. But, luckily, this much is [...]

Pingback by | March 19, 2011 | Reply

12. Regarding the bit about doing topology horizontally and algebra vertically: somewhere I read a riff on this as follows. ‘Sheaf’ in ordinary English can mean something like a bundle of stalks, the stalks being aligned vertically. Or, it can mean a sheaf of papers, each leaf lying horizontally (like a section of a sheaf). The word play is supposed to be even better with the French equivalent faisceau.

I wish I could remember where I read this; I’m pretty sure it was in some categorical discussion about sheaves.

Comment by | March 20, 2011 | Reply

13. [...] Direct Image Functor So far our morphisms only let us compare presheaves and sheaves on a single topological space . In fact, we have a category of sheaves (of sets, by [...]

Pingback by | March 21, 2011 | Reply

14. [...] more construction we’ll be interested in is finding the “stalk” of a presheaf over a point . We want to talk about how a presheaf behaves at a single point, but a single point [...]

Pingback by | March 22, 2011 | Reply

15. [...] of Functions on Manifolds Now that we’ve talked a bunch about presheaves and sheaves in general, let’s talk about some particular sheaves of use in differential [...]

Pingback by | March 23, 2011 | Reply

16. [...] subspace we can restrict a vector field on to one on , which means we’re talking about a presheaf. In fact, it’s not hard to see that we can uniquely glue together vector fields which agree [...]

Pingback by | May 23, 2011 | Reply

17. Iterated integration of multi-integrals seems to constitute a presheaf where the topological space is \$\{ 1, …, n \}\$ and \$F(U)\$ is the set of integrable functions \$f(x_k | k \in U)\$ (e.g. \$f(x_2, x_3, x_5)\$ for \$U = \{ 2, 3, 5 \}\$). The restriction map \$U \rightarrow V\$ is integration over \$(x_k | k \in U \ V)\$.

Comment by | November 7, 2012 | Reply

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