For anyone who has done a bit of work in modern geometry (primarily the notion of a (smooth) manifold), we want objects to be “locally” trivial, or easy to study. The global structure might be this crazy awesome geometric shape, but locally it’s going to look like boring old or something like that. How much it’s supposed to “look like” depends on what you want to study. For example, a smooth manifold is a set together with an atlas of “smooth” charts, such that for any point , there is an open neighborhood of that is diffeomorphic to an open subset of .
The idea is that although the global structure of some object might be hard to study, local behavior should be easy. Think of looking at say… a torus (doughnut). For any point on the torus, if you look close enough, it looks pretty much flat. Even though the global shape is decidedly not flat.
Think now of something like a smooth function on a smooth manifold , say . We don’t really have to define everywhere, we just have to know that behaves smoothly with respect to the atlas of . That is, for any point , there is a neighborhood , and chart , such that is a smooth, real-valued function.
Most people don’t go this deep down the rabbit hole, but there is a unifying principle behind extending local data to global data. This is given by the notion of a “sheaf.” Most of the time, people first encounter these things in an algebraic geometry or algebraic topology class, in the context of “cohomology with local coefficients” which are usually abelian groups or something similar.
First, presheaves (of abelian groups) on a topological space . A presheaf on constists of the data of:
- For every open set , an abelian group .
- For every inclusion of open sets , a “restriction” homomorphism .
- , the trivial group.
A sheaf is all this, subject to a nice “gluing” condition. That is:
- For every open set and open cover of , if is such that for all , then .
- For every open set and open cover of , if are sections such that for all we have , then there exists a section such that for all .
Note here that the former condition implies that the section in the latter condition is unique.
That was a bit of a mouthful. So complicated a definition. I don’t really like this way of defining it, but it’s okay.
Let’s start again. Let be a topological space, and make the category X whose objects are the open sets of and morphisms are those induced by the obvious poset structure. Then a presheaf of abelian groups is just a functor . Simple!
is a sheaf if, for every open set and cover ,
is an equalizer diagram.
So now we have sheaves. What are the maps? Well, the sheaves are just functors, so the obvious choice is that they’re natural transformations of functors. Hence, we have a category of sheaves! Denote this by if the sheaves have values in a category C.
Why should I care?
tends to retain a lot of the structure of the category C. The most encountered example is that is an abelian category whenever C is (I’ll revisit these neat abelian categories in a later post. They basically “behave like abelian groups” enough for us to do homological algebra.). The example I want to pursue is that is a topos whenever C is (I’ll DEFINITELY do a post on these things later).
They come up everywhere in geometry. Smooth function on a smooth manifold? Sheaf. Continous functions on a topological space? Sheaf. Measurable functions on a a measure space? Sheaf. Regular functions on a variety? Sheaf.
I’m still learning this stuff, and I’m continually amazed at how pervasive the idea is. Turns out that you can also define sheaves on a category by giving a the category a certain “topology” called a “Grothendieck topology.”
Wherever there is the study of local vs. global behavior, there is sheaf theory. Even in physics now, where one studies the structure of “quantum events” via covers of boolean reference frames, or where “locality and contextuality” is the cohomology of sheaves. So. Fucking. Cool.
Until next time.
9 thoughts on “What are Sheaves, and why should I care?”
I don’t think a presheaf is required to satisfy $F(\varnothing)=0$. A sheaf does satisfy that, but it follows from the sheaf axiom, so you don’t have to list that requirement explicitly.
I suppose you’re right. It does also follow from the presheaf axioms, since is the initial object in X, then is terminal in abelian groups, since is contravariant. The terminal object is the 0 group, so I guess it does come for free.
It only comes for free if you stipulate that $F$ is continuous (or cocontinuous, depending on how you define continuity for contravariant functors), which we generally do not do. In general a functor may take an initial object to one that is not. For example inclusion of pointed spaces into all spaces. Just consider a constant presheaf: it does not generally satisfy $F(\varnothing)=0$. The sheafification, however, always will.
The quickest description of the presheaf axioms are this: a presheaf on $X$ is a contravariant functor Open(X)->C. So the axioms of a presheaf are just the axioms of a functor. The axioms of a functor do not require it to be continuous, so neither do the presheaf axioms. Seen in this light, one might even be moved to suggest that you’ve missed two axioms: the composition law and the identity law. Of course those are tautological if your restriction morphisms are actual restrictions of the domains of functions. Is that the only case under consideration?
You’re right; that’s all contained when I make the second definition, but I forgot to include those axioms in the first definition (it’s been so long since I’ve thought of a presheaf as anything but a contravariant functor!). Also, Hartshorne DOES list as an axiom for presheaves, though this does not follow from the case where is just a contravariant functor. Continuity of would indeed yield , as is terminal in , but that’s a rather strong requirement.
Replying here because I don’t get the option to reply below.
Anyway, yup, your definition does match Hartshorne. I guess it’s a matter of convention. I was gonna complain that Hartshorne’s definition rules out the constant presheaf, but he points out that his definition is constant on connected open sets. Ok, whatever.
I still do like it better without that additional axiom. Then a presheaf is just a functor. A constant presheaf really is constant. Also you get to do the fun bit of vacuous logic category theory to show that a sheaf does satisfy the requirement: the empty set is covered by the empty cover, and the product over an empty category is the terminal object t, and by the sheaf axiom since t–>t equalizes the diagram, there is a unique arrow t–>F(0), hence F(0)=t.
But yeah, if it’s a matter of convention, then I withdraw my objection.
You gave as an example that measurable functions on a measure space $X$ comprise a sheaf. That’s new to me. What is the domain category for these sheaves? Is it still the topology of open sets in $X$? Seems like it might be more natural to make it the $\sigma$-algebra of measurable sets, but I’m not familiar with sheaf theory without topology.
Hmm you’re right. I gleamed that off of this thesis http://etd.library.pitt.edu/ETD/available/etd-04202006-065320/unrestricted/Matthew_Jackson_Thesis_2006.pdf (I”m currently sort of reading through this paper now with my analysis prof. It’s really interesting stuff.).
As I learn more measure theory, I’ll post about this. But that link is where I got that fact.
Thank you for that link. Looks very interesting.