Last time I defined sheaves on a topological space this time I’ll say how to get these sheaves from ‘bundles’ over You may or may not have heard of bundles of various kinds, like vector bundles or fiber bundles. If you have, be glad: the bundles I’m talking about now include these as special cases. If not, don’t worry: the bundles I’m talking about now are much simpler!

A **bundle** over is simply a topological space equipped with a continuous map to say

You should visualize as hovering above and as projecting points down to their shadows in This explains the word ‘over’, the term ‘projection’ for the map and many other things. It’s a powerful metaphor.

Bundles are not only a great source of examples of sheaves; in fact *every* sheaf comes from a bundle! Conversely, every sheaf—and even every presheaf—gives rise to a bundle.

But these constructions, which I’ll explain, do not give an equivalence of categories. That is, sheaves are not just another way of thinking about bundles, and neither are presheaves. Instead, we’ll get *adjoint functors* between the category of presheaves on and the category of bundles and these will restrict to give an equivalence between the category of ‘nice’ presheaves on —namely, the sheaves—and a certain category of ‘nice’ bundles over which are called ‘etale spaces’.

Thus, in the end we’ll get two complementary viewpoints on sheaves: the one I discussed last time, and another, where we think of them as specially nice bundles over In Sections 2.8 and 2.9 Mac Lane and Moerdijk use these complementary viewpoints to efficiently prove some of the big theorems about sheaves that I stated last time.

Before we get going, a word about a word: ‘etale’. This is really a French word, ‘étalé’, meaning ‘spread out’. We’ll see why Grothendieck chose this word. But now I mainly just want to apologize for leaving out the accents. I’m going to be typing a lot, it’s a pain to stick in those accents each time, and in English words with accents feel ‘fancy’.

### From bundles to presheaves

Any bundle over meaning any continuous map

gives a sheaf over Here’s how. Given an open set define a **section** of **over** to be a continuous function

such that

In terms of pictures (which I’m too lazy to draw here) maps each point of to a point in ‘sitting directly over it’. There’s a presheaf on that assigns to each open set the set of all sections of over

Of course, to make into a presheaf we need to say how to restrict sections over to sections over a smaller open set, but we do this in the usual way: by restricting a function to a subset of its domain.

**Puzzle.** Check that with this choice of restriction maps is a presheaf, and in fact a sheaf.

There’s actually a *category* of bundles over Given bundles

and

a morphism from the first to the second is a continuous map

making the obvious triangle commute:

I’m too lazy to draw this as a triangle, so if you don’t see it in your mind’s eye you’d better draw it. Draw and as two spaces hovering over and as mapping each point in over to a point in over the same point

We can compose morphisms between bundles over in an evident way: a morphism is a continuous map with some property, so we just compose those maps. We thus get a category of bundles over which is called

I’ve told you how a bundle over gives a presheaf on Similarly, a morphism of bundles over gives a morphism of presheaves on Because this works in a very easy way, it should be no surprise that this gives a functor, which we call

**Puzzle.** Suppose we have two bundles over say and and a morphism from the first to the second, say Suppose is a section of the first bundle over the open set Show that is a section of the second bundle over Use this to describe what the functor does on morphisms, and check functoriality.

### From presheaves to bundles

How do we go back from presheaves to bundles? Start with a presheaf

on To build a bundle over we’ll start by building a bunch of sets called one for each point Then we’ll take the union of these and put a topology on it, getting a space called There will be a map

sending all the points in to and this will be our bundle over

How do we build these sets Our presheaf

doesn’t give us sets for *points* of just for *open sets*. So, we should take some sort of ‘limit’ of the sets over smaller and smaller open neighborhoods of Remember, if our presheaf gives a restriction map

So, what we’ll actually do is take the *colimit* of all these sets as ranges over all neighborhoods of That gives us our set

It’s good to ponder what elements of are actually like. They’re called **germs** at which is a nice name, because you can only see them under a microscope! For example, suppose is the sheaf of continuous real-valued functions, so that consists of all continuous functions from to By the definition of colimit, for any open neighborhood of we have a map

So any continuous real-valued function defined on any open neighborhood of gives a ‘germ’ of a function on But also by the definition of colimit, any two such functions give the same germ iff they become equal when restricted to some open neighborhood of So the germ of a function is what’s left of that function as you zoom in closer and closer to the point

(If we were studying *analytic* functions on the real line, the germ at would remember exactly their Taylor series at that point. But smooth functions have more information in their germs, and continuous functions are weirder still. For more on germs, watch this video.)

Now that we have the space of germs for each point we define

There is then a unique function

sending everybody in to So we’ve almost gotten our bundle over We just need to put a topology on

We do this as follows. We’ll give a basis for the topology, by describing a bunch of open neighborhoods of each point in Remember, any point in is a germ. More specifically, any point in is in some set so it’s the germ of some where is an open neighborhood of But this has lots of other germs, too, namely its germs at all points We take this collection of all these germs to be an open neighborhood of A general open set in will then be an arbitrary union of sets like this.

**Puzzle.** Show that with this topology on the map is continuous.

Thus any presheaf on gives a bundle over

**Puzzle.** Describe how a morphism of presheaves on gives a morphism of bundles over and show that your construction defines a functor

### Etale spaces

So now we have functors that turn bundles into presheaves:

and presheaves into bundles:

But we have already seen that the presheaves coming from bundles are ‘better than average’: they are *sheaves!* Similarly, the bundles coming from presheaves are better than average. They are ‘etale spaces’.

What does this mean? Well, if you think back on how we took a presheaf and gave a topology a minute ago, you’ll see something very funny about that topology. Each point in has a neighborhood such that

restricted to that neighborhood is a *homeomorphism*. Indeed, remember that each point in is a germ of some

for some open We made the set of all germs of into an open set in Call that open set

**Puzzle.** Show that is a homeomorphism from to

In class I’ll draw a picture of what’s going on. is a space sitting over has lots of open sets that look exactly like open sets down in In terms of our visual metaphor, these open sets are ‘horizontal’, which is why we invoke the term ‘etale’:

**Definition.** A bundle is **etale** if each point has an open neighborhood such that restricted to is a homeomorphism from to an open subset of We often call such a bundle an **etale space** over

So, if you did the last puzzle, you’ve shown that any presheaf on gives an etale space over

(By the way, if you know about covering spaces, you should note that every covering space of is an etale space over but not conversely. In a covering space we demand that each point *down below*, in has a neighborhood such that is a disjoint union of open sets homeomorphic to with restricting to homeomorphism on each of these open sets. In an etale space we merely demand that each point *up above*, in has a neighborhood such that restricted to is a homeomorphism. This is a weaker condition. In general, etale spaces are rather weird if you’re used to spaces like manifolds: for example, will often not be Hausdorff.)

**Sheaves versus etale spaces**

Now things are nicely symmetrical! We have a functor that turns bundles into presheaves

but in fact it turns bundles into sheaves. We have a functor that turns presheaves into bundles

but in fact it turns presheaves into etale spaces.

Last time we defined to be the full subcategory of having sheaves as objects. Now let’s define to be the full subcategory of having etale spaces as objects. And here’s the punchline:

**Theorem.** The functor

is left adjoint to the functor

Moreover, if we restrict these functors to the subcategories and we get an equivalence of categories

The proof involves some work but also some very beautiful abstract nonsense: see Theorem 2, Corollary 3 and Lemma 4 of Section II.6. There’s a lot more to say, but this seems like a good place to stop.

There’s some \Lambda(F)_x and \Lambda(F) that are not rendering properly.

Thanks… I’ll keep cleaning things up.

Without the accents, you don’t have to remember that, while the maps are étale, the spaces are étalé! (At least this is so in the original French; a lot of English writers seems to say that the spaces are also étale.) However you write it, this affects the pronunciation too; an E with an accent is pronounced, but an E without an accent is silent. (This is not a general rule in French, but it works for these two words.) See https://ncatlab.org/nlab/show/%C3%A9tal%C3%A9+space#grammar for why.

My theory is that one shouldn’t have to learn anything about French to learn topos theory, so we English speakers shouldn’t feel ashamed to write “etale” and pronounce it however the hell we want.

However, these points are very interesting—thanks! I think it’s sort of wild that Grothendieck, or whoever, would choose two similar but apparently ‘distantly related’ words for these two connected concepts.

“so it’s the germ of some where is an open neighborhood of .” Shouldn’t that be “?

Hi! You’re right, I’ll fix that.

I keep wanting to call an element of a ‘section’ of the presheaf over the open set but I don’t think Mac Lane and Moerdijk use that term, and when playing with both sheaves and bundles it could be confusing.

On the other hand, they speak of a ‘cross-section’ of a bundle rather than a ‘section’, but I think this is a bit unusual—I’m saying ‘section’.

If Mac Lane and Moerdijk don’t have a term for an element of , then you can always call that a section and then follow the book by calling the other thing a cross-section. The term ‘cross-section’ is a more concretely geometric term, so it's appropriate in the more concretely geometric setting.

In the definition of an etale bundle, the condition “ restricted to is a homeomorphism” should be modified to read “ restricted to is a homeomorphism onto “. I also think it is helpful to point out that this sort of map is usually called a “local homemorphism” (as is done as part of the definition of an etale space in both the Mac Lane and Moerdijk textbook as well as the nlab).

That’s what I meant, of course. But you’re right, we are not only restricting the domain, we are ‘corestricting’ the codomain of It’s good to make that explicit, since we’re doing category theory here… so I will.

Thanks for clarifying. Note that your definition still differs from the one given in Mac Lane and Moerdjik, since they require to be open. Are you sure that extra condition is not required in order to obtain the sheaf etale bundle equivalence? For an example showing that your definition really is different than the one from the textbook, note that all inclusion maps are etale (per your definition here) whereas only the inclusion maps of open subsets are etale per the textbook definition.

I was just trying to copy the textbook definition, and failing.

You mentioned how the open sets of the etale space are “horizontal”, but not how the stalks are “vertical”. In fact, did you mention stalks at all? They’re the source of the term “sheaf”, in its original meaning.

I like to say that “stalks” and “sheaves” come from an agricultural analogy, while the open sets come from a mineralogical one (like mica schist).

No, I didn’t mention stalks… trying to keep the terminology down to a minimum. Stalks are related to ‘fibers’, both mathematically and in terms of the general agricultural metaphor. A ‘sheaf’ is a bit like a ‘fiber bundle’, agriculturally speaking.

Here’s an analytic example of a bundle where the projection is a local homeomorphism, but not a covering map. Let . Then maps onto (fun exercise), and is a locally 1-1 map, in fact a local homeomorphism.

(HInt for the exercise: little Picard).

What’s the difference between and ? I’m understanding both as the category of presheaves on the category of opens of X.

They’re the same thing; I experimented with both ways of writing it but I intend to use every time. I’ll see what needs fixing and fix it.

Thanks! Nice complement to the last chapter of Seven Sketches (arXiv:1803.05316).

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

[…] Remember, a

bundleover a topological space is a topological space equipped with a continuous mapWe say it’s an

etale spaceover if it has a special property: each point has an open neighborhood such that restricted to this neighborhood is a homeomorphism from this neighborhood to an open subset of In Part 2 we defined the category of bundles over which is called and the full subcategory of this whose objects are etale spaces, called […]You wrote to consider as a set of functions defined on

We defined as the

colimitof for all .Then is a set of functions defined on the

smallestopen subset (that is, and and not ).Then you consider the elements of

germsat .This is what I don’t get. Because the definition which I remember for a germ of function at is a

class of equivalenceof functions, for which there is a subset containing $x$ where they areequal.But the restricted functions in are not necessarily equal in that smallest .

So how come that they are called germs?

Kvantumo wrote:

Right.

That’s true if there exists a smallest open set containing But in most examples that’s not true! For example if is the real line, or the complex plane, there is no smallest open set containing a point.

So, you need to stop thinking about the case where there exists a smallest open set containing the point and start thinking about these other examples. Then you’ll see that a germ at defined as I defined it, is indeed an equivalence class of functions defined on open sets containing where two functions are equivalent if they become equal when restricted to a small enough open set. This is how colimits work in this example.

Thank you for pointing it out! I was totally blind to the possibility that a smallest open subset could not exist in some topology.

As John said, usually there is no smallest open set owning (that is, there is no smallest

neighbourhoodof ). But let's consider when there is. Then as you said, can simply be , a set of functions on . Butthisset is not meant to be a germ; it’s anindividualelement of that's meant to be a germ. And that works, because even though a germ is normally taken to be a set of functions (an equivalence class), in this case where is the smallest neighbourhood of , each equivalence class consists of a single function on . (Part of the reason is that in the definition of a germ, two functions are only equivalent if there is aneighbourhoodof on which they are equal.)Indeed, if every point in every topological space had a smallest neighbourhood, then there would be no need to define a germ as an equivalence class! It’s precisely because there usually is no smallest neighbourhood that we instead have to define a germ in a relatively complicated way. A germ at is what

wouldbe a function on the smallest neighbourhood of if there were such a thing. (In nonstandard analysis, you can talk aboutinfinitesimalneighbourhoods, and then a germ at can simply be defined as a standard function on an infinitesimal neighbourhood of .)