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An example of a presheaf without an associated sheaf – UPDATED

[This was originally posted on Google+ on 19 June 2013. This has been lightly edited to fit the new format and for clarity. ADDED Dec 2025: it’s also wrong! I explain at the end]

I may have shared this blog posting before, but this is a really good example of when we have a large site which isn’t constrained by a small amount of data for each object: the category of (affine) schemes with the pretopology of flat surjections. One can define a presheaf on which has no sheafification for this pretopology, but its definition in the linked blog post explicitly uses von Neumann ordinals. I should like to write down a more structural version of this. I have some points I’d like to clear up, if you want to chip in, namely 2.-4. under ‘Some final comments’. The original source for this material is

William C. Waterhouse, Basically bounded functors and flat sheaves.
Pacific J. Math. Volume 57, Number 2 (1975), 597-610, http://projecteuclid.org/euclid.pjm/1102906018

The example as given by Waterhouse

Given an affine scheme , assign to it the set of locally constant functions from to the von Neumann cardinal of the set

the supremum of the cardinalities of residue fields at points of , such that the value at any point (which is a cardinal less than ) is smaller than the cardinality of the residue field at that point. This gives a functor , using the fact maps of fields are injective.

Simplifying the example

In fact, one can take the site to be merely the full subcategory of affine schemes which are spectra of fields, since one arrives at a contradiction assuming the existence of a sheafification by using a flat covering for and fields (in fact any field extension gives such a cover). Then locally constant functions are merely elements of , or in other words, can be taken as itself. In other words, restricts to the forgetful functor . This is a very natural presheaf to consider (see comment 3 below regarding the pretopology on this subcategory).

Calculation

Let denote the constant sheaf on corresponding to a well-ordered set of the same name. Then there is a map of presheaves which is just the inclusion for and the retract sending to the bottom element of otherwise. Then by the universal property of sheafification, there must be a unique map making the obvious triangle commute, where is the sheafification of , for any . In particular, factors through . Now for any given take so that is injective, which implies that is injective, and hence that is a mono.

Now we use the fact that for any map (necessarily a flat cover) we have that the equaliser of the two maps [here we’ve embedded into , see comments below]

injects into (We can check this by applying the natural transformation to the diagram

(using for a pair of parallel arrows) and remembering is an equaliser.) But is a point, so this equaliser is itself (can we see this directly without going via the spectrum?)

The upshot of the preceding two paragraphs is this: must have an injective set map into , but can be as large as we like, independent of (take say the function field over on the power set of , which is certainly a field larger that ). Thus cannot have a sheafification.

Some final comments

I find this a nicer example, as one then doesn’t have to mess around with descriptions of cardinals and specially constructed bounded functions. Then one should be able to show without too much effort that given a presheaf on , or even , which restricts to the full subcategory as the forgetful functor has no sheafification for the flat pretopology, because it would give one for .
It would be nice to say that the presheaf on above was a (nice) Kan extension of the presheaf , then one wouldn’t have to fiddle with existence of presheaves restricting to . This seems not unlikely, given the definition of using bounds on residue fields. Alternatively, we could perhaps extend point 1. to work for any Kan extension of (left/right as appropriate), rather than an extension up to isomorphism.
We find that all we are doing is taking the fact that the restriction to of the flat pretopology of is just the maximal pretopology (actually see (*) below), where all maps are covers, and there is no weakly initial set in . Such conditions would probably give a non-sheafifiable presheaf on any for a concrete category whose image in is unbounded.
(*) This is being a little slack, as isn’t a field, but can find a map from it to a field (as -algebras), and so we get a coverage on , rather than a Grothendieck pretopology (this shouldn’t change the calculation above). The inclusion functor is flat, so I think this means it is a morphism of sites as in Remark 2.3.7 of Sketches of an Elephant (certainly covers in are sent to covers in ). Any thoughts on this?
On a more foundational/structural note, I would like to be able to define the maps without choosing a well-ordering of every field, but I’m not sure I can do that, as one might not be able to get enough maps . Really one just needs, for any field , a sheaf and a map such that is injective. I don’t know how to supply this if I don’t have AC, but I haven’t thought very hard. Ideas? Perhaps we can prove this by contradiction.

….and why this is wrong.

Sadly, the above reasoning falls over right at the part where I claimed “(this shouldn’t change the calculation above)”. It’s true that one can get a singleton coverage on as described, where every inclusion of fields gives a cover in the opposite category (‘singleton’ in the sense that every covering family consists of a single arrow). The definition of (singleton) coverage doesn’t require pullbacks, merely ‘filler’ squares with the same ‘parallel arrow also is a cover’ condition. For the replacement of the “consider the pullback of along itself”, we can actually fill the square with the identity map of (in the opposite cat) on the remaining two sides. More generally, any maps can be used to fill in the square, using the fact every arrow is a cover. From now on I’ll drop all the ‘op’ business for simplicity.

So when we come to consider the sheaf condition for , the correct definition to use for the general coverage we have here is that for any filling square as suitable, and descent data for the presheaf, it glues uniquely. I have a small quibble in that the Elephant only defines a compatible family (i.e. descent data) for a general coverage as a subordinate clause in the definition of sheaf. But it amounts to this in the current context: descent data for is an element , such that when a span fills the square (i.e. ) then . The sheaf condition is then that given such descent data, we have that there is a unique such that . Since this condition quantifies over fillers of the square, we can consider the subclass where , and thus that . The only way that we can have for all elements of the Galois group is that comes from the (underlying set of the) base field .

Thus while Waterhouse’s example is not a sheaf, and it doesn’t have a sheafification, its restriction to the category of fields is already a sheaf.

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