An example of a presheaf without an associated sheaf

[This was originally posted on Google+ on 19 June 2013. This has been lightly edited to fit the new format and for clarity]

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 $\mathbf{Aff}$ of (affine) schemes with the pretopology of flat surjections. One can define a presheaf on $\mathbf{Aff}$ 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 $\mathrm{Spec}(A)$ , assign to it the set of locally constant functions from $\mathrm{Spec}(A)$ to the von Neumann cardinal of the set

$L_A := \sup{ |k(p)| \mid p \in \mathrm{Spec}(A) },$

the supremum of the cardinalities of residue fields at points of $\mathrm{Spec}(A)$ , such that the value at any point (which is a cardinal less than $L_A$ ) is smaller than the cardinality of the residue field at that point. This gives a functor $F \colon \mathbf{Aff}^{op} \to \mathbf{Set}$ , using the fact maps of fields are injective.

Simplifying the example

In fact, one can take the site to be merely the full subcategory $\mathbf{Field}^{op}$ 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 $\mathrm{Spec}(k’) \to \mathrm{Spec}(k)$ for $k’$ and $k$ fields (in fact any field extension $k \to k’$ gives such a cover). Then locally constant functions $\mathrm{Spec}(k) \to L_k$ are merely elements of $L_k$ , or in other words, $F(\mathrm{Spec}(k))$ can be taken as $k$ itself. In other words, $F$ restricts to the forgetful functor $U \colon (\mathbf{Field}{op}){op} = \mathbf{Field} \to \mathbf{Set}$ . This is a very natural presheaf to consider (see comment 3 below regarding the pretopology on this subcategory).

Calculation

Let $X$ denote the constant sheaf on $\mathbf{Field}^{op}$ corresponding to a well-ordered set of the same name. Then there is a map of presheaves $i_X \colon U \to X$ which is just the inclusion for $|k| \leq X$ and the retract $k \to X$ sending $|k| - X$ to the bottom element of $X$ otherwise. Then by the universal property of sheafification, there must be a unique map $X \to U^+$ making the obvious triangle commute, where $U^+$ is the sheafification of $U$ , for any $X$ . In particular, $U \to U^+$ factors through $X$ . Now for any given $k$ take $X > |k|$ so that $U(k) \to X$ is injective, which implies that $U(k) \to U^+(k)$ is injective, and hence that $U \to U^+$ is a mono.

Now we use the fact that for any map $\mathrm{Spec}(k’) \to \mathrm{Spec}(k)$ (necessarily a flat cover) we have that the equaliser of the two maps [here we’ve embedded into $\mathbf{Ring}$ , see comments below]

$k’ \rightrightarrows k’ \otimes_k k’$

injects into $U^+(k)$ (We can check this by applying the natural transformation $U \to U^+$ to the diagram

$k \to k’ \rightrightarrows k’ \otimes_k k’$

and remembering $U^+(k) \to U^+(k’)$ is an equaliser.) But $\mathrm{Spec}(k’)$ is a point, so this equaliser is $k’$ itself (can we see this directly without going via the spectrum?)

The upshot of the preceding two paragraphs is this: $k’$ must have an injective set map into $U^+(k)$ , but $k’$ can be as large as we like, independent of $U^+$ (take say the function field over $k$ on the power set of $U^+(k)$ , which is certainly a field larger that $k$ ). Thus $U$ 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 $\mathbf{Aff}$ , or even $\mathbf{Sch}$ , which restricts to the full subcategory $\mathbf{Field}^{op}$ as the forgetful functor $U$ has no sheafification for the flat pretopology, because it would give one for $U$ .
It would be nice to say that the presheaf $F$ on $\mathbf{Aff}$ above was a (nice) Kan extension of the presheaf $U$ , then one wouldn’t have to fiddle with existence of presheaves restricting to $U$ . This seems not unlikely, given the definition of $F$ using bounds on residue fields. Alternatively, we could perhaps extend point 1. to work for any Kan extension of $U$ (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 $\mathbf{Field}^{op}$ of the flat pretopology of $\mathbf{Aff}$ is just the maximal pretopology (actually see (*) below), where all maps are covers, and there is no weakly initial set in $\mathbf{Field}^{op}$ . Such conditions would probably give a non-sheafifiable presheaf on any $C^{op}$ for $C$ a concrete category whose image in $\mathbf{Set}$ is unbounded. (*) This is being a little slack, as $k’ \otimes_k k’$ isn’t a field, but can find a map from it to a field (as $k$ -algebras), and so we get a coverage on $\mathbf{Field}^{op}$ , rather than a Grothendieck pretopology (this shouldn’t change the calculation above). The inclusion functor $\mathbf{Field}^{op} \to \mathbf{Aff}$ 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 $\mathbf{Field}^{op}$ are sent to covers in $\mathbf{Aff}$ ). Any thoughts on this?
On a more foundational/structural note, I would like to be able to define the maps $i_X \colon U \to X$ 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 $i_X$ . Really one just needs, for any field $k$ , a sheaf $Z_k$ and a map $U \to Z_k$ such that $U(k) \to Z_k(k)$ 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.

The example as given by Waterhouse

Simplifying the example

Calculation

Some final comments

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