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I am slightly confused about sheafification at the moment.

I first learned sheaves defined as a subcategory of presheaves, then I was told that sheaves are also a localisation of presheaves, then I was told this was a common feature of localisations (i.e. they are often reflective), but then I was told that not every presheaf can be fpqc-sheafified.

So, what's the deal? What is the correct notion of an fpqc sheaf (localisation vs subcategory)? Or is the problem that the localisation doesn't exist (as can sometimes happen)? But then again, can't one always put a model structure on presheaves so that the homotopy category is sheaves?

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    $\begingroup$ The category of sheaves is a reflective subcategory (see e.g. ncatlab.org/nlab/show/reflective+subcategory) and as such is a localisation of the category of presheaves. One problem with the question as stated is that fpqc is not a noun. :-) Otherwise I could say more. $\endgroup$
    – David Roberts ♦
    Commented Oct 24, 2012 at 13:50
  • $\begingroup$ ok, so I don't understand how it's possible that some presheaves cannot be fpqc-sheafified: you can always take the image in the localisation, and then remember that localisation was a subcategory. $\endgroup$
    – Jacob Bell
    Commented Oct 24, 2012 at 13:53
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    $\begingroup$ I could be wrong, but I believe there are set-theoretic issues (of size, bigness and so forth). In the Stacks Project chapter Topologies on Schemes, the section on the fpqc topology, it is explained that given a ring $R\neq 0$, there is no set $A$ of fpqc coverings of $R$ such that every fpqc covering of $R$ is refined by one in $A$. Also, in Milne's book on etale cohomology, he mentions an example (I think he attributes it to Waterhouse but I don't have the book at hand to check) of a presheaf with no associated sheaf, but it's unclear to me what topology he's talking about. $\endgroup$
    – Keenan Kidwell
    Commented Oct 24, 2012 at 14:04
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    $\begingroup$ @Keenan - yes, it is due to Waterhouse, see my answer for the link. $\endgroup$
    – David Roberts ♦
    Commented Oct 24, 2012 at 14:05
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    $\begingroup$ @Martin - see lemma tag 000R (stacks.math.columbia.edu/tag/000R) which details closure properties of small categories of schemes. Tag 021A (stacks.math.columbia.edu/tag/021A) gives the definition of the etale topology, compare with the discussion around the fpqc topology (stacks.math.columbia.edu/tag/022A) where Johan says "For these reasons we do not introduce fpqc sites and we will not consider cohomology with respect to the fpqc-topology." $\endgroup$
    – David Roberts ♦
    Commented Oct 24, 2012 at 15:20

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$\newcommand\Set{\mathit{Set}}\newcommand\Aff{\mathit{Aff}}$An fpqc sheaf is exactly what you think it is: a functor from the opposite of the category of schemes (or relative schemes) to $\Set$ (i.e. a presheaf) such that the usual glueing conditions hold for fpqc covers.

You may be thinking of the fact remarked on here: http://ncatlab.org/nlab/show/fpqc+site that the collection of fpqc covers of a scheme doesn't have a cofinal set, and so one cannot just assume that the site at hand is small.

Even though sites that people work with can be large (say $\Aff$) ‘nice’ Grothendieck pretopologies are given by a set of covering families for each object, or at the very least have a cofinal set of covering families (this means there is a coverage given by a set of covering families, and this is enough to define sheaves, though generally weaker than a pretopology). Without this ‘local smallness’ condition (called WISC), the category of sheaves may not be locally small.

There is an example of a functor on schemes which admits no fpqc sheafification:

In the case of large sites without the condition WISC, the appropriate thing to consider is small sheaves, namely sheaves that are small colimits of representable sheaves:

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  • $\begingroup$ so, is the category of fpqc-sheaves not reflective then? $\endgroup$
    – Jacob Bell
    Commented Oct 24, 2012 at 14:09
  • $\begingroup$ and, if so, does the localisation exist? $\endgroup$
    – Jacob Bell
    Commented Oct 24, 2012 at 14:16
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    $\begingroup$ The category of fpqc sheaves may not be locally small (i.e. there can be a proper class of maps between two sheaves) and so usual category theoretic results need to be examined to make sure they work. And the category of presheaves on the category of schemes is definitely not locally small. The reason that the stacks project puts in so much effort in set theory at the beginning is so that we can work with small sites. But for the fpqc topology this is not possible. $\endgroup$
    – David Roberts ♦
    Commented Oct 24, 2012 at 14:17
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    $\begingroup$ According to some definitions, the 'category' of fpqc sheaves is not a category, but (following e.g. MacLane) a metacategory. It is most likely, but I haven't checked the proof, that the treatment of sheaves as reflective subcategories of categories of presheaves needs local smallness to work. $\endgroup$
    – David Roberts ♦
    Commented Oct 24, 2012 at 14:19

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