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?



Fpqc sheafification and localisation
$\newcommand\Set{\mathit{Set}}\newcommand\Aff{\mathit{Aff}}$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](http://ncatlab.org/nlab/show/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](http://ncatlab.org/nlab/show/WISC#in_other_sites__external_version_15)), the category of sheaves may not be locally small.


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


* William C. Waterhouse *Basically bounded functors and flat sheaves*, Pacific Journal of Mathematics, Pacific J. Math. 57(2), 597-610, (1975), [Project Euclid](http://projecteuclid.org/euclid.pjm/1102906018), [DOI 10.2140/pjm.1975.57.597](http://dx.doi.org/10.2140/pjm.1975.57.597).


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:


* Mike Shulman, *Exact completions and small sheaves*, Theory and Applications of Categories, Vol. 27, 2012, No. 7, pp 97-173, <http://www.tac.mta.ca/tac/volumes/27/7/27-07abs.html>.


