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?

reflectivesubcategory (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. – David Roberts Oct 24 '12 at 13:50