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Daniel Schaeppi’s new paper on the arxiv contains a more conceptual proof (see Appendix A) of the theorem that stackification with respect to the singleton covers in a superextensive site preserves sheaves for the extensive topology, hence produces stacks for the whole superextensive topology. Section 5 also contains a proof that at least in one case, and probably more generally, stackification of internal categories is a bicategorical localization at weak equivalences, and hence (by David Roberts’s analogous theorem for anafunctors) produces a bicategory equivalent to that of internal categories and anafunctors.
(The rest of the paper is about very concrete algebrogeometric objects that I don’t pretend to understand, but Daniel very helpfully pointed me to the parts that I would be interested in. (-: )
Hmm, I’d better get a wriggle on and write up some stuff. Thanks for pointing this out. I was vaguely interested in reading this, but felt no pressing need.
Vistoli’s notes here have some relevant stuff, in particular lemmas 4.25-4.27
That seems to be only about characterization of stacks in terms of the two generating subtopologies, which is pretty easy. The tricky bit is to show that stackification with respect to the singleton topology preserves stacks for the extensive topology.
In a slightly unrelated way, what do you think of this idea, which I’m sure is not hugely deep. Let $S$ be a superextensive site. Then $S^{op}$ has finite (or small) products. Isn’t a sheaf or a stack for the extensive topology just a product-preserving functor on $S^{op}$? This seems to me to be like some vague shadow of $S$ being dual to a generalisation of Lawvere theory, and extensive sheaves/stacks being algebras for that theory. Then sheaves for the full topology on $S$ are those algebras which additionally preserve coequalisers. There should probably be some sort of doctrinal interpretation of this.
Consider the site which is the coproduct completion of Urs’ favourite site CartSp. The opposite of this category looks very much like a theory.
Indeed, the opposite of a site is just a particular kind of limit sketch.
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