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I have split morphism of sites off from site, and in the process made bold to change the definition to the one I think is inarguably correct, using covering-flatness rather than representable-flatness (which is equivalent in case the sites have finite limits and subcanonical topologies). I made corresponding changes to the statements of the theorems relating morphisms of sites to geometric morphisms, but I don’t have the time or energy to change the proofs, so I left them as-is with pointers to my paper for proofs in the general case. It would be nice to have proofs of these theorems in the general case that don’t require the reader to understand $\kappa$-ary exact completion for general $\kappa$, but I can’t think when I might have time to write out such proofs in the foreseeable future myself.
What is a morphism of (∞,1)-sites? nLab does not seem to have an article about them, nor could I find anything in Lurie’s HTT.
Presumably it should be a functor that preserves covering families and satisfies the appropriate analog of being a covering-flat functor. But what is it?
Good question. Of course, for $(\infty,1)$-sites with finite limits, the answer is obvious: the functor has to preserve finite limits. In general, one could turn Proposition 2.15 on the page morphism of sites into a definition. It’s unclear to me whether there will be a useful reformulation of this like there is in the 1-categorical case; $(\infty,1)$-sites without finite limits generally seem to be less well-behaved. For instance, the enveloping $(\infty,1)$-topos of a presheaf 1-topos need not be the corresponding presheaf $(\infty,1)$-topos if the (trivial) site in question lacks finite limits.
Where’s Proposition 2.15 on the page morphism of sites? Section 2 only has Definition 2.1 and Remark 2.2.
(∞,1)-sites without finite limits are quite important, e.g., the site of smooth manifolds or cartesian spaces.
Do we at least know that an ordinary morphism of sites without finite limits induces a geometric morphism of (∞,1)-toposes?
Oops, I meant the page covering-flat functor, sorry.
Offhand I’m not sure we even know that.
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