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created strongly infinity-connected site with the analogous definition, analogous proposition and analogous proof as at strongly connected site.
Only difference is that I define it as a cosifted $\infty$-connected site instead of as a cosifted locally $\infty$-connected site, because I am currently not quite sure about the definition of the latter because my model-category theoretic proofs rely on the existence of a terminal object, without that my standard Quillen adjunction model for the terminal $\infty$-geometric morphism fails and I’d need to think harder.
But all these entries of higher connected and local sites currently have the issue that they give sufficient conditions and don’t prove necessary conditions, so I think it’s okay, but we should keep in mind that there might be refinements of these definitions.
Quick general comment: based on what I remember from the Elephant, in general the “necessary” condition will probably be of the form “if a topos has X property, then there exists a site with Y property which defines that topos.” Since all sorts of wacky sites can define the same topos, it seems unlikely one could find a condition on a particular site which is equivalent to its topos of sheaves having some property. Maybe that is obvious, but I thought I would recall it.
Right, okay I’ll think about it more.
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