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Some stuff that Zoran wrote on recollement reminded me that I had been long meaning to write Artin gluing, which I’ve done, starting in a kind of pedestrian way (just with topological spaces). Somewhere in the section on the topos case I mention a result to be found in the Elephant which I couldn’t quite find; if you know where it is, please let me know.
Very nice, thanks!! I added the Elephant reference (A4.5.6) and modified the theorem in the topos case so it states the result for toposes that aren’t necessarily localic.
I have added some material to Artin gluing. Unfortunately, it mostly spells out things said there before explicitly or implicitly but I nevertheless thought this would improve comprehension and parsing for readers that are not interested in all the details. I am not quite sure it does achieve that effect though since there is already a wealth of material in place. An idea to consider seems to me to lift out a concrete description of $Gl(f)$ to an ’appetizer concise introduction to topos part’ section right before the topology section starts this would probably accomodate readers who are interested in the construction primarily as a tool.
These look like useful additions to me. Thanks!
Has gluing been defined for higher toposes? I could not find this in Lurie’s book (nor in the nlab), but I may well be overlooking something.
I don’t remember seeing it written down anywhere.
Since gluing is constructed by the topos+of+coalgebras+over+a+comonad, I guess the generalization to higher toposes stated there would give a variant of gluing.
Ah, yes! (In case anyone else has trouble finding it on the page, the mention of ∞-toposes just links to here. And we might hope to modify that argument to work in the elementary case too, by removing accessibility but checking other conditions as well. We should have pages about these.
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