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• CommentRowNumber1.
• CommentAuthorTodd_Trimble
• CommentTimeJan 5th 2012
• (edited Jan 5th 2012)

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.

• CommentRowNumber2.
• CommentAuthorMike Shulman
• CommentTimeJan 6th 2012

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.

• CommentRowNumber3.
• CommentAuthorTodd_Trimble
• CommentTimeJan 10th 2012
Thanks for that, Mike!
• CommentRowNumber4.
• CommentAuthorThomas Holder
• CommentTimeJul 17th 2016

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.

• CommentRowNumber5.
• CommentAuthorTodd_Trimble
• CommentTimeJul 17th 2016

These look like useful additions to me. Thanks!

• CommentRowNumber6.
• CommentAuthorspitters
• CommentTimeMar 20th 2017

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.

• CommentRowNumber7.
• CommentAuthorMike Shulman
• CommentTimeMar 20th 2017

I don’t remember seeing it written down anywhere.

• CommentRowNumber8.
• CommentAuthorspitters
• CommentTimeMar 20th 2017

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.

• CommentRowNumber9.
• CommentAuthorMike Shulman
• CommentTimeMar 20th 2017

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.

• CommentRowNumber10.
• CommentAuthorThomas Holder
• CommentTimeJul 12th 2019

Added a reference to

• Peter F. Faul, Graham R. Manuell, Artin Glueings of Frames as Semidirect Products , arXiv:1907.05104 (2019). (abstract)
• CommentRowNumber11.
• CommentAuthorJohn Baez
• CommentTimeApr 29th 2020

I said it’s enough for $\Phi$ to preserve pullbacks to get a topos via Artin gluing. Todd assured me this is the case.

• CommentRowNumber12.
• CommentAuthorThomas Holder
• CommentTimeDec 11th 2020