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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeOct 10th 2013
• (edited Oct 10th 2013)

I have turned logos from a redirect to Heyting category into a stand-alone disambiguation entry, to account for Joyal’s proposal from 2008 (maybe meanwhile abandoned?) to say “logos” for “quasi-category”.

• CommentRowNumber2.
• CommentAuthorDavidRoberts
• CommentTimeMay 3rd 2019

Added another item for the Joyal/Anel–Joyal notion, and linked to the new preprint by Anel–Joyal, and Joyal’s Topos à l’IHÉS videos.

• CommentRowNumber3.
• CommentAuthorDavid_Corfield
• CommentTimeMay 3rd 2019

Judging from the bibliography of the preprint, e.g.,

M.Anel,The geometry of ambiguity – an introduction to derived geometry. Chapter in this volume.

this will appear in New Spaces for Mathematics and Physics.

• CommentRowNumber4.
• CommentAuthorDavidRoberts
• CommentTimeMay 3rd 2019

Aha. There’s also to some extent Anel’s talk from Topos à l’IHÉS, treating toposes as analogues of commutative rings, but I think it might best fit on a more dedicated page.

What I find odd is when people think unbounded Set-toposes are somehow super weird and need a new name, just because they are not Grothendieck toposes. And then you get Barwick saying the category of condensed sets is not a topos, whereas the category of pyknotic sets is (relative to some larger universe), just because (I think!) the former is an unbounded elementary topos.

• CommentRowNumber5.
• CommentAuthorMike Shulman
• CommentTimeMay 3rd 2019

I don’t have any idea what condensed sets or pyknotic sets are, but there are substantial differences between Grothendieck toposes and unbounded elementary Set-toposes.

I think I’ve heard some people use the term “Giraud frame” for an object of $GrTop^{op}$. Of course one can also just say “locally presentable infinitary-pretopos”.

• CommentRowNumber6.
• CommentAuthorDavidRoberts
• CommentTimeMay 3rd 2019

Sure, but to ignore the fact they are elementary toposes seems odd to me

• CommentRowNumber7.
• CommentAuthorDavidRoberts
• CommentTimeMay 3rd 2019

Dmitri P made condensed set and pyknotic set this week.

• CommentRowNumber8.
• CommentAuthorMike Shulman
• CommentTimeMay 3rd 2019

Why then are condensed sets not a Grothendieck topos? According to that page, it’s defined as the category of sheaves on some site. Is the site not small? Sheaves on an arbitrary large site aren’t in general even an elementary topos…

• CommentRowNumber9.
• CommentAuthorDavidRoberts
• CommentTimeMay 4th 2019
• (edited May 4th 2019)

It’s sheaves on the large category of Stone spaces, but presented in a careful way in Scholze’s notes to be the limit of a diagram of Grothendieck toposes, much like in my 2015 IHES talk. Barwick cheats a bit and has three universes lying around.

• CommentRowNumber10.
• CommentAuthorMike Shulman
• CommentTimeMay 4th 2019

Sounds to me like the only difference between the two situations is a different choice of how to deal with size issues.

Do I guess rightly that the small-set-valued concrete pyknotic sets are the quasi-topological spaces?

• CommentRowNumber11.
• CommentAuthorDavidRoberts
• CommentTimeMay 4th 2019

That’s probably true.

• CommentRowNumber12.
• CommentAuthorDavid_Corfield
• CommentTimeAug 12th 2019

Added more to the Anel-Joyal sense of logos.

• CommentRowNumber13.
• CommentAuthorMike Shulman
• CommentTimeAug 23rd 2019

Added reference to Anel’s HoTTEST talk.

• CommentRowNumber14.
• CommentAuthorDavid_Corfield
• CommentTimeAug 23rd 2019
• (edited Aug 23rd 2019)

Presumably the bullet point you added concerns the ’$\infty$-logos’ of the preceding point.

• CommentRowNumber15.
• CommentAuthorMike Shulman
• CommentTimeAug 23rd 2019

Yes, he’s also talking about $\infty$-logoi (although his definition is different from that of the preceding point). I wasn’t sure how to say this while retaining the same grammar as the other bullets.