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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”.
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.
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.
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”.
Sure, but to ignore the fact they are elementary toposes seems odd to me
Dmitri P made condensed set and pyknotic set this week.
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…
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.
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?
That’s probably true.
Presumably the bullet point you added concerns the ’$\infty$-logos’ of the preceding point.
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.
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