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Leopold Schlicht (May 03 2021 at 12:40):Is there an agreed precise definition of the notion of an elementary (∞,1)-topos?
Morgan Rogers (he/him) (May 03 2021 at 12:53):@Mike Shulman may be able to answer that one!
Nathanael Arkor (May 03 2021 at 13:00):I believe the answer is no. However, Nima Rasekh gave a talk at CT 2019 about a proposed definition: you can see the slides here.
Fawzi Hreiki (May 03 2021 at 13:18):Perhaps this is a naive question, but is it possible to say what an (infinity, 1)-category is in first order terms (allowing quantification over the natural numbers)?
Morgan Rogers (he/him) (May 03 2021 at 13:27):Nathanael Arkor said:
I believe the answer is no. However, Nima Rasekh gave a talk at CT 2019 about a proposed definition: you can see the slides here.
In case you find yourself wondering: the reason this definition isn't definitive, despite ostensibly being a direct generalization of the notion of elementary topos, is that elementary (1-)toposes have various extra properties which are deducible from the definition, and some of these deductions fail in the higher category context. Without a lot of non-trivial examples of categories which "should" be elementary toposes (but which aren't, say, Grothendieck -toposes), it's hard to decide which properties should be manually added to the definition, and which others should be viewed as 1-categorical coincidences which can be allowed to fail at the -level.
Mike Shulman (May 03 2021 at 14:32):I proposed a definition in this blog post, and Rasekh (one, two) and Lo Monaco have since written papers about it.
Mike Shulman (May 03 2021 at 14:34):I'm still reasonably convinced by this definition, but the question of groupoid quotients remains open, and the treatment of universes is still a bit unsatisfying although I'm doubtful there will be anything better.
Leopold Schlicht (May 03 2021 at 15:25):Thank you all!