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Stream: community: general

Topic: Toposes as categorifications

John Baez (Apr 05 2020 at 23:05):

A sub-rant on how topoi categorify Heyting algebras:

https://twitter.com/johncarlosbaez/status/1246921021367574528

@Francis16833887 @mathemensch @CreeepyJoe @_julesh_ Yes, a topos is a categorification of a Heyting algebra. A Heyting algebra is a cartesian closed *poset* with finite limits and colimits, while a topos is a specially nice cartesian closed *category* with finite limits and colimits. Emily Riehl: (1/n) https://twitter.com/johncarlosbaez/status/1246921021367574528/photo/1

- John Carlos Baez (@johncarlosbaez)

John Baez (Apr 05 2020 at 23:07):

I would like to know what the expected pattern is as we go beyond 2-topoi to n-topoi.

Todd Trimble (Apr 05 2020 at 23:26):

There is a point of view on Grothendieck toposes that views them as categorifications of locales. Namely, a poset $P$ is a frame if its Yoneda embedding or principal down-set embedding $P \to [P^{op}, 2]$ has a left exact (= meet-preserving) left adjoint, and the correct notion of frame morphisms is that they are poset maps that are left exact (they preserve meets) left adjoints (they preserve sups). A locale is the same thing as a frame, except the morphisms point in the opposite direction -- you could take them to be the right adjoints of frame maps.

Analogously, a locally small category $C$ is (more or less) a Grothendieck topos if its Yoneda embedding has a left exact left adjoint. The analogous appropriate morphisms are again left exact left adjoints if we are thinking "algebraically", but we take the right adjoints to those if we are thinking "geometrically", and we get the notion of geometric morphism this way.

(Of course, this is not what you were asking about; it was more about the Twitter comment.)

Matteo Capucci (he/him) (Apr 06 2020 at 07:44):

I'm very much into this lately:
AnelJoyal_Topo-logie.pdf

Matteo Capucci (he/him) (Apr 06 2020 at 07:45):

It does topos theory from the point of view of this analogy, plus a few more. A really fun one is the analogy with commutative algebra/algebraic geometry.

Joe Moeller (Apr 06 2020 at 17:13):

I was really into this paper in the summer, but then I got distracted and haven't returned yet.

John Baez (Apr 06 2020 at 17:46):

Yes, I was influenced by the Anel-Joyal paper - it helped me realized topos theory can be seen as categorified ring theory with a few extra twists.

Gershom (Apr 06 2020 at 20:26):

In the Grothendieck case (and specifically the case of a localic topos), we can go up from n=1 to higher n, via this: https://ncatlab.org/nlab/show/n-localic+%28infinity%2C1%29-topos

Gershom (Apr 06 2020 at 20:28):

there's a result from Joyal and Tierney that every Grothendieck topos can be presented as a localic _groupoid_ but I've never dug too deep into that fact. I do wonder if we can "n-ify" that as well?

John Baez (Apr 06 2020 at 20:31):

The nLab page is about $(\infty,1)$-topoi, which are in some ways a lot more like 1-topoi than $n$-topoi. A while back I asked a question here about what the definition of $n$-topos might be like. I don't think I've seen anything on this past $n = 2$.

John Baez (Apr 06 2020 at 20:31):

Still, I should learn more about $n$-localic $(\infty,1)$-topoi!

John Baez (Apr 06 2020 at 20:33):

I've glanced at that Joyal-Tierney result on how every Grothendieck topos is equivalent to a topos of sheaves on a localic groupoid. I don't understand it yet. :frown: