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Stream: learning: questions

Topic: Pos-enrichment

Mike Stay (Dec 09 2024 at 20:25):

Given a Pos-enriched category $C$, what can be said about the (Pos-enriched) functor category ${\rm Pos}^{C^{\rm op}}$? Does it form something topos-like?

Mike Shulman (Dec 09 2024 at 20:28):

Yes, it's a (1,2)-topos!

Mike Stay (Dec 09 2024 at 20:29):

Nice, thanks! Is there a standard reference for this case? nlab has the (2,2) and (∞,2) cases, but not the (1,2) case.

Mike Shulman (Dec 09 2024 at 20:34):

I don't think I know of a published reference. In my old musings about 2-categorical logic I tried to treat the cases of $(m,n)$-toposes for $0\le m,n\le 2$ even-handedly. For example, a (1,2)-topos has quotients of "(1,2)-congruences".