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Alyssa Renata (Aug 12 2024 at 17:48):Hi, I hope this is the right stream, but I'm wondering if someone knows more about Heyting-valued groupoids, or can point me in the right direction.
There is a notion of H-valued sets whose category is equivalent to the topos of sheaves on H, where H is a complete heyting algebra. I'm wondering if there have been work on generalizing this idea to H-valued groupoids in some appropriate sense of the word. More generally, the construction of H-valued sets is generalized by the tripos-to-topos construction, and so I'm also wondering if there's any work on some sort of tripos-to-(2,1)-topos construction.
Perhaps related to this question: The (2, 1)-category of groupoids should be a (2, 1)-topos. So, it is reasonable to expect a correct notion of H-valued groupoid to also yield a (2, 1)-topos. But is there a more accessible account of (2,1)-toposes that does not require wading through infinity-toposes as in Lurie's Higher Topos Theory?
Fernando Yamauti (Aug 12 2024 at 19:21):What's even an H-valued groupoid? If it's just an internal groupoid in H-sets, you will get an internal groupoid in sheaves over H. But I don't think that describes all 2-sheaves over H...
A definition making the equivalence work would be nontrivial, I think. It seems worthwhile trying to work it backwards from sheaves to H-sets and, then, try copying the procedure starting with sheaves valued in groupoids in order to get the correct definition... but I don't even remember how was the equivalence and I'm too lazy to search for it.
Alyssa Renata (Aug 12 2024 at 20:38):I suppose my question is exactly that I'm not sure what an H-valued groupoid should be formally, and was just wondering if anyone has encountered something along these lines. Thank you for the ideas though, I will try out the sheaves valued in groupoids.
Morgan Rogers (he/him) (Aug 13 2024 at 09:07):This sounds like something @Joshua Wrigley could easily answer?
Joshua Wrigley (Aug 13 2024 at 10:36):Morgan Rogers (he/him) said:
This sounds like something Joshua Wrigley could easily answer?
I'm not sure I can easily answer this.
Fernando Yamauti said:
What's even an H-valued groupoid?
I would agree with Fernando that answering this question would clarify what you are searching for. You seem to want a 2-topos out at the end, which for me would suggest that when generalising the tripos-to-topos construction, you should be replacing partial equivalence relations with some kernel pair groupoids. I'm not aware of any work in this direction.
Morgan Rogers (he/him) (Aug 13 2024 at 12:42):Sorry Josh!
Alyssa Renata (Aug 13 2024 at 15:09):I think I am indeed quite iffy on what H-valued groupoid means, and I am working on clarifying what I mean of this notion. In any case, thanks for the perspective!