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This page just had a couple of references, so I’ve added the idea and more references.
I came to this subject via Lurie’s MO question. Isn’t it a shame that such a highly regarded mathematician reaching out to the categorical logic community doesn’t receive an answer from them?
Generational difference. I don’t see Johnstone, Hyland or their peers on MO, and the categories mailing list, which they read, is stagnating, possibly people are dissuaded by well-known vocal doyens.
But there is a younger generation as I indicate there.
Given your G+ expressed interest in the pro-etale site, note how it crops up in this comment to me.
It seems reasonably likely to me that his question is not one that has been considered before, in which case the lack of an answer would be exactly the expected behavior.
Well I guess. On the other hand, there’s usually a comment or two to be seen even when nobody has the answer, and you might expect that logical scheme approach to be able to say something.
Anyway, I was just taking it as instance of lack of communication, rather than something to single out, even if, given that it involves JL, recently seen doubting that other piece of categorical logical known as HoTT, a missed opportunity.
I agree that it would be nice if the community had an answer for him.
I’ve expanded the page at conceptual completeness, and have remarked on the relation to Makkai duality (which I’ll update later) and the logical scheme approach of Awodey and Breiner. (Grey links to be fleshed out).
if $f : T_1 \to T_2$ is a pretopos morphism and $- \circ f : \operatorname{Hom}_{\mathbf{Pretop}}(-, T_2) \to \operatorname{Hom}_{\mathbf{Pretop}}(-,T_1)$ is an equivalence, then $f$ was also.
You mean
if $f : T_1 \to T_2$ is a pretopos morphism and $- \circ f : \operatorname{Hom}_{\mathbf{Pretop}}(T_2, Set) \to \operatorname{Hom}_{\mathbf{Pretop}}(T_1, Set)$ is an equivalence, then $f$ was also?
Or am I missing something?
And above
$\operatorname{Hom}_{\mathbf{Pretop}}(-,X) : \mathbf{Pretop} \to \mathbf{Cat}$ reflects equivalences.
is
$\operatorname{Hom}_{\mathbf{Pretop}}(-,Set) : \mathbf{Pretop} \to \mathbf{Cat}$ reflects equivalences?
Ah, that’s right.
Corrected! Thanks, David.
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