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Moved
Morally, this says that Type behaves like an object classifier.
to accompany the discussion of the name “univalence”,
and dropped the “morally” since “behaves like” seems to already be sufficiently hedged.
This is a belated follow-up to #59-60.
Numbers 59-60 of univalence, that is.
Colin, I haven’t tracked down who put the “morally” there – maybe he/she should be the one to respond – but I wouldn’t have removed it. The term has a particular currency with mathematicians; you might find interesting these remarks by Eugenia Cheng.
I’m not really opposed to restoring the use of “morally” there.
The main point of the edit was to group the mention of the object classifier with the discussion of the fibration of types. The sentence then required some rephrasing.
I’ll let others decide. But it sounds to me that putting “morally” there had a more positive value than hedging.
I didn’t put “morally” back in; it doesn’t seem to me necessary here because we do have precise statements about how univalence corresponds to object classifiers. However, I won’t object if anyone else wants to put it back.
I added
A proof of the full result has been announced by Simon Huber and Krzysztof Kapulkin (Huber 19).
Details may require adjustments to the succeeding paragraphs.
So all open issues of HoTT as a foundation are being sorted out these days:
Mike showed that all $\infty$-topos are models, Eric Finster gave a strategy for internalizing infinitary algebraic structure., Kapulkin-Sattler claim proof of homotopy canonicity.
But I heard in Paris the other week that somebody found a gap in Eric’s argument and apparently it wasn’t immediate how to fix it. But I don’t know any details.
Is there anyone we know that was actually in Oslo today at Sattler’s talk?
Carlo Angiuli reports from the talk (here). Or almost. At least on the title of the talk.
Re: #11, in Paolo Capriotti’s HoTTEST talk in April he mentioned that Eric’s definition (which was always just a plausible-sounding proposal, subject to confirmation or refutation) seems to be wrong (has some extra junk of some sort) starting in some dimension around $n=4$ (I don’t remember exactly). But he didn’t give any details, and in particular it’s unclear to me how serious the problem, whether it blows the whole thing out of the water or whether there might be a quick fix.
Thanks, Mike.
Now Darin Morrison reminds me that he still thinks he has a solution, we had briefly talked about it here, but that he has not received feedback from anyone yet. Maybe there is a language barrier, as he is somewhat from a different community, I gather. But might you be able to digest what exactly his proposal is?
Maybe in an alternate universe where I had several days available to stare at it… (-:O
Right, if all there is available is his Agda code, he should write some prose to go with it. I have suggested it to him, hope he will look into it.
Has he announced this anywhere, other than through you on the nForum? Or asked around in the community for feedback?
Maybe not. All that I am aware of is a Twitter message of his from months ago, where he essentially just pointed to his github repository (and now I don’t find that tweet anymore – not that it would matter much). But he says he is not in academia anymore and didn’t find time to make a prose writeup. Hm.
Unfortunately, it seems that Paolo’s flaw will turn out to be fatal for my approach, at least in vanilla Martin-Lof type theory. I had a number of possible “quick-fixes” in mind, but working through them in Stockholm with Peter and Guillaume, we could not seem to get any of them to go through. A rough way to describe the problem is that, while my definition generates an infinite number of coherences, these coherences are not “compatible” enough with the coherences of the universe (which is the terminal example of the structure i was trying to axiomatize). In order to make them compatible, we would have to finish the construction of the universe itself and hence we end up with the usual kind of circularity. So it looks bad … :(
Also, I submitted an issue on Darin’s github page asking for an example of how to use his definition to define the interchange law. Perhaps a simple example like this will serve to clarify his approach.
Hi Eric, regarding #20, this sounds interesting. In #14, it was mentioned that one might be able to see the issue around the level of 4-categories. Are you able to say a little bit about that concretely?
Could it not be that the gadgets you define are interesting in their own right?
At a quick glance, your approach seems reminiscent of Penon’s approach to higher categories. Did his work have any influence on yours?
Just to say that Darin has sent a reply
github.com/freebroccolo/agda-nr-cats/issues/4#issuecomment-503789479
to Eric’s request from #21 above.
Added pointer to univalent foundations for mathematics.
added publication details and links to:
After pointing out that the univalence axiom is due to Hofmann and Streicher, the References section here claims that, 7 years later:
The univalence axiom in its modern form was introduced and promoted by Vladimir Voevodsky around 2005. (?)
Can anyone replace that question mark by a pointer?
The next references offered is from 12 years later:
I have added a remark “see Section 4” to that, but even with such a remark it won’t be easy for outsiders to recognize the description of an axiom in that text.
Digging around on vv’s old webpage, I see that next, 16 years later, there is this, which I have added now:
But again, the uninitiated will have a hard time recognizing the advertized axiom in this text.
If prose is not the venue of choice here, maybe there is a time-stamped Coq-file which one could reference, where the “modern” axiom is first coded.
I have added pointer to
and correspondingly expanded the parenthesis below the pointer to Bousfield 06
After saying (here) that the univalence axiom is “almost” due to Hofman & Streicher, the entry continued to say, somewhat mysteriously, that:
the only difference is the lack of a coherent definition of equivalence.
I have now expanded this out as follows:
The only issue is that these authors refer to a subtly incorrect type of equivalences in homotopy type theory (see there for details).
It is this notion of equivalence in homotopy type theory which was fixed by Vladimir Voevodsky (…reference?), ever since the univalence axiom is widely attributed to him.
added pointer to:
P.S.: Though the section title is debatable: The univalence axiom was stated by Hofmann & Steicher in 1996 (§5.4 p. 23-24) under the name “universe extensionality” (arguably the more appropriate naming). Vladimir’s contribution was not the axiom as such, but to fix a subtlety in the definition of the type of type equivalences.
re #42:
finally found a reference by a type theorist who admits that the univalence axiom is due to Hofmann & Streicher:
(though still without concretely referencing their statement…)
added pointer to:
I don’t think it’s really correct to cite Hofmann-Streicher for “the univalence axiom”. Their universe extensionality axiom is stated only for a universe of sets, and would be incorrect if generalized naively beyond that domain. I would say they stated a version of the univalence axiom that applies to h-sets. Note that the version of the univalence axiom that applies to h-propositions is already true in set theory.
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