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• CommentRowNumber1.
• CommentAuthorspitters
• CommentTimeApr 24th 2018

Updating reference to cubical type theory. This page need more work.

1. 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.

• CommentRowNumber3.
• CommentAuthorColin Zwanziger
• CommentTimeSep 16th 2018
• (edited Sep 16th 2018)

Numbers 59-60 of univalence, that is.

• CommentRowNumber4.
• CommentAuthorTodd_Trimble
• CommentTimeSep 16th 2018

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.

• CommentRowNumber5.
• CommentAuthorColin Zwanziger
• CommentTimeSep 16th 2018
• (edited Sep 16th 2018)

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.

• CommentRowNumber6.
• CommentAuthorTodd_Trimble
• CommentTimeSep 17th 2018

I’ll let others decide. But it sounds to me that putting “morally” there had a more positive value than hedging.

• CommentRowNumber7.
• CommentAuthorMike Shulman
• CommentTimeSep 18th 2018

Clarify the relationship between univalence and object classifiers.

• CommentRowNumber8.
• CommentAuthorMike Shulman
• CommentTimeSep 18th 2018

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.

• CommentRowNumber9.
• CommentAuthorDavid_Corfield
• CommentTimeJun 12th 2019

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.

• CommentRowNumber10.
• CommentAuthorDavid_Corfield
• CommentTimeJun 12th 2019

Wrong person, Sattler not Huber.

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeJun 12th 2019

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.

• CommentRowNumber12.
• CommentAuthorDavidRoberts
• CommentTimeJun 12th 2019

Is there anyone we know that was actually in Oslo today at Sattler’s talk?

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeJun 12th 2019

Carlo Angiuli reports from the talk (here). Or almost. At least on the title of the talk.

• CommentRowNumber14.
• CommentAuthorMike Shulman
• CommentTimeJun 12th 2019

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.

• CommentRowNumber15.
• CommentAuthorUrs
• CommentTimeJun 13th 2019

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?

• CommentRowNumber16.
• CommentAuthorMike Shulman
• CommentTimeJun 13th 2019

Maybe in an alternate universe where I had several days available to stare at it… (-:O

• CommentRowNumber17.
• CommentAuthorUrs
• CommentTimeJun 13th 2019

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.

• CommentRowNumber18.
• CommentAuthorMike Shulman
• CommentTimeJun 13th 2019

Has he announced this anywhere, other than through you on the nForum? Or asked around in the community for feedback?

• CommentRowNumber19.
• CommentAuthorUrs
• CommentTimeJun 13th 2019
• (edited Jun 13th 2019)

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.

• CommentRowNumber20.
• CommentAuthorericfinster
• CommentTimeJun 16th 2019

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 … :(

• CommentRowNumber21.
• CommentAuthorericfinster
• CommentTimeJun 16th 2019

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.

2. 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?

3. At a quick glance, your approach seems reminiscent of Penon’s approach to higher categories. Did his work have any influence on yours?

• CommentRowNumber24.
• CommentAuthorUrs
• CommentTimeJun 20th 2019
• (edited Jun 20th 2019)

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.

• CommentRowNumber25.
• CommentAuthorUrs
• CommentTimeAug 4th 2020
• CommentRowNumber26.
• CommentAuthorUrs
• CommentTimeMay 13th 2021
• (edited May 13th 2021)

• CommentRowNumber27.
• CommentAuthorUrs
• CommentTimeMay 13th 2021
• (edited May 13th 2021)

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.

• CommentRowNumber28.
• CommentAuthorcisinski
• CommentTimeMay 13th 2021
• (edited May 13th 2021)
There is this source: https://www.ias.edu/ideas/2014/voevodsky-origins
Quote from Voevodsky:
"I have been working on the ideas that led to the discovery of univalent models since 2005 and gave the first public presentation on this subject at Ludwig-Maximilians-Universität München in November 2009. While I have constructed my models independently, advances in this direction started to appear as early as 1995 and are associated with Martin Hofmann, Thomas Streicher, Steve Awodey, and Michael Warren."

However, the date of birth of univalence is arguably May 1st 2006, since this is when Bousfield sketched a proof of a baby version of it (express any homotopy equivalence between two Kan complexes as a Kan fibration to \Delta^1). Grayson did post a copy of a series of E-mail exchanges between Voevodsky, Grayson, May and Bousfield here: https://groups.google.com/g/homotopytypetheory/c/K_4bAZEDRvE/m/FSfFdoJ3AAAJ
• CommentRowNumber29.
• CommentAuthorUrs
• CommentTimeMay 13th 2021

Thanks!

I have worked that into the list of references, here.

• CommentRowNumber30.
• CommentAuthorUrs
• CommentTimeJun 24th 2022

I have added pointer to

and correspondingly expanded the parenthesis below the pointer to Bousfield 06

4. Adding a section on the univalence axiom for Tarski universes and its equivalence to an “external” version of the univalence axiom where the canonical transport function is an equivalence.

Anonymous

5. Adding a section about the relation between univalence and axiom K.

Anonymous

6. Added redirect for universe extensionality

Anonymous

Anonymous

8. Adding section on definitional univalence.

Anonymous

9. Added redirect for definitional univalence

Anonymous

10. added motivation for the transport definition of univalence: strongly predicative Tarski universes which are not closed under function types.

Anonymous

11. added an explanation how different notions of sameness of types leads to different notions of univalence

Anonymous

12. separated definitions from its categorical semantics

Anonymous

• CommentRowNumber40.
• CommentAuthorUrs
• CommentTimeDec 27th 2022
• (edited Dec 27th 2022)

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.

• CommentRowNumber41.
• CommentAuthorUrs
• CommentTimeDec 27th 2022

• CommentRowNumber42.
• CommentAuthorUrs
• CommentTime3 days ago
• (edited 3 days ago)