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• CommentRowNumber1.
• CommentAuthorDavid_Corfield
• CommentTimeAug 18th 2018

Since I gathered them for my recent talk, I may as well provide a list here of work in this area. I need to add names, etc.

• CommentRowNumber2.
• CommentAuthorDavid_Corfield
• CommentTimeAug 18th 2018

I was wondering, famously because the internal language of a topos is not classical it is possible to add classically inconsistent axioms, such as the Kock-Lawvere axiom. Can something similar occur with HoTT?

At this new axioms section it points out that we may add things like Whitehead’s axiom, but this goes in the direction of making things more classical. How about something which relies on the constructive aspect of HoTT?

Hmm, or might modalities play this role?

• CommentRowNumber3.
• CommentAuthorDavidRoberts
• CommentTimeAug 19th 2018

Fixed incorrect link to Felix Wellen’s abstract for HoTTUF18

• CommentRowNumber4.
• CommentAuthorMike Shulman
• CommentTimeAug 19th 2018

Yes, “Axiom R” in my realcohesion paper is a nonclassical axiom, inconsistent with full LEM. It doesn’t require higher homotopy, but it becomes much more powerful in the presence of it.

• CommentRowNumber5.
• CommentAuthorDavid_Corfield
• CommentTimeJul 8th 2019

• Kristina Sojakova, The equivalence of the torus and the product of two circles in homotopy type theory, (arXiv:1510.03918)
• CommentRowNumber6.
• CommentAuthorDavid_Corfield
• CommentTimeDec 19th 2019

• CommentRowNumber7.
• CommentAuthorDavid_Corfield
• CommentTimeJul 14th 2020

• CommentRowNumber8.
• CommentAuthorDavid_Corfield
• CommentTimeJul 24th 2020

• Ayberk Tosun, Formal Topology in Univalent Foundations, (pdf)
1. Added “Lawvere-Tierney sheafification in Homotopy Type Theory”

Anthony Hart

2. Add thesis link for Analysis in Univalent Type Theory by Auke Booij.

Anthony Hart

• CommentRowNumber11.
• CommentAuthorDavid_Corfield
• CommentTimeAug 31st 2020
• (edited Aug 31st 2020)

• Daniel Carranza, Jonathan Chang, Chris Kapulkin, Ryan Sandford, 2-adjoint equivalences in homotopy type theory, (arXiv:2008.12433)

• Martin E. Bidlingmaier, Florian Faissole, Bas Spitters, Synthetic topology in Homotopy Type Theory for probabilistic programming((arXiv:1912.07339))

Anthony Hart

• CommentRowNumber13.
• CommentAuthorDavid_Corfield
• CommentTimeDec 6th 2020

• Andrew Swan, On the Nielsen-Schreier Theorem in Homotopy Type Theory, (arXiv:2010.01187)
• CommentRowNumber14.
• CommentAuthorDavid_Corfield
• CommentTimeDec 6th 2020

And another

• Håkon Robbestad Gylterud, Elisabeth Bonnevier, Non-wellfounded sets in homotopy type theory, (arXiv:2001.06696)
• CommentRowNumber15.
• CommentAuthorUrs
• CommentTimeDec 6th 2020

Let’s not forget to add these references to the pertaining entries! This one should presumeably also be listed at set theory.

• CommentRowNumber16.
• CommentAuthorDavidJaz
• CommentTimeDec 7th 2020

I’ve added my paper on modal fibrations to the modal homotopy type theory section.

• CommentRowNumber17.
• CommentAuthorDavid_Corfield
• CommentTimeDec 17th 2020

There are other UniMath papers to include.

• CommentRowNumber18.
• CommentAuthorDavid_Corfield
• CommentTimeJan 5th 2021

• CommentRowNumber19.
• CommentAuthorUrs
• CommentTimeJan 6th 2021

Let’s add the items of the list also to the subjects that they are about, to increase the chance of finding them. This one could go to cubical type theory and synthetic homotopy theory. Have now added it there.

• CommentRowNumber20.
• CommentAuthormartinescardo
• CommentTimeJan 6th 2021

Added publication details of arxiv preprint.

• CommentRowNumber21.
• CommentAuthormartinescardo
• CommentTime3 days ago

• CommentRowNumber22.
• CommentAuthorUrs
• CommentTime3 days ago

Allow me to make some suggestions on how to go about these references

• if you hyperlink author names, do so through their nLab entries, i.e. by simply by including the author names in double square brackets. This has the advantage that it’s more stable than the direct links (which may change) and that it helps the GoogleBrain to understand what is related to what on the nLab.

• don’t say “published in” in a bibitem (we don’t do that in journals or books either), unless in one of the rare cases where the publication whereabout do deserve extra comments

I suggest to stick to the formatting of bibitems in a usual form

  * AuthorNames, _Title_, Journal Name-Volume-year (HyperlinkedArXivNumber, HyperlinkedDoiNumber, HyperlinkedISBNNumber  )


Finally, I keep feeling that this page “mathematics presented in HoTT” is not doing itself a favor: Before long the list here will be either too long or too incomplete to be of much use, since which mathematics is not going to be presented in HoTT?

More useful both for the readers as well as for authors who want to be read is to add the references to the entry about their subject. Your article reference on domain theory in HoTT should be included at domain theory, for maximal visibility

• CommentRowNumber23.
• CommentAuthorDmitri Pavlov
• CommentTime3 days ago
• (edited 3 days ago)

Re #22: Would you mind copy-pasting your remark about bibitems to the appropriate place on the wiki, perhaps the FAQ?

It would be nice to include one concrete example of a “perfectly formatted bibitem” for our reference.

I have a script that formats bibitems for me and I’d like to adapt it to nLab’s format too.

4. Finally, I keep feeling that this page “mathematics presented in HoTT” is not doing itself a favor: Before long the list here will be either too long or too incomplete to be of much use, since which mathematics is not going to be presented in HoTT?

I can see that happening at some point, but I think it’s useful for the moment when numbers are still very low.

• CommentRowNumber25.
• CommentAuthorUrs
• CommentTime2 days ago

Dmitri, I think I once wrote this at HowTo, let me see… yes, here. But we could expand there, maybe giving more concrete examples.

• CommentRowNumber26.
• CommentAuthorUrs
• CommentTime2 days ago

David, maybe I am being a pain here, sorry.

I am just feeling that with the effort already being invested into your page, it seems a shame not to apply the few keystrokes to copy-and-paste the items also into the relevant entries (and to their author’s pages, for that matter).

But maybe we need to wait for some bibtex like functionality for such exhaustive distribution of reference items to be easier on us users…

5. Yes, sure, we should do that too.

I did at least go through the process of thinking where the entries should go, and now we have a page synthetic homotopy theory that’s an obvious destination. But then I got lost in an internal debate as to where ’synthetic homotopy theory’ begins and ends. If in the modern spirit, as Barwick put it, “Homotopy theory is not a branch of topology…I think of homotopy theory as an enrichment of the notion of equality”, then it would be rather odd to limit synthetic homotopy theory to be whatever’s proved in HoTT of the kind of thing you’d find in the old ’homotopy theory is a part of algebraic topology’ picture.

Since I couldn’t then come to a principled way to separate the part of that which is proved in HoTT which should not count as synthetic homotopy theory, I gave up. Is there a way?

• CommentRowNumber28.
• CommentAuthorUrs
• CommentTime2 days ago

I simply mean that any article of the type “XYZ in HoTT” should (also) go to the page XYZ!

For instance the article on domain theory in HoTT should go to domain theory, as in de Jong-Escardo 21, etc.

• CommentRowNumber29.
• CommentAuthorUrs
• CommentTime2 days ago

I’ll give another concrete example of what I am suggesting:

  [[Andrew Swan]], _On the Nielsen-Schreier Theorem in Homotopy Type Theory_ ([arXiv:2010.01187](https://arxiv.org/abs/2010.01187))


Andrew Swan, On the Nielsen-Schreier Theorem in Homotopy Type Theory (arXiv:2010.01187)

I have now proliferated that as follows, and I suggest to do the analogue with every item in the list (at least with every new item):

First, in mathematics presented in HoTT I have expanded to

  * {#Swan20} [[Andrew Swan]], _On the Nielsen-Schreier Theorem in Homotopy Type Theory_ ([arXiv:2010.01187](https://arxiv.org/abs/2010.01187))

> (on the [[Nielsen-Schreier theorem]])


Then I have copied to Nielsen-Schreier theorem, there giving it a cross-link back to HoTT/UV:

  Discussion in [[homotopy type theory]]/[[univalent foundations]] (see also [[mathematics presented in HoTT]]):

* {#Swan20} [[Andrew Swan]], _On the Nielsen-Schreier Theorem in Homotopy Type Theory_ ([arXiv:2010.01187](https://arxiv.org/abs/2010.01187))


Discussion in homotopy type theory/univalent foundations (see also mathematics presented in HoTT):

Finally, I have added it to the author’s page Andrew Swan as:

  On the [[Nielsen-Schreier theorem]] in [[homotopy type theory]]/[[univalent foundations]]:

* {#Swan20} [[Andrew Swan]], _On the Nielsen-Schreier Theorem in Homotopy Type Theory_ ([arXiv:2010.01187](https://arxiv.org/abs/2010.01187))

6. I’ve just done that for an article on the torus. I guess the issue is that to do it properly, there should then be a section on the XYZ page that speaks to the HoTT treatment, and that takes some work, even in the simple case of the torus.

• CommentRowNumber31.
• CommentAuthorUrs
• CommentTime2 days ago

Thanks!

I think if the references-section says “Discussion of this topic in HoTT/UV is in:…” then that’s a first step for the entry to talk about the HoTT treatment! If that line prompts people to add some actual discussion to the entry, then all the better. That’s why such cross-linking is important, I think, the interconnection makes it all be more fruitful.