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• CommentRowNumber1.
• CommentAuthorUlrik
• CommentTimeJan 16th 2019

Initial stub to record some references. Wanted by type theoretic model category

• CommentRowNumber2.
• CommentAuthorUlrik
• CommentTimeJan 16th 2019

• CommentRowNumber3.
• CommentAuthorUlrik
• CommentTimeJan 16th 2019

Change the page name to the one with hyphen and add the other versions as redirects.

• CommentRowNumber4.
• CommentAuthorUlrik
• CommentTimeJan 17th 2019

The assumptions and some instances of these. More later.

• CommentRowNumber5.
• CommentAuthorMike Shulman
• CommentTimeJan 17th 2019

Is this talk also related?

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeJan 17th 2019

Suggestions for better names:

• interval-induced model category
• interval-generated model category
• cubical-type model category
• univalence-induced model category
• CommentRowNumber7.
• CommentAuthorAli Caglayan
• CommentTimeJan 18th 2019

Even cubical model category might do. Although it could be confused with the model structure on cubical sets.

• CommentRowNumber8.
• CommentAuthorMike Shulman
• CommentTimeJan 18th 2019

I think by analogy to “simplicial model category”, the term “cubical model category” would be assumed to mean a model category enriched over cubical sets.

• CommentRowNumber9.
• CommentAuthorUlrik
• CommentTimeJan 19th 2019
• (edited Jan 19th 2019)

What about “univalent model category”? That’s reasonably short and hints at the key ingredient (and a raison-d’être). Otherwise, “univalence-induced” might do, but it is a cumbersome. I’ll ask Thierry and Christian for their opinions.

Re #5, yes, that’s very relevant, and I realized that the current assumptions on this page (type-theoretic model structure) are not general enough. In particular, once we have fibrant dependent products and fibrant univalent universes, then we should get a model structure worthy of the name. But then we might get a bit closer to type-theoretic model category, which AFAICS doen’t discuss univalent universes at the moment.

• CommentRowNumber10.
• CommentAuthorAli Caglayan
• CommentTimeJan 19th 2019

“univalent model category” seems a bit greedy since there could be other models of univalence in the future. What about something silly like “cubivalent model category”?

• CommentRowNumber11.
• CommentAuthorMike Shulman
• CommentTimeJan 19th 2019

In fact there are already other models of univalence, such as Voevodsky’s very first model of simplicial sets and diagrams of such. So “univalent model category” won’t cut it.

• CommentRowNumber12.
• CommentAuthorspitters
• CommentTimeFeb 15th 2019

Remarks about equivalence with simplicial sets.

• CommentRowNumber13.
• CommentAuthorMike Shulman
• CommentTimeFeb 15th 2019

Thanks. I’d like to change the name of this page now to something less conflicty and confusing, before the term starts to get used too much. I suggest “cubical-type model category” for now, does anyone have a better suggestion?

• CommentRowNumber14.
• CommentAuthorAli Caglayan
• CommentTimeFeb 15th 2019

I think “cubical-type model category” is a reasonable improvement.

• CommentRowNumber15.
• CommentAuthorMike Shulman
• CommentTimeFeb 15th 2019

Renamed page to something that’s a bit better. But if we come up with an even better name we can of course rename it again.

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeFeb 3rd 2021

I have hyperlinked more of the technical terms

Where it says “the model in simplicial sets” a sensible hyperlink should be given.

Looking around for what we have, I notice the entry with the sweet title

That goes in the direction to what one should link to here. But maybe this could be brought into a little more standard format.

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