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The simplicial type theory of Riehl and Shulman is not the only setting for synthetic (infinity,1)-category theory. There is also the bicubical directed type theory by Weaver and Licata:
Matthew Z. Weaver and Daniel R. Licata. “A Constructive Model of Directed Univalence in Bicubical Sets”. In: Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science. LICS ’20. Saarbrücken, Germany: Association for Computing Machinery, 2020, pp. 915–928. doi: 10.1145/3373718.3394794.
Matthew Weaver. ([Directed] Higher) Inductive Types in Bicubical Directed Type Theory. Presentation at HoTT/UF, part of FSCD 2021. 2021.
@2 Weaver talks about directed univalence rather than the core topics of synthetic $(\infty, 1)$-category theory as his primary focus, how much of that is actually related to synthetic $(\infty, 1)$-category theory?
For what it’s worth, the abstract of Weaver & Licata 2020 claims a close relation to Riehl & Shulman, the difference highlighted being just a switch from bisimplicial to bicubical sets.
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By the way, it’s easy to make hyperlinks here and on the nLab, the syntax is the same in both cases: [link text](linkurl)
For example the above link is made by typing
[Weaver & Licata 2020](https://dl.acm.org/doi/abs/10.1145/3373718.3394794)
See also the source code of the entry, where I have implemented hyperlinks for the references now. For example:
Ulrik Buchholtz, Jonathan Weinberger, Synthetic fibered $(\infty,1)$-category theory $[$arXiv:2105.01724, talk slides$]$
* [[Ulrik Buchholtz]], [[Jonathan Weinberger]], *Synthetic fibered $(\infty,1)$-category theory* $[$[arXiv:2105.01724](https://arxiv.org/abs/2105.01724), [talk slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Weinberger-2022-01-20-HoTTEST.pdf)$]$
I have taken the liberty of expanding out the Idea-section to provide more context:
In general, by synthetic $(\infty,1)$-category theory one will want to mean some formulation of (∞,1)-category theory in the spirit of synthetic mathematics, here specifically relating to synthetic homotopy theory as $\infty$-category theory relates to homotopy theory.
One implementation of this idea was proposed by Riehl & Shulman 2017, based on a variant of homotopy type theory called simplicial type theory. This is further developed by Buchholtz & Weinberger 2021.
Another thought regarding #2, #3, #4:
Would it in fact be more accurate to re-name the entry simplicial type theory to directed simplicial type theory?
Because the entry does not rhyme on the usual sense of simplicial homotopy theory: The interval type introduced in would-be “simplicial type theory” is crucially a “directed” interval. No?
The bicubical approach is very similar to the bisimplicial one, especially in motivation, but it’s definitely different. Directed univalence should be one of the basic aspects of synthetic higher category theory, just as ordinary univalence is one of the basic aspects of synthetic homotopy theory.
Urs, the interval in ordinary simplicial sets is also just as directed. Ordinary simplicial sets can be used to do either $\infty$-groupoid theory, with Kan complexes, or $(\infty,1)$-category theory, with quasicategories. Similarly, simplicial type theory is a formal system that could be used to do synthetic $\infty$-groupoid theory, with the “discrete” types, or synthetic $(\infty,1)$-category theory, with the Segal/Rezk types. It’s just that the latter is a more interesting application, because we already have a way to do synthetic $\infty$-groupoid theory with unaugmented HoTT.
Is it possible to augment higher observational type theory with the shapes, extension types, and directed interval of simplicial MLTT to get “simplicial higher observational type theory”?
There seems to remain a language issue in that “simplicial homotopy type theory” is not to “type theory” as “simplicial homotopy theory” is to “homotopy theory”. I don’t want to further belabor the way things are named, but I understood #2 as making a valid point on the entry missing some information that still deserves to be added in.
Adding talk to references section
Anonymous
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