added this remark:

The type-theory-literature traditionally refers to such categories generically as *display map/fibration categories with such-and-such types*, eg.

The definitions in this section are relatively standard in the literature. A display map category which models $\Sigma$-types coincides with Joyal’s notion of

clan, and a display map category which models $\sum$-types and $\prod$-types coincides with his notion of$\pi$-clan. […]A

tribein the sense of Joyal is a display map category which models $\Sigma$-types and Paulin-Mohring Id-types.

- Paige Randall North,
*Identity types and weak factorization systems in Cauchy complete categories*, Mathematical Structures in Computer Science**29**9 (2019) 1411-1427 [doi:10.1017/S0960129519000033, arXiv:1901.03567]

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I have adjusted wording and referencing to make it clearer that the terminology may be new, but the basic idea isn’t (as far as I am aware).

]]>However, you won’t find these things under any other names in the HoTT Book either. The reason is that the HoTT Book is not about semantics at all, only about doing mathematics *internal* to HoTT.

Thanks for these. I hadn’t yet come across categorical model of dependent types.

]]>I’d also think that the statement in #3 follows by immediate inspection of the definition. For a more explicit quote from a random reference on the matter:

]]>A tribe in the sense of Joyal is a display map category which models Σ-types and Id-types

Maybe the references at tribe will help.

]]>As far as I am aware, the terms “tribe” et al. are essentially synonyms for what most people had already called by other names, such as “display map categories” and others. Probably best to start with the $n$Lab entry *categorical model of dependent types*.

I was just fed this great presentation:

https://www.youtube.com/watch?v=HRBShaxIblI

I just did a ctrl+f in the HoTT book and didn’t find “clan” or “tribe”. I also don’t see any pages on it here. Have these concepts proven to be useful in understanding the trinity? Any chance on getting an update on these pages?

]]>A stub with a few references.

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