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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?
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.
Maybe the references at tribe will help.
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
Thanks for these. I hadn’t yet come across categorical model of dependent types.
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.
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 tribe in the sense of Joyal is a display map category which models $\Sigma$-types and Paulin-Mohring Id-types.
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