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• CommentRowNumber1.
• CommentAuthorMike Shulman
• CommentTimeJan 14th 2019

A stub with a few references.

• CommentRowNumber2.
• CommentAuthorsamwinnick
• CommentTimeJan 17th 2022

I was just fed this great presentation:

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?

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeJan 17th 2022

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.

• CommentRowNumber4.
• CommentAuthorDavid_Corfield
• CommentTimeJan 17th 2022

Maybe the references at tribe will help.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeJan 17th 2022

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

• CommentRowNumber6.
• CommentAuthorsamwinnick
• CommentTimeJan 17th 2022

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

• CommentRowNumber7.
• CommentAuthorMike Shulman
• CommentTimeJan 17th 2022

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.