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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeJan 14th 2019
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexA stub with a few references. <a href="https://ncatlab.org/nlab/revision/tribe/1">v1</a>, <a href="https://ncatlab.org/nlab/show/tribe">current</a>

   A stub with a few references.

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorsamwinnick
   • CommentTimeJan 17th 2022
   • PermaLink
   Author: samwinnick
   Format: MarkdownItexI 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?

   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?

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJan 17th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexAs 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]]*.

   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.

4. • CommentRowNumber4.
   • CommentAuthorDavid_Corfield
   • CommentTimeJan 17th 2022
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexMaybe the references at [[tribe]] will help.

   Maybe the references at tribe will help.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeJan 17th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexI'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 ([arXiv:1901.03567](https://arxiv.org/abs/1901.03567), [p. 6](https://arxiv.org/pdf/1901.03567.pdf#page=6))

   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

   (arXiv:1901.03567, p. 6)

6. • CommentRowNumber6.
   • CommentAuthorsamwinnick
   • CommentTimeJan 17th 2022
   • PermaLink
   Author: samwinnick
   Format: MarkdownItexThanks for these. I hadn't yet come across [[categorical model of dependent types]].

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

7. • CommentRowNumber7.
   • CommentAuthorMike Shulman
   • CommentTimeJan 18th 2022
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexHowever, 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.

   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.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeAug 24th 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexI 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). <a href="https://ncatlab.org/nlab/revision/diff/tribe/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/tribe/3">v3</a>, <a href="https://ncatlab.org/nlab/show/tribe">current</a>

   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).

   diff, v3, current

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeAug 24th 2023
   • (edited Aug 24th 2023)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded this remark: *** The type-theory-literature traditionally refers to such categories generically as *[[display map]]/[[fibration categories]] with such-and-such types*, eg. [North 2019, Rem. 2.4](#North19): > The definitions in this section are relatively standard in the literature. A display map category which models [[dependent pair type|$\Sigma$-types]] coincides with Joyal’s notion of *clan*, and a display map category which models [[dependent pair type|$\sum$-types]] and [[dependent function type|$\prod$-types]] coincides with his notion of *$\pi$-clan*. &lbrack;...&rbrack; > A *tribe* in the sense of Joyal is a display map category which models [[dependent pair type|$\Sigma$-types]] and Paulin-Mohring [[identification types|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 &lbrack;[doi:10.1017/S0960129519000033](https://doi.org/10.1017/S0960129519000033), [arXiv:1901.03567](https://arxiv.org/abs/1901.03567)&rbrack; *** <a href="https://ncatlab.org/nlab/revision/diff/tribe/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/tribe/3">v3</a>, <a href="https://ncatlab.org/nlab/show/tribe">current</a>

   added this remark:

   ---

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

   North 2019, Rem. 2.4:

   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.

   • 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]

   ---

   diff, v3, current