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• CommentRowNumber1.
• CommentAuthorspitters
• CommentTimeJan 16th 2019

Adding model structure from cubical type theory.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJun 27th 2020

• CommentRowNumber3.
• CommentAuthorHurkyl
• CommentTimeNov 24th 2020

Corrected the mention of etale geometric morphisms.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeOct 2nd 2021
• (edited Oct 2nd 2021)

added comment/argument (here) that the global section geometric morphism of a Grothendieck $\infty$-topos is essentially unique.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeOct 14th 2021

• Carlos Simpson, A Giraud-type characterization of the simplicial categories associated to closed model categories as $\infty$-pretopoi (arXiv:math/9903167)

which has been missing also in a number of related entries.

Is this the first use of the terminology “$\infty$-topos”, by a margin? And it seems all the ideas are in there, in a kind of model topos-form (am adding the pointer there, too).

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeOct 14th 2021

while I was at it, I have boosted the list of early reference a little more (here). For instance where a “letter to Grothendieck” was mentioned, I added the point to Joyal’s letter, and then also added pointer to Jardine’s early article, for completeness on this point of model categories of simplicial presheaves.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeOct 18th 2021

I have replaced the (brief) section on powering (which essentially just said it exists) and half of the following section on cartesian closure (the half that talked about the internal hom preserving limits also in the first argument) with the new !include-section powering of ∞-toposes over ∞-groupoids – section.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeNov 13th 2021