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• CommentRowNumber1.
• CommentAuthorspitters
• CommentTimeFeb 4th 2017

equifibered natural transformation misses references. I’ve traced cartesian natural transformation back to: Street - the petit topos of globular sets which refers to: Carboni, Johnstone - Connected limits, familial representability and Artin glueing unfortunately, I do not have access to the latter.

Is this the best source?

• CommentRowNumber2.
• CommentAuthorTodd_Trimble
• CommentTimeFeb 4th 2017

It’s a good source, yes. My memory is that Tom Leinster’s book on higher categories also treats cartesian transformations, and it’s easily available.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeFeb 4th 2017

Hopefully one of you finds a minute to add the reference to the entry!

• CommentRowNumber4.
• CommentAuthorCharles Rezk
• CommentTimeFeb 4th 2017
• (edited Feb 4th 2017)

I’ve used the phrase “equifibered map” (where the map is itself a natural transformation) in various places, with this meaning (but in an $\infty$-context). For instance, in my old model topos notes http://www.math.uiuc.edu/~rezk/homotopy-topos-sketch.pdf, or here: http://www.math.uiuc.edu/~rezk/i-hate-the-pi-star-kan-condition.pdf.

• CommentRowNumber5.
• CommentAuthorTodd_Trimble
• CommentTimeFeb 5th 2017

• CommentRowNumber6.
• CommentAuthorspitters
• CommentTimeFeb 5th 2017

Thanks! To be sure, is this notion indeed different from the notion of cartesian natural transformation in the theory of fibred categories. E.g. Streicher.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeFeb 5th 2017

Thanks, Todd!

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeFeb 5th 2017
• (edited Feb 5th 2017)

I have also added pointer to Rezk 10 and Rezk 14.

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeJun 30th 2020
• (edited Jun 30th 2020)

I have added the statement (here) of descent in $\infty$-toposes characterized via equifibered transformations of colimiting diagrams

But I am going to give this sub-section its stand-alone entry now, and then re-!include it, because the same discussion ought to be included in various other relevant entries