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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?
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.
Hopefully one of you finds a minute to add the reference to the entry!
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.
References now added.
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.
Thanks, Todd!
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
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