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• CommentRowNumber1.
• CommentAuthorTim Campion
• CommentTimeFeb 14th 2021

Added some basic examples from HTT. There doesn’t seem to be a page for the corresopnding 1-categorical notion. This notion is used pretty heavily in $\infty$-category theory, but it’s not so familiar from 1-category theory. But I’d have to think a 1-categorical treatment exists somewhere, right?

• CommentRowNumber2.
• CommentAuthorAlexanderCampbell
• CommentTimeFeb 14th 2021
• (edited Feb 14th 2021)

I know these from Schumacher & Street’s paper on Some parametrized categorical concepts as “cartesian cones”. Also, in the case of diagrams indexed by discrete categories, I think these are the “initial lifts” of “structured sources” from The Joy of Cats.

• CommentRowNumber3.
• CommentAuthorTim Campion
• CommentTimeFeb 15th 2021
• (edited Feb 15th 2021)

Thanks! “Roughly equivalent” notions probably appear in different guises elsewhere too, of course.

• CommentRowNumber4.
• CommentAuthorTim Campion
• CommentTimeFeb 27th 2021

Added a better example which Rune Haugseng explained to me.