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Fawzi Hreiki (Mar 06 2021 at 14:01):I'm learning about (co)ends but it's a little difficult to see the big picture because there are multiple entrance points to the concept (universal wedge, (co)equaliser of endo-profunctor actions, hom-weighted (co)limit, etc..). What is the 'minimal'/'canonical' structure a 2-category needs to get an internal notion of (co)end? Obviously a Yoneda structure suffices since it provides hom-arrows and weighted (co)limits, but this seems to be overkill.
Mike Shulman (Mar 06 2021 at 15:03):I don't know of anything significantly less than a structure for formal category theory (Yoneda structure, proarrow equipment, etc.) that suffices to define (co)ends in a 2-category. I suppose if you have a Weber 2-topos then you can define twisted arrow categories, which suffice to define (co)ends, but that's pretty close to a Yoneda structure anyway, and it's also a purely cartesian approach that doesn't work for 2-categories of enriched categories.
Fawzi Hreiki (Mar 06 2021 at 15:12):Ah thanks
Nathanael Arkor (Mar 07 2021 at 00:18):A Yoneda structure is really quite minimal structure to ask for anyway; it's just equivalent to having a KZ doctrine, i.e. a certain kind of pseudomonad on a 2-category.