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Written by HarrisonGrodin on 2024-10-23 at 23:29

What is a (co)limit in a (generalized) multicategory?

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Written by Zanzi @ Monoidal Cafe on 2024-10-23 at 23:31

@HarrisonGrodin I have been asking this not too long ago on monoidal cafe, I dont think it's fully worked out even for multicategories, let alone generalized ones!

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Written by HarrisonGrodin on 2024-10-23 at 23:41

@zanzi Interesting! I was only able to find this minor reference, but nothing else (even in books about multicategory theory): https://math.stackexchange.com/a/3647208/412385

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Written by Boarders on 2024-10-24 at 01:45

@zanzi @HarrisonGrodin do you think it relates to weighted limits/colimits?

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Written by Zanzi @ Monoidal Cafe on 2024-10-24 at 18:55

@boarders @HarrisonGrodin I'm not sure tbh! That might be the right way to generalize it, but it's not something I've worked through!

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Written by ohad on 2024-10-24 at 04:22

@HarrisonGrodin Levy and Staton POPL paper develops this for pre-multicategories, I guess a slightly simpler development for multicategories would work.

Multicategories have a Yoneda structure (Hermida develops that one) that would give you one class. You can try finding a Yoneda structure for co-multicategories, it might give you the other. But there's something funny happening for those, so there might be more than one yoneda structure, or maybe none at all and you'd need to work with proarrow equipments.

Just an intuition!

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