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Stream: theory: category theory

Topic: Unit cells in a virtual double category

Evan Patterson (Jul 06 2023 at 05:02):

In Cruttwell-Shulman's paper on virtual equipments and generalized multicategories, units in a virtual double category are defined by saying that each object $X$ has a proarrow $U_X: X \not\to X$ along with a nullary globular cell $()_X \to U_X$ with the universal property of being opcartesian.

We also want that for each arrow $f: X \to Y$ there is a cell $U_f: U_X \to U_Y$ bounded by $f$ on either side. This is needed for a virtual double category with units to have an underlying 2-category and also for a virtual equipment to have companions and conjoints, which are important for doing formal category theory. However, I don't see how to obtain this cell $U_f$. How can this be done?

Nathanael Arkor (Jul 06 2023 at 05:32):

You use the opcartesian 2-cell associated to $U_Y$ together with nullary composition of 2-cells, to obtain a 2-cell with nullary domain, $f$ on both sides, and $U_Y$ as the codomain. Finally, you use the universal property of $U_X$.

Nathanael Arkor (Jul 06 2023 at 05:34):

(I'm on mobile, so it's a little awkward to share the relevant diagram right now, but you can see it at the top of page 9 here.)

Evan Patterson (Jul 06 2023 at 06:05):

Ah, so the figure on p. 9, which interpreted naively doesn't make sense because there appears to be a cell with nullary codomain, is actually a nullary composite of 2-cells. Very sneaky. I didn't realize that was allowed, but in retrospect it makes sense. Thanks!

Nathanael Arkor (Jul 06 2023 at 06:11):

Ah, yes, that's just a notational convention to make everything align to a grid, which I find is helpful to keep things readable. If I remember correctly, Leinster is quite explicit in drawing attention to the nullary composition of 2-cells in his papers on generalised enrichment, because it is a subtle point.

Mike Shulman (Jul 06 2023 at 06:28):

I would draw the opcartesian 2-cell, like any nullary 2-cell, as a triangle (we did this a lot in our paper), which can be "whiskered" at the top by a vertical morphism to get another nullary 2-cell (apparently we never drew this in our paper!).

Evan Patterson (Jul 06 2023 at 07:07):

Right, it was this "whiskering" move that I was missing, but it makes sense now!