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Stream: theory: category theory

Topic: Enriched laxness and tensor products

Reuben Stern (they/them) (Jun 17 2020 at 12:29):

Let $\mathcal{V}$ be a monoidal category, and $\mathcal{C}$ and $\mathcal{D}$ be $\mathcal{V}$-categories. Is there a good notion of a lax $\mathcal{V}$-functor $F: \mathcal{C} \to \mathcal{D}$? Thereafter, is there a good notion of the lax Gray tensor product for $\mathcal{V}$-categories, corepresenting pairs of lax $\mathcal{V}$-functors? Ultimately I'm looking toward the enriched $(\infty, 1)$-category world, where 2-categories are just categories enriched in categories, so this would recover the usual lax Gray tensor product.

Reid Barton (Jun 17 2020 at 14:17):

Seems to me you need some sort of extra structure on $\mathcal{V}$ for this--the first one that comes to mind being a 2-category.

Reuben Stern (they/them) (Jun 17 2020 at 14:28):

If $\mathcal{V}$ is presentably monoidal, then we can view $\mathcal{V}$ as a $\mathcal{V}$-category, so maybe that's enough?

Reid Barton (Jun 17 2020 at 14:28):

Well, what do you expect to happen for $\mathcal{V} = \mathrm{Set}$, say?

Reid Barton (Jun 17 2020 at 14:29):

I think viewing $\mathcal{V}$ as a $\mathcal{V}$-category only helps if you then have some other reason to be able to treat $\mathcal{V}$ as basically Cat. For example, a directed interval object or something similar.

Reid Barton (Jun 17 2020 at 15:31):

Or from another point of view, how do you expect to get out lax functors as opposed to oplax functors?

Reuben Stern (they/them) (Jun 17 2020 at 15:34):

ah I see what you mean. I guess I have been thinking of $\mathcal{V}$ as a 2-category secretly

John Baez (Jun 17 2020 at 19:36):

I don't think there should be a notion of lax $\mathcal{V}$-functor unless $\mathcal{V}$ is a 2-category or more. After all, lax preservation of composition of 1-morphisms demands something like 2-morphisms.

John Baez (Jun 17 2020 at 19:39):

If $\mathcal{V}$ has something like an interval object you could probably use that to define lax $\mathcal{V}$-functors. I say "something like" an interval object because there are various notions of interval object, and you'll need to choose one with the right properties to get the job done.