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Stream: theory: category theory

Topic: a lemma about reindexing monoidal presheaves

Matteo Capucci (he/him) (Mar 28 2025 at 15:59):

Does anybody know if this is known already?

Let $\cal V$ be a cocomplete monoidal category, let $\cal A$ and $\cal B$ be monoidal categories. Suppose $F^* \dashv F_*$ is a doctrinal adjunction in $\bf MonCat_{lax}$, so that $F^*$ is strong and $F_*$ is lax. Then the induced inverse image functor ${\cal V}^{F_*}:\cal V^B \to \cal V^A$ is lax monoidal.

In fact I believe one can also give $\cal V$ a colax structure, though I don't know what the compatibility between the two turns out to be (I'd be very surprised if they are inverse to each other).

Matteo Capucci (he/him) (Mar 28 2025 at 16:01):

I can show there exists a laxator by end-fu, and in the posetal case I'm actually interested in that's enough (since it's automatically coherent). I was wondering if it holds more generally though, and how coherence would be proven in that case.

Nathanael Arkor (Mar 28 2025 at 17:28):

Since $F_*$ is lax monoidal, it is a functor between multicategories, hence $V^{F_*}$ is the corresponding functor of multicategories induced by the universal property of the exponential, so that it is also lax monoidal.

Nathanael Arkor (Mar 28 2025 at 17:30):

For the same reason, $V^{F^*}$ is lax monoidal, and $V^{F_*} \dashv V^{F^*}$, so $V^{F_*}$ is furthermore pseudo monoidal by doctrinal adjunction.

Kevin Carlson (Mar 28 2025 at 17:31):

Using that representable multicategories are exponentiable?

Nathanael Arkor (Mar 28 2025 at 19:51):

Yes, making use of Pisani's work on exponentiable multicategories.

Matteo Capucci (he/him) (Mar 30 2025 at 09:55):

What a sleek argument! Thank you very much. Also good observation that strong monoidality follows by doctrinal adjunction.

John Baez (Mar 30 2025 at 15:59):

(Are you saying "sleek" as a deliberate variant of "slick"? I kind of like it. People always talk about slick arguments, and I've never heard of a sleek argument. Some non-native speakers pronounce "slick" as "sleek". But an argument can, in fact, seem sleek.)

Matteo Capucci (he/him) (Apr 02 2025 at 09:26):

well I've never realized they are two different words until now lol so no, very much not deliberate

Matteo Capucci (he/him) (Apr 02 2025 at 09:26):

btw sleek sounds sleeker to me than slick :P

John Baez (Apr 02 2025 at 15:39):

They mean different things: a cat can be sleek, an oil spill is slick. Sleek things are thin and elegant, slick things are wet and slippery. People often say a salesman is slick if they have a rapid and persuasive way of talking - it's not a good thing. Nonetheless you may be the first person in the world to say a proof is "sleek"; people say a "slick proof" as a kind of compliment, though perhaps with an undertone of "efficient but perhaps not very insightful".

Matteo Capucci (he/him) (Apr 02 2025 at 16:18):

Well then I somehow used it correctly---I do think Nathanael's proof is thin and elegant!

John Baez (Apr 02 2025 at 16:44):

It's correct: it's just the first time anyone has ever said it.