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Stream: theory: category theory

Topic: partially monoidal functors and extensions

Jonathan Beardsley (Dec 21 2023 at 22:57):

In light of my complete failure to prove what seemed like a straightforward thing (cf. #theory: category theory > ✔ pullback in functor 2-category), maybe someone here has an idea. Maybe since people around here like applied category theory, where partially monoidal categories sometimes show up, this will be generally interesting to folks.

The question is the following: Say that $A$ is a symmetric monoidal category and $C\subseteq A$ is a full subcategory containing the unit $1_A$. Then $C$ inherits a partially monoidal structure, and the inclusion functor $i\colon C\hookrightarrow A$ is (strong) partially monoidal. Now suppose I've got some other symmetric monoidal category $B$ and a (strong) partially monoidal functor $F\colon C\to B$. I would like to right Kan extend $F$ and get a (lax) monoidal functor $RF\colon A\to B$. Obviously this might just not be true in general, and I think the linked topic suggests it's not....but maybe there's something simple I'm just not seeing?

Nathanael Arkor (Dec 21 2023 at 23:01):

This is essentially the topic of algebraic Kan extensions, which has been widely studied (though usually one assumes full monoidal structure, rather than partial monoidal structure). For instance, see @Tobias Fritz and @Paolo Perrone's A Criterion for Kan Extensions of Lax Monoidal Functors and the related work section therein.

Nathanael Arkor (Dec 21 2023 at 23:02):

(In general, it is not true that the Kan extension of a monoidal functor along another monoidal functor is monoidal.)

Jonathan Beardsley (Dec 21 2023 at 23:03):

Nathanael Arkor said:

This is essentially the topic of algebraic Kan extensions, which has been widely studied (though usually one assumes full monoidal structure, rather than partial monoidal structure). For instance, see Tobias Fritz and Paolo Perrone's A Criterion for Kan Extensions of Lax Monoidal Functors and the related work section therein.

Thanks. My investigation of this sort of thing seems to always land in people studying left Kan extensions. For instance, there's a paper (I think of Day and Street, or maybe Lack and Street?) looking at left Kan extensions of promonoidal functors, but it says nothing about right Kan extensions.

Jonathan Beardsley (Dec 21 2023 at 23:06):

My question is spelled out in horrifying detail here: https://mathoverflow.net/questions/452394/lax-monoidal-structure-on-the-right-kan-extension-of-a-partially-monoidal-%ce%93-set

Nathanael Arkor (Dec 21 2023 at 23:13):

Ah, I recall now there was an earlier thread.

Jonathan Beardsley (Dec 21 2023 at 23:43):

heh yeah.... god I guess I really am repeating myself. In that situation I thought that one could replace a partially monoidal category with a promonoidal one freely. But @James Hefford and @Aleks Kissinger disabused me of that notion via email.

Jonathan Beardsley (Dec 21 2023 at 23:44):

Given a symmetric monoidal category there's always the induced ``trace promonoidal structure'' on the subcategory, but I sort of think that's not the right thing. Though I'm not sure.

Jonathan Beardsley (Dec 21 2023 at 23:50):

But the other convo with Mike Shulman indicates that pulling back along a strong symmetric monoidal functor isn't strong symmetric monoidal (w/r/t Day convolution) anyway, so there's no way it's going to come from some formal reason (as I hypothesized in my MO post). But maybe one gets lucky if one is actually Kan extending from a subcategory.