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Stream: theory: category theory

Topic: Indexed monoidal categories over a semicartesian base

Brendan Murphy (Sep 23 2023 at 03:51):

The title is sort of a lie, I really want to think about an opindexed monoidal category on a semicocartesian base. (But I'll be working with the indexed category on a semicartesian base in the rest of this post.)

It seems like you can still construct the "external tensor product" on the total space of the indexed category from the fiberwise ones, although you can't go the other direction anymore. And with this tensor product, the fibration you get via the grothendieck construction is a monoidal fibration. Additionally in my case of interest, the fibers are actually cartesian closed. I'd like to produce a proarrow equipment from this (under some conditions on the fibration, but not the base!). In the paper I'm looking at (Framed bicategories and monoidal fibrations) this requires the base catrgory to be cartesian monoidal. Does anyone know if it's been worked out & written with the weaker condition (semicartesian), assuming there is some way to get it to work?

Brendan Murphy (Sep 23 2023 at 04:01):

My semicocartesian category is naturally a wide (but not full) subcategory of a cocartesian category, and maybe i should be doing everything with that and then restricting my arrows. But I'm still curious about this level of generality

Mike Shulman (Sep 23 2023 at 04:33):

I haven't thought about this before, but it's possible you might get something to work. The original construction uses the diagonals to define composition, taking the external tensor product of $M$ over $A\times B$ with $N$ over $B\times C$ to get $M\boxtimes N$ over $A\times B\times B\times C$, pulling back along the diagonal to $A\times B\times C$ and then pushing forwards to $A\times C$. But if you have the fiberwise products, you could perhaps first pull $M$ and $N$ both back to $A\times B\times C$ (using the semicartesian projections), tensor them fiberwise there, and then push forward to $A\times C$ (again using the semicartesian projections).

Brendan Murphy (Sep 23 2023 at 04:38):

Oh nice, that makes sense! I was indeed getting stuck trying to think about how to define composition, but this at least seems very reasonable in my case of interest

Brendan Murphy (Sep 23 2023 at 04:43):

I'm thinking about constructing a framed double category from a (strict, for simplicity) monoidal category $(\mathsf{M}, \otimes, k)$ via the following process: view $\mathsf{M}$ it as a monoid object of $(\mathsf{Cat}, \times, 1)$ and classify it by a strong monoidal functor $\Delta \to \mathsf{Cat}$. Turn this into an obfibration $\mathsf{E} \to Δ$, observe that this is a monoidal opfibration when we give $\mathsf{E}$ the external tensor product, then contruct a double category from this as in your paper

Brendan Murphy (Sep 23 2023 at 04:44):

Interestingly, I believe $\mathsf{E}$ is the free monoidal category generated by the represented multicategory of $\mathsf{M}$

Brendan Murphy (Sep 23 2023 at 04:45):

I'm probably just going to write down this construction explicitly instead of going through the monoidal fibration machinery, but it's helped me organize my thoughts

Brendan Murphy (Sep 23 2023 at 04:48):

Hm although I think there might be no unit for the composition operation you've suggested, so maybe it's just a virtual proarrow equipment or something

Mike Shulman (Sep 23 2023 at 04:50):

Yes, I don't offhand see any way to get a unit without a diagonal.

Mike Shulman (Sep 23 2023 at 04:51):

If you just want a virtual equipment, then in some ways it's even easier as you don't even need to have an opfibration.

Brendan Murphy (Sep 23 2023 at 04:52):

Well I did initially want an honest double category, it just seems like I might not have one

Brendan Murphy (Sep 23 2023 at 04:53):

I started thinking about this stuff when I noticed that E had a sort of "internal" vs "external" tensor product and it seemed like a sort of double category thing. And I guess it is, since the external tensor product can be constructed from the internal ones as in your paper