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Stream: theory: category theory

Topic: Join / Slice for multicategories

Noah Chrein (Aug 10 2024 at 01:06):

Is anyone aware of a join/slice construction for T-multicategories? That is, an operation $\star$ such that

$[M\star M',N]_P \cong [M,N_{/P} ]$ (for $M'\overset{P}\to N)$

Would be useful for a homotopy theory of virtual equipments. There are clear issues that arise when one attempts to do this, and I am trying to resolve them, but just wondering if anyone has done such a thing?

Kevin Carlson (Aug 15 2024 at 20:47):

I’m pretty sure it can’t exist. Let $P: *\to \top$ be the inclusion of the plain multicategory representing objects into the terminal multicategory. Then in the 2-category of plain multicategories the comma object $\top/P$ is $*$, so if also $M=\top$ then the right-hand side of your desired isomorphism is empty while the left-hand side is presumably a singleton.

Noah Chrein (Aug 16 2024 at 16:09):

Interesting counter, perhaps the "adjunction" above doesn't generalize directly from the case of simplicial sets, though I still think something that can be done.

Can you elaborate on $\ast$ and $\top$?
I know $\top$ is $\mathbb N \overset{\text{id}}\leftarrow \mathbb N \rightarrow \ast$.
Is $\ast$ something like $\mathbb N \overset{\text{0}}\leftarrow \ast \rightarrow \ast$?

Kevin Carlson (Aug 16 2024 at 21:11):

$*$ is one object with just an identity morphism, so if I’m following your notation I would shift the left leg of the second span to $1$.

Kevin Carlson (Aug 16 2024 at 21:14):

I think it’s a Real Problem

Noah Chrein said:

Interesting counter, perhaps the "adjunction" above doesn't generalize directly from the case of simplicial sets, though I still think something that can be done.

I think there’s probably a real problem here, myself. Slicing for multicategories is weird. If you slice over a plain object (a multifunctor from $*$) the resulting multicategory has only unary morphisms, for instance. The existence of nontrivial subterminals, in other words, feels like a big difference from categories or from simplicial stuff.

Noah Chrein (Aug 19 2024 at 15:54):

I think this is probably just the issue, the slice, as a comma category in the 2-category of multicategories is probably just the wrong concept to capture over-category. 2-categories don't, apriori, "see" the multicategorical nature of their objects

Noah Chrein (Aug 19 2024 at 15:55):

I've been approaching it internally, trying to generalize a form of Day convolution