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Stream: theory: category theory

Topic: Characterization of promonoidal cats among multicats

Kevin Carlson (Oct 16 2025 at 22:54):

Oncentersandlaxcenters-3.pdf
In On centres and lax centres for promonoidal categories, Day, Panchadcharam, and Street consider a multicategory as a category $C$ equipped with profunctors $P_n:C^n\nrightarrow C$ for all $n\ge 0.$ A multicategory arises from a promonoidal category when every multimorphism is "essentially uniquely" a composite of binary and nullary morphisms. In particular, we must have the coend identity $C(-,-)=\int^X P_2(X,-;-)\times P_0(X;),$ so every unary morphism arises by deleting a domain of a binary morphism, in a way that's unique up to unary morphisms in $X.$ You also have on Page 2a coend identity for unary morphisms with the opposite handedness, plus two analogous identities saying that each ternary morphism can be decomposed into two binaries essentially uniquely up to a handedness choice.

Kevin Carlson (Oct 16 2025 at 22:56):

Now, the authors don't write quite what I said about promonoidality; rather, they write, deleting some enrichment decorations "A promonoidal category is a category for which [the four morphisms I mentioned above] are invertible. In this case, $P_n$ is determined up to isomorphism by $P_0,C,P_2.$" I'm not crazy in observing that this is clearly not what they meant to say, right?

Kevin Carlson (Oct 16 2025 at 23:02):

Specifically, I don't think there's any way to decompose the quaternary morphisms just from decompositions of the ternary and unary ones, is there? For instance, suppose you start with a multicategory generated by five unital magmas $a,b,c,d,e$, to get the promonoidality condition on unary morphisms via the magma operations. Now freely add a quaternary morphism $a,b,c,d \to e.$ I have no way to decompose this thing into smaller morphisms.

Kevin Carlson (Oct 16 2025 at 23:04):

Now, this example still isn't promonoidal, even according to DPS, because I can delete one of the domains to find an indecomposable ternary morphism. So I can't either prove or disprove DPS's claim yet. But it seems intuitively false to me. Does anybody know whether the unary and ternary decomposition conditions on a multicategory really suffice to pick out the promonoidals?

Kevin Carlson (Oct 16 2025 at 23:08):

It seems like I could fix the failed counterexample above by starting with a monoid $x,$ then adding an extra quaternary operation $\tau: x,x,x,x\to x,$ then just asserting that all the deletions of $\tau$ along the unit of $x$ coincide with the (correctly decomposable) ternary multiplication of $x.$

Matt Earnshaw (Oct 17 2025 at 07:32):

It's nascent folklore that the statement there is incorrect exactly as you suggest. See section 3/proposition 3.10 of @Mario Román's thesis for a proper treatment of this claim

Nathanael Arkor (Oct 17 2025 at 08:50):

Since I imagine it may be the context in which you came across this claim, it is worth noting that @Claudio Pisani's proof of the characterisation of exponentiable multicategories must be adapted because of this mistake in Day–Panchadcharam–Street (i.e. one must consider pushouts with the free-standing $n$-ary multimorphism for each $n$, rather than just $n \in \{ 0, 2 \}$), though the argument remains the same.

Kevin Carlson (Oct 17 2025 at 20:27):

Yep, that’s right. Thanks for the confirmations, and indeed I had just found my way to Mario’s thesis after posting this yesterday, Matt!