Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: community: our work

Topic: Magmal characterisations of cocartesian categories

Nathanael Arkor (Aug 18 2025 at 07:55):

I have a new preprint on arXiv: Magmal characterisations of cocartesian categories. Here's the abstract:

We present a survey of characterisations of cocartesian categories in terms of monoidal categories – and, more generally, magmal categories – satisfying additional properties. In particular, we show that the following are equivalent for a unital magmal category $(\mathcal M, \otimes)$ , sharpening several classical characterisations.

• $(\mathcal M, \otimes)$ is cocartesian monoidal.
• Every object of $\mathcal M$ admits the structure of a unital magma with respect to $\otimes$ , such that every morphism is a homomorphism, and a single compatibility condition holds between the magma structures and $\otimes$ .
• The tensor product functor ${\otimes} \colon \mathcal M \times \mathcal M \to \mathcal M$ admits a right adjoint.

This paper is a result of a rabbit-hole I fell down a little while ago concerning characterisations of (co)cartesian monoidal categories that are typically called "Fox theorems". There are many occurrences in the literature of theorems that give sufficient conditions for a monoidal category to be cocartesian. Usually these involve equipping every object with commutative monoid structure such that every morphism is a homomorphism and various compatibility conditions hold.

However, it turns out that many of the compatibility conditions that authors typically impose are redundant. This means that one often does much more work than is necessary to prove a given monoidal category is cocartesian (especially in examples where the tensor product structure is complicated). Conversely (and concerningly), some authors impose fewer assumptions than are needed to guarantee the monoidal structure is cocartesian. Since I was unable to find any references for the most general statement (and a proof that the conditions in the statement were indeed necessary), I decided it would be helpful to write a short article setting the record straight. As far as I know, a "Fox-type theorem" at this level of generality (even in terms of monoidal categories, rather than the magmal categories I work with) has not been written down in the literature, but let me know if I've missed any relevant references!

Kevin Carlson (Aug 18 2025 at 18:02):

Very cool! There's something quite reminiscent of the Eckmann-Hilton argument, which makes two unital magmas distributing over each other into a single commutative monoid. Is there anything more than an analogy there?

Nathanael Arkor (Aug 18 2025 at 18:39):

It's a good question; I had wondered the same, but I didn't manage to make the analogy precise (I'd be glad to hear any thoughts on the topic). In the presence of symmetry, the single compatibility condition is exactly stating that the multiplications of two different magma structures on $A \otimes B$ coincide, which seems related, but a little different from the usual EH set up (where one deduces the two multiplications coincide). And in the absence of symmetry, the single compatibility condition looks even less like the one appearing in the EH argument (though it is derived from the condition for the symmetric case). So the short answer is that I don't know :)

Joe Moeller (Aug 18 2025 at 18:43):

I've always had a sense that Eckmann-Hilton and Fox's theorem were right next door. I'd also be interested in hearing a very nice story connecting the two. I think stabilization should also be an ingredient to the very nice story, which I think is hinted at in Nathanael's answer.

David Michael Roberts (Aug 19 2025 at 03:59):

I humbly point out my paper https://doi.org/10.32408/compositionality-5-8 where I call magmal categories "magmoidal categories", and use a variant that has diagonals. One example that I gave was that of cocommutative cosemigroup objects in a symmetric monoidal category. But I gave some other examples that were more exotic.

Nathanael Arkor (Aug 19 2025 at 07:24):

Thanks, that would be a good reference to add!