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: learning: questions

Topic: Cocartesian multicategories

Dylan Braithwaite (Dec 04 2023 at 00:34):

I'm interested in a structure I would call a 'cartesian comulticategory', or dually a 'cocartesian multicategory'. Essentially I have a 'comulticategory' with a way to combine multimaps $f : A \to \Gamma$, $g : A \to \Delta$ into a multimap $\langle f, g \rangle : A \to \Gamma, \Delta$, and a family of maps with empty codomain $!_A : A \to \epsilon$ such that $\langle !_A, f \rangle = f = \langle f , !_A \rangle$.

I'm wondering if something like this has appeared somewhere in literature about multicategories or polycategories already? I'm aware of cartesian multicategories, but I don't see an obvious way to dualise that to get to this structure, so I'm not sure if cartesian is the right label for this, or if there is a better term for this structure?

Mike Shulman (Dec 04 2023 at 01:09):

If $\Gamma,\Delta$ has the full universal property of a product, so that $\hom(A; \Gamma,\Delta) \cong \hom(A;\Gamma) \times \hom(A;\Delta)$, then it seems to me that the structure would be uniquely determined by its underlying ordinary category, since $\hom(A; (B_1,\dots,B_n)) \cong \hom(A;B_1) \times\cdots \times\hom(A;B_n)$.

Dylan Braithwaite (Dec 04 2023 at 01:28):

Yeah, I definitely don't have a full isomorphism between the hom-sets, which is why I was reluctant to call it cartesian. But requiring the full universal property seems too restrictive since, as you say, such a definition would just be a complicated way to describe an ordinary category

Mike Shulman (Dec 04 2023 at 08:50):

So you have a way to combine maps, but no axioms about that operation? Or what?

Dylan Braithwaite (Dec 04 2023 at 10:27):

Well I wouldn't say no axioms: as I said there's a family of discard maps that act as a unit for the product, and the pairing commutes with composition. In fact I think the only barrier to this satisfying the full universal property is that the function $\mathrm{hom}(A; \Gamma) \times \mathrm{hom}(A; \Delta) \to \mathrm{hom}(A; \Gamma, \Delta)$ is not surjective. So I guess it satisfies a weak universal property in that any operation $\mathrm{hom}(A; \Gamma) \times \mathrm{hom}(A; \Delta) \to \mathrm{hom}(B, \Theta)$ defined using the multicat structure must factor through the product.

Maybe a better analogy is with Markov/semicartesian CD categories in that both have a wide cartesian subcategory in some sense.

Nathanael Arkor (Dec 04 2023 at 10:39):

It sounds like the structure is weakly cartesian in the sense of a [[weak limit]] (i.e. satisfying the existence aspect of a universal property, but not uniqueness).