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.
Cole Comfort (Sep 07 2020 at 01:08):Monoidal categories are pseudomonoids in the monoidal bicategory
*-autonomous categories are frobenius pseudomonoid in the monoidal bicategory , with some extra representability conditions.
Does anyone know if bimonoidal categories are pseudobimonoids in a particular monoidal bicategory?
Antonin Delpeuch (Sep 07 2020 at 08:13):My impression is that you can just internalize the definition of a bimonoidal category in a similar way, but because of distributivity you need the ambient product to be cartesian, not just monoidal. In any cartesian bicategory you should be able to define the notion of "rig object" (or "bimonoidal object"), as a structure with two pseudomonoids, one being commutative, with certain 2-cells for distributivity, and a lot of axioms about these 2-cells to get coherence. In Prof, if you require all morphisms involved to be representable, I expect you just get a bimonoidal category. But I am not sure how useful that is, since it is a bit ad hoc!
Antonin Delpeuch (Sep 07 2020 at 08:23):For me, the beauty of the case of *-autonomous categories is that the notion of Frobenius pseudomonoid is relatively concise, and then unpacks into a relatively rich structure. For bimonoidal categories it seems harder to get (especially for coherence).
John Baez (Sep 10 2020 at 23:40):Not ?
Antonin Delpeuch (Sep 11 2020 at 07:25):Hmm yes indeed - I don't remember exactly why we use Prof with representability conditions for *-autonomous categories btw.