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 02 2023 at 23:00):In Lack's paper composing props he works in the bicategory of profunctors internal to the category of monoids; regarding distributive laws of props as certain distributive laws of monads in Mon-Prof^op. It seems to me like this is an internal, strict monoidal version of what would be the bicategory of Tamabara bimodules. Can anyone confirm/does anyone have a reference for the (Set-enriched) bicategory of Tambara modules between (nonstrict) monoidal categories?
Nathanael Arkor (Sep 03 2023 at 09:06):In §3.2 of Spivak–Schultz–Rupel's String diagrams for traced and compact categories are oriented 1-cobordisms, they describe a double category of monoidal categories and monoidal distributors/profunctors, which ought to give the non-strict variant of the construction Lack considers.
However, my understanding is that the term "Tambara bimodule" refers to strong distributors rather than monoidal distributors. Paré and Román define strong distributors in §3.5 of Dinatural numbers, though they don't define a bicategory or double category.
Nathanael Arkor (Sep 03 2023 at 09:08):(For what it's worth, I think the terminology "Tambara module" is needlessly confusing/ambiguous, when there is already a systematic naming scheme for such structures. Furthermore, Tambara was not the first to consider such structures, so it is not accurate to name them after him.)
Cole Comfort (Sep 03 2023 at 15:52):Thank you!
Matteo Capucci (he/him) (Sep 12 2023 at 09:14):This was the topic of @Dylan Braithwaite's poster at last CT... I guess me and him rediscovered some of the stuff in the papers Nathanael cited