Timeline for Is the tensor product of symmetric pseudomonoids their coproduct?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 4, 2023 at 14:10 | vote | accept | John Baez | ||
| Sep 19, 2020 at 22:06 | comment | added | John Baez | Whoops. You're right. | |
| Sep 18, 2020 at 21:54 | comment | added | Alexander Campbell | @JohnBaez This isn't true. The universal property of a coproduct of objects in a bicategory involves an equivalence of categories (not just groupoids): $Hom(\sum_i A_i,B) \simeq \prod_i Hom(A_i,B)$. However, if the bicategory admits cotensors by the walking arrow, then coproducts are detected at the level of the underlying (2,1)-category. | |
| Sep 17, 2020 at 19:48 | comment | added | John Baez | The definition of coproduct of objects in a bicategory (or 2-category) doesn't mention any non-invertible 2-morphisms, so it doesn't hurt, for this purpose, to assume all the 2-morphisms are invertible. | |
| Sep 17, 2020 at 17:46 | answer | added | John Baez | timeline score: 6 | |
| Sep 16, 2020 at 14:31 | comment | added | Maxime Ramzi | @TimCampion : Yes, I agree, I expect so too; but I definitely don't have enough experience in the $(2,2)$-world to be sure | |
| Sep 16, 2020 at 14:27 | comment | added | Tim Campion | @MaximeRamzi Ah, good point. Although typically I should expect that coproducts in a 2-category can be detected at the (2,1) level, and similarly with the tensor product. | |
| Sep 16, 2020 at 9:26 | comment | added | Maxime Ramzi | @TimCampion : in (1), don't you mean a general (2,1)-symmetric monoidal category ? Which would not be enough if John is interested in general 2-categories with | |
| Sep 15, 2020 at 22:25 | comment | added | Tim Campion | Two ideas: (1) It's surely been shown in general that if $C$ is a symmetric monoidal $\infty$-category, then in $Alg_{E_\infty}(C)$ the coproduct and tensor product coincide. This should specialize to what you want by taking $C$ to be the (2,1)-category of categories (maybe the appropriate comparisons (still!) haven't been done). (2) Perhaps this can be deduced representably from Fong-Spivak: a symm.pseudomonoid in $C$ should be a certain kind of lift of a representable 2-functor $C \to Cat$ through the forgetful 2-functor $SymPsMon \to Cat$, with coproduct and tensor product defined levelwise | |
| Sep 15, 2020 at 20:21 | history | asked | John Baez | CC BY-SA 4.0 |