Timeline for Strictification for closed monoidal categories
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Sep 25 at 15:29 | answer | added | varkor | timeline score: 3 | |
| Sep 21, 2021 at 16:02 | comment | added | Mike Shulman | One possible abstract approach would be to express closed monoidal categories as pseudoalgebras for a 2-monad (on the 2-category of categories, functors, and natural isomorphisms, because there is contravariance) and try to apply a general strictification theorem for pseudoalgebras for 2-monads. | |
| Sep 21, 2021 at 14:37 | comment | added | varkor | @ZhenLin: yes, exactly. | |
| Sep 21, 2021 at 14:37 | comment | added | varkor | @TimCampion: that's an interesting approach. Coherence of the double involution on ∗-autonomous categories gives a coherence result for (symmetric) *-autonomous categories. Perhaps this can be reflected to give one for symmetric closed monoidal categories (it certainly provides further evidence that a coherence theorem holds). | |
| Sep 21, 2021 at 14:09 | comment | added | Zhen Lin | @varkor I suppose you must be doing something like this: take the set of objects of the strictification to be the set of formal expressions generated by the set of objects of the original category and the operations $I$, $\otimes$, and $[-, -]$, modulo the equations you want to hold; prove a normal form theorem for this set; then define the morphisms by "pullback" along the "evaluation" from the set of formal expressions to the original set of objects. This certainly sounds like something that should be an instance of a general theorem... | |
| Sep 21, 2021 at 12:43 | history | edited | varkor | CC BY-SA 4.0 |
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| Sep 21, 2021 at 12:17 | comment | added | Tim Campion | The first thing that comes to mind for me is a theorem of Mike Shulman's: every closed monoidal category embeds fully faithfully into a star-autonomous monoidal category. For a star-autonomous category, I'm tempted to say that some sort of strictification theorem will follow from the usual strictification of monoidal categories... It seems relevant that the star-autonomous structure is almost definable in terms of the monoidal structure. | |
| Sep 19, 2021 at 17:49 | comment | added | Fernando Muro | So, how would you do it for the category of sets? I'm really curious. | |
| Sep 19, 2021 at 17:17 | comment | added | varkor | I'm reasonably sure it is true, and think I can prove it using a syntactic argument, but there are quite a number of conditions to check, and I would hope there was either a more elegant argument, or a proof already in the literature. | |
| Sep 19, 2021 at 15:07 | comment | added | Fernando Muro | I find this unlikely | |
| Sep 19, 2021 at 14:10 | history | asked | varkor | CC BY-SA 4.0 |