Timeline for Inverting objects in a symmetric monoidal category
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 9, 2021 at 18:30 | vote | accept | user374433 | ||
| Nov 8, 2021 at 0:16 | comment | added | Simon Henry | @AlecRhea Yes absolutely, it is not even monoidal. | |
| Nov 8, 2021 at 0:14 | comment | added | Simon Henry | @მამუკაჯიბლაძე : I don't think so as these relations are also satisfied by objects with duals while the collaps of the action of (123) is only automatic for invertible objects. One way to think about it is that if X is invertible, then $X \otimes \_ $ is an equivalence of categories and so $Hom(X,X)$ is equivalent (as a monoid) to $Hom(1,1)$ where $1$ is the unit. In particular, $Hom(X,X)$ has to be commutative. It follows that as $X^{\otimes 3}$ is also invertible the action of $S_3$ on it takes values in a commutative monoids and hence has to factor through the signature map... | |
| Nov 7, 2021 at 19:45 | comment | added | Alec Rhea | I'm confused -- the monoid of scalars on the unit in a monoidal category is always commutative, even if the category isn't symmetric (e.g. lemma 2, page 14 here); how does this mesh with your claim that the monoids of endomorphisms must all be non-commutative? Ah, the colimit you describe must not even be monoidal, got it. My bad. | |
| Nov 7, 2021 at 16:13 | comment | added | მამუკა ჯიბლაძე | Could this (123) condition be also related to the requirement that the composites $Y\otimes X\to Y\otimes X\otimes X^{-1}\otimes X\to Y\otimes X$ and $Y\otimes X^{-1}\to Y\otimes X^{-1}\otimes X\otimes X^{-1}\to Y\otimes X^{-1}$ must be identities? | |
| Nov 7, 2021 at 15:51 | comment | added | Simon Henry | True. I've also added a small explanation of where the (123) condition comes in. | |
| Nov 7, 2021 at 15:51 | history | edited | Simon Henry | CC BY-SA 4.0 |
added 520 characters in body
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| Nov 7, 2021 at 15:30 | comment | added | Maxime Ramzi | The proof of Proposition 6 in Thomas Nikolaus' note on group completion also contains an explanation of this $(123)$-phenomenon | |
| Nov 7, 2021 at 15:18 | comment | added | user374433 | Thank you very much for this wonderful answer! …I look forward to the paper :) | |
| Nov 7, 2021 at 14:47 | history | answered | Simon Henry | CC BY-SA 4.0 |