Timeline for Duals and direct summands in an abelian monoidal category
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 21 at 7:48 | comment | added | Maxime Ramzi | In a closed monoidal category, $X$ is dualizable if and only if the canonical map $Y\otimes \hom(X,1) \to \hom(X,Y) $ is an isomorphism for all $Y$ (I am being sloppy about left/right here but there is a way to make this correct), and since this map is natural in $X,Y$, this condition is clearly closed under retracts | |
| Nov 21 at 7:14 | vote | accept | Jannik Pitt | ||
| Nov 21 at 7:14 | comment | added | Jannik Pitt | Ah, I also tried to show it in the idempotent complete case, but got stuck at showing the triangle identified. I'll try again and post it as an answer when I got it. What would the dualisability condition be in a closed monoidal category? I know that the dual of $X$ has to be given by the internal Hom from $X \to 1$, but what else needs to be satisfied? | |
| Nov 19 at 21:59 | history | answered | Maxime Ramzi | CC BY-SA 4.0 |