Timeline for When does a monoidal functor between ribbon categories preserve cups and caps, but not necessarily braidings?
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| Sep 16, 2014 at 15:48 | comment | added | Manuel Bärenz | It's also possible to understand it like this: In a pivotal category, $X^* =\prescript{*}{}{X}$ canonically, right? But braided categories have a canonical pivotal element, I thought? Then the question is basically "Are there pivotal functors that are not braided?". | |
| Jul 22, 2014 at 16:17 | comment | added | Theo Johnson-Freyd | Right. The strong monoidal functor from supervector spaces to $(\mathbb Z/2)$-modules (with the usual symmetric structure) does not preserve quantum dimension. And any two (right, say) duals are canonically isomorphic, but that isomorphism often is not the identity for some looks-convenient coordinates. | |
| Jul 21, 2014 at 15:15 | comment | added | Manuel Bärenz | Ok, so every strong monoidal functor maps a duality onto a duality, but not necessarily the one I've chosen beforehand in the target category, right? That means that a strong monoidal functor need not preserve quantum dimensions, for example. | |
| Feb 5, 2014 at 18:34 | history | answered | Theo Johnson-Freyd | CC BY-SA 3.0 |