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Mike Shulman (Apr 05 2022 at 18:35):Is there an established name for a compact closed category in which every object is self-dual? (I suspect that unlike ordinary compact closedness this is additional structure on a symmetric monoidal category.)
Cole Comfort (Apr 05 2022 at 19:07):In the literature this is often called a "self dual compact closed category".
I think this terminology is justified as opposed to merely saying a compact closed category with self dual objects because the transpose becomes a dagger functor, thus it is actually isomorphic to its opposite category. However, I don't know if a self dual category which happens to be compact closed is automatically "self dual compact closed" in this sense.
Oscar Cunningham (Apr 05 2022 at 20:04):Cole Comfort said:
However, I don't know if a self dual category which happens to be compact closed is automatically "self dual compact closed" in this sense.
The category of dualizable (i.e. finitely-generated and projective) modules over a ring is equivalent to its own opposite category, but it's not always the case that the dual of a module is isomorphic to itself (although I don't have any good examples of this in my head).
Mike Shulman (Apr 05 2022 at 21:12):Any compact closed category is equivalent to its opposite by the dualization functor, and by fattening up the category you can make that equivalence into an isomorphism. So I don't think "self dual compact closed category" is a good name. I had thought of "locally-self-dual compact closed category".
Mike Shulman (Apr 05 2022 at 21:12):(By analogy with "locally presentable category", whose objects are presentable.)
Nathanael Arkor (Apr 06 2022 at 01:24):"Locally" is one of the most egregiously overloaded terms in category theory. That said, I don't have an alternative suggestion for the moment :)