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I added a note to compact closed category on the fact that the inclusion from compact closed categories into SMCCs has a left adjoint, pointing to an article by Day where he describes the free compact closed category over a closed symmetric monoidal category as a localization. Question: this left adjoint is not full, but I believe it is faithful – does anyone know how to prove that?
actually, I’m not so sure the functor is faithful, because I don’t see what rules out Kelly & Mac Lane’s counterexample from “Coherence in closed categories”: the morphism $[[[A,I],I],I] \to [[[A,I],I],I]$ defined as the composition $[[[A,I],I],I] \to^{[\eta_A,id]} [A,I] \to^{\eta_{[A,I]}} [[[A,I],I],I]$ is not the identity in an arbitrary SMCC $C$, but if I’m not mistaken it must be mapped to the identity in the free compact closed category over $C$. Perhaps my question was not so relevant.
Actually, I wasn’t even sure what you meant. Reading your last comment, I think you may have meant that the unit of the adjunction $M \to U F M$, where $M$ is an smcc, $F$ is the free (2-)functor, and $U$ is the forgetful (2-)functor from compact closed cats to smcc’s, is faithful, and that’s what you’re calling into question now with the Kelly-Mac Lane example. (The comment in #1 made me wonder whether you meant $F$ itself was faithful in some 2-categorical sense.)
Oh, you’re right, that’s what I actually had in mind. And in thinking that the unit $M \to UFM$ was faithful, I was overlooking the Kelly-Mac Lane example.
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