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  1. 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?

  2. 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][[[A,I],I],I][[[A,I],I],I] \to [[[A,I],I],I] defined as the composition [[[A,I],I],I] [η A,id][A,I] η [A,I][[[A,I],I],I][[[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 CC, but if I’m not mistaken it must be mapped to the identity in the free compact closed category over CC. Perhaps my question was not so relevant.

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 13th 2015

    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 MUFMM \to U F M, where MM is an smcc, FF is the free (2-)functor, and UU 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 FF itself was faithful in some 2-categorical sense.)

  3. Oh, you’re right, that’s what I actually had in mind. And in thinking that the unit MUFMM \to UFM was faithful, I was overlooking the Kelly-Mac Lane example.

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