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nLab > nLab General Discussions: Reference for a characterization of cartesian monoidal categories

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  1.  
    • CommentRowNumber1.
    • CommentAuthorBruno Stonek
    • CommentTimeApr 13th 2017

    In cartesian monoidal categories, Paragraph 2, a result is stated: “Moreover, one can show that any symmetric monoidal category…”.

    Does anyone have a reference where this result is explicitly stated?

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 13th 2017
    • (edited Apr 13th 2017)

    It could be ancient as folklore, but I think Tom Fox is one name I’d check for an explicit reference to this statement. It might be for example here, but this is behind a paywall for me at the moment. It certainly plays a role in Cartesian Bicategories by Carboni and Walters, so I’d check there too.

  3. It’s in Chris Heunen and Jamie Vicary’s notes on Categorical Quantum Mechanics, for example page 84 of the copy available here. They view it as a form of the no-cloning theorem; if a category of processes has cloning (a diagonal map) then it must be classical (cartesian).

    • CommentRowNumber4.
    • CommentAuthormaxsnew
    • CommentTimeApr 14th 2017

    This is also a theorem that is taken for granted in Programming Languages research, where the syntactic interpretation is that Linear Logic is a refinement of Intuitionistic Logic in that you can translate intuitionistic logic into Linear Logic + weakening (drop) + contraction (duplication), so you may find a citation in early work on linear logic.

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