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    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeAug 22nd 2022

    Created stub

    v1, current

    • CommentRowNumber2.
    • CommentAuthorGuest
    • CommentTimeAug 22nd 2022

    does the given definition violate the principle of equivalence?

    • CommentRowNumber3.
    • CommentAuthormaxsnew
    • CommentTimeAug 22nd 2022

    Yes, but see the article identity on objects functor for a discussion.

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeAug 29th 2022
    • CommentRowNumber5.
    • CommentAuthorRodMcGuire
    • CommentTimeAug 29th 2022
    • (edited Aug 29th 2022)
    • A functor return:𝒱𝒞return : \mathcal{V}\to\mathcal{C} that is the identity on objects and preserves central morphisms.

    “central morphisism” really needs to be defined here. I checked the Stanton & Levy paper, http://www.cs.ox.ac.uk/people/samuel.staton/papers/popl13.pdf where they define it.

    If tt commutes with all uu, then we say that tt is central. In other words, tt is central if it doesn’t matter when it is executed. A premulticategory is a multicategory if all morphisms are central. (Lambek [21] calls this the axiom of commutativity.)

    This makes the nLab definition somewhat confusing. I guess it is just saying the returnreturn functor maps all morphisms of 𝒱\mathcal{V} to central morphisms of 𝒞\mathcal{C}.

    ( the referenced Lambeck paper can be found as the first reference on our Joachim Lambek page. )

    ( this distinction between multicategory and premulticategory seems to be the defining difference and really should be mentioned somewhere )