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    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 1st 2012

    I’ve been inactive here for some months now; I hope this will significantly change soon.

    I have written a stubby beginning of iterated monoidal category, with what is admittedly a conjectural definition that aims to be slick. I am curious whether anyone can help me with the following questions:

    • Is the definition correct (i.e., does it unpack to the usual definition)? If so, is there a good reference for that fact?

    • Assuming the definition is correct, it hinges on the notion of normal lax homomorphism (between pseudomonoids in a 2-category with 2-products). Why the normality?

    In other words (again assuming throughout that the definition is correct), it would seem natural to consider the following type of iteration. Start with any 2-category with 2-products C, and form a new 2-category with 2-products Mon(C) whose 0-cells are pseudomonoids in C, whose 1-cells are lax homomorphisms (with no normality condition, viz. the condition that the lax constraint connecting the units is an isomorphism), and whose 2-cells are lax transformations between lax homomorphisms. Then iterate Mon(), starting with C=Cat. Why isn’t this the “right” notion of iterated monoidal category, or in other words, why do Balteanu, Fiedorowicz, Schwänzel, and Vogt in essence replace Mon() with Monnorm() (where all the units are forced to coincide up to isomorphism)?

    Apologies if these are naive questions; I am not very familiar with the literature.

    • CommentRowNumber2.
    • CommentAuthorvarkor
    • CommentTimeJan 15th 2025

    Mention that 2-fold monoidal categories are the same as normal duoidal categories.

    diff, v8, current