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    • CommentRowNumber1.
    • CommentAuthormetaweta
    • CommentTimeDec 23rd 2021
    • (edited Dec 23rd 2021)

    The term “linear category” currently redirects to “algebroid”, but Benton uses the term for a very different idea. For Benton, a linear category is

    • a symmetric monoidal closed category (L,,I,)(L, \otimes, I, \multimap) and

    • a symmetric monoidal comonad (!,ε,δ,q)(!, \epsilon, \delta, q) on LL equipped with

    • monoidal natural transformations e,de, d with components e A:!AIe_A:!A \to I and d A:!A!A!Ad_A: !A \to !A \otimes !A

    such that

    • each (!A,e A,d A)(!A, e_A, d_A) is a commutative comonoid,

    • e Ae_A and d Ad_A are coalgebra maps, and

    • all coalgebra maps between free coalgebras preserve the comonoid structure.

    He showed that this idea is equivalent to a symmetric monoidal adjunction between a symmetric monoidal closed category and a cartesian closed category.

    Is Benton the only one who uses the term that way?

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 26th 2022

    I expect he isn’t the only one, but I’d have to delve into the literature to be sure. I was glancing at Benton’s work not long ago.

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