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nLab: Is one of our definitions of “Cartesian multicategory” incorrect?

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    • CommentRowNumber1.
    • CommentAuthorgoblin
    • CommentTimeFeb 24th 2015
    • (edited Feb 25th 2015)

    If I understand nLab’s definition of a Cartesian multicategory correctly, a one-object Cartesian multicategory is basically the same as a Lawvere theory, despite that the latter usually has infinitely many objects, while the former only has one. So far so good, however, later, the linked page says:

    A cartesian multicategory can also be defined as a category with specified finite products whose set of objects under the “product” operation is a free monoid on specified generators.

    This doesn’t seem consistent with the previous definition, because it seems to require that we have either 00 objects, or infinitely many; so in particular, one-object Cartesian multicategories do not exist under this definition.

    What do you guys think? Is this “definition” mistaken?

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 24th 2015

    The definitions produce equivalent notions, but not literally identical. The “objects” according to the former definition correspond to the “generating objects” under the latter one.

    • CommentRowNumber3.
    • CommentAuthorgoblin
    • CommentTimeFeb 25th 2015
    • (edited Feb 25th 2015)

    Are you sure? What would be an example of a monogenerated Type II cartesian multicategory (the one with specified finite products) that isn’t representable?

    (Lets call them Type I and Type II respectively.)

    • CommentRowNumber4.
    • CommentAuthorSridharRamesh
    • CommentTimeFeb 25th 2015

    Well, “representability” in the sense of having finite products is immediate for Type II cartesian multicategories. It is only a nontrivial notion if we ask whether these finite products of generating objects are themselves generating objects. And to find an example where this fails is easy enough: just use your favorite non-trivial Lawvere theory, which will not represent the empty product (i.e., the terminal object will not be the generating object).

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