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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 $0$ 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?
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.
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.)
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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