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?

]]>