hi john,
i think mike used the example of "monoids in the category of abelian groups" as an example of "algebras in the category of algebras". the original question was about "categories in the category of algebras" and "algebras over categories". that places the
questions in the realm of functorial semantics. functorial semantics has been developed in terms of product-preserving functors, finite-limit preserving functors, etc. caregories are finie-limit preserving functors. there is no version, i think, of categories
that are monoidal functors.
so in the framework of the original question, there doesn't seem to be any ambiguity. "categories of monoids" and "monoids over categories" do not involve tensor products and do not depend on the monoidal structure.
((there is no such thing as "functorial semantics with respect to tensor product". the correspondence between algebras for a monad and product-preserving functors, referred to in the original question, does not lift to a correspondence with monoidal functors...
pawel and i tried to develop a relational version of functorial semantics, and the only references that we could find were two papers by aurelio carboni... and we used some stuff from joyal-street's tannakian categories. didn't find much else and got stuck
on basic questions...))
i guess the facts that monoids as algebras for the monoid monad share the name with algebras in a monoidal category is a terminological clash. a double terminological clash. so much for the hope of categories tidying stuff up :)))
all the best,
-- dusko