Re: categories of models of cartesian PROPs

["Fred E.J. Linton" <fejlinton@usa.net>]

You ask,

> Suppose you have a category with finite products, say T, and a symmetric
> monoidal category, say C.   Let [T,C] be the category where
> 
>     objects are symmetric monoidal functors from T to C,
>     morphisms are monoidal natural transformations.
> 
> ...
> 
> Should [T,C] also have some sort of "comultiplication"?  What extra
> benefits do we get from T being cartesian?

Not entirely analogous, but the fundamental group of an Abelian group C
is just an Abelian group ... no comultiplication, in general, even though
the circle (the counterpart of T) is both a comonoid (one of the reasons 
[T,C] gets a monoid structure) and a monoid. Makes me wonder about your
question.

Cheers, -- Fred



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