yeah, something like that, but i think there might be a few subtleties
when you choose colors you're choosing a set of objects that generates under the monoidal product, and that corresponds to choosing some forgetful functors for algebras over the PROP (which are jointly faithful? or something); you might want the choice of colors to be part of the data
i think when people say "colored PROP" the colors are usually a choice of generators under monoidal product, not all of the objects
e.g. the colored PROP describing algebras and modules has two colors, one for the algebra and one for the module, but as a symmetric monoidal category it has countably many objects
i guess one way to say it is, you can think of a colored PROP as a presentation of a strict symmetric monoidal category whose underlying (commutative) monoid is free on the set of colors, and then you add some morphisms in
it's not obvious to me which symmetric monoidal categories are equivalent to a symmetric monoidal category presented in this way. this is independent of the issue about whether a choice of colors is part of the data
There is the additional constraint that Hom(A_1 \otimes ... \otimes A_n, B_1 \otimes ... \otimes B_m) decomposes as a coproduct of products of factors of the form Hom(A_{i_1} \otimes... \otimes A_{i_k}, B_j). For example Vect under tensor product is not the symmetric monoidal category associated to any PROP, because there are maps into B_1 \otimes B_2 which don't arise from a map into B_1 and a map into B_2.
@TimCampion wait is that really true for an arbitrary PROP? that seems like it looks like the category of operators for a multicategory. is every PROP really obtained in that way?
I mean, there should be PROPs that have, for instance, comultiplication operators $m\to (m,a)$ which doesn't seem to fit that framework.
I might be wrong!
The only reason I feel anxious about what you wrote down is that what you described is essentially a collection of trees (though possibly with their trunks braided together in some non-trivial way), but a PROP should include corollas, i.e. vertices with $n$ inputs and $m$ outputs, and I can't see how they would decompose into trees.
Yeah, what I said would be true for symmetric multicategories, but not for PROPs. I think I never actually looked at the definition of a PROP before :)
