categories of models of cartesian PROPs

[John Baez <baez@math.ucr.edu>]

Hi -

Here are two questions:

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.

*1.  What structure beyond a mere category does [T,C] automatically get in
this sort of situation?*

*2.  What further structure do we get if C has some particular class of
limits or colimits?*

I haven't thought about this much.  Even if T were just symmetric monoidal,
I think [T,C] should get a symmetric monoidal structure due to "pointwise
multiplication", just as the set of homomorphisms from one commutative
monoid to another becomes a commutative monoid where

fg(x) := f(x) g(x)

Should [T,C] also have some sort of "comultiplication"?  What extra
benefits do we get from T being cartesian?

Here's why I care:

My student Brendan Fong wrote a masters' thesis about Bayesian networks,
which he's trying to polish up and publish.

In the new improved version, he'll associate to any Bayesian network a
category with finite products, say T.   This plays the role of a "theory".
An assignment of probabilities to random variables consistent with this
theory is a symmetric monoidal functor from T to C, where C is some
symmetric monoidal category - but not cartesian! - category of probability
measure spaces and stochastic maps.  So, [T,C] plays the role of the
"category of models of T in C".

It would be nice to know the properties of [T,C] that follow instantly from
what I've said, not reliant on any more detailed information about T and C.

Best,
jb

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