Re: categories of models of cartesian PROPs

[Aleks Kissinger <aleks0@gmail.com>]

Some of this is now superseded by Richard's remarks, but I'll repost
nonetheless, since it offers a slightly different perspective...


The functor category is indeed symmetric monoidal. In fact, its really
only the monoidal structure on C that plays a role (by analogy to your
pointwise multiplication example, where you only need structure on the
codomain).

To ensure the functor is monoidal, the symmetry from C comes in to play:

(F @ G)(X @ Y) := F(X @ Y) @ G(X @ Y)
~= FX @ FY @ GX @ GY
~= FX @ GX @ FY @ GY
~= (F@ G)(X) @ (F @ G)(Y)

Probably more interesting is that, since the base category is
cartesian, so too is the functor category. Each object is endowed with
natural "copy" and "delete" maps, which are again natural in [T,C].
This might seem surprising, but being a monoidal natural
transformation is actually pretty restrictive when it comes to PROPs.
E.g. for frobenius algebras, it forces the map to be iso.


On Wed, Aug 19, 2015 at 3:13 PM John Baez <baez@math.ucr.edu> wrote:
>
> 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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