Dear John, Caveat lector - this is all rather off the cuff so there may be some mistakes in what follows:

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?*

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?

Yes, this pointwise structure exists; it's part of a monoidal bicategory structure on the 2-category of symmetric monoidal closed categories and symmetric strong monoidal functors. More generally if T is any symmetric pseudomonoidal 2-monad on Cat, then the 2-category of T-algebras and strong morphisms is a monoidal bicategory; see: Hyland, Power "Pseudo-commutative monads and pseudo-closed 2-categories" (2002) In the case you describe it seems that this pointwise structure on StrMon[T,C] is actually cartesian. The point is that, since each t in T is a cocommutative comonoid naturally in T, if F: T--->C is strong monoidal, then each Ft is a cocommutative comonoid, naturally in T. But this means that we have maps F ---> F * F for the pointwise tensor product on StrMon[T,C], making it into a cocommutative comonoid therein. Since these maps are actually natural in F, each object of StrMon[T,C] is naturally a cocommutative comonoid, so that the monoidal structure must be cartesian. There's some details I haven't checked here, because various things need to be strong monoidal functors and transformations, but my immediate impression is that all this should work.

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

For any symmetric monoidal T, it's easy to see that LaxMon[T,C] will inherit any pointwise limits existing in the mere functor category [T,C]. Likewise OplaxMon[T,C] will inherit any colimits. In order for some class of these pointwise limits or colimits to restrict back to StrMon[T,C], it seems that what you need is for the tensor functor C x C ----> C and the unit functor I:1---->C to preserve limits or colimits in that class. In particular, if C had filtered colimits or reflexive coequalisers (or more generally any kind of sifted colimit) and these were preserved by the tensor product functor in each variable separately (which would happen if, for example, C were symmetric monoidal closed) then you could deduce that, actually, C x C ---> C and 1 ---> C preserved them, and so conclude that StrMon[T,C] had filtered colimits / reflexive coequalisers computed pointwise. Dually, if C had coreflexive equalisers preserved by tensor in each variable (for example, if C = Vect) then StrMon[T,C] would again have pointwise coreflexive equalisers. In particular, if T is cartesian, so that we have finite products given by the pointwise tensor product, then in this case we have all finite limits. Richard [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]