Re: categories of models of cartesian PROPs

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

Thanks to everyone for your replies!

I was surprised to hear that StrMon[T,C] becomes cartesian if T is, because
I usually think of the hom of commutative monoids as inheriting its
multiplicative properties from the target monoid.  But Richard Garner
convinced me it's true.   (Brendan will check, though, if he uses this.)

I was reassured by a decategorified analogue: if T and C are commutative
monoids and we make the set of monoid homomorphism T -> C into a
commutative monoid by pointwise multiplication, any one-variable identity
(like x^2 = x) obeyed by* either C or T*  will be inherited by CommMon[T,C].
It seems identities with more variables only get inherited from C.

Best,
jb


On Thu, Aug 20, 2015 at 8:28 AM, Richard Garner <richard.garner@mq.edu.au>
wrote:

> 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

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