Re: categories of models of cartesian PROPs

Re: categories of models of cartesian PROPs

[Fred E.J. Linton]

You ask,

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

Not entirely analogous, but the fundamental group of an Abelian group C
is just an Abelian group ... no comultiplication, in general, even though
the circle (the counterpart of T) is both a comonoid (one of the reasons 
[T,C] gets a monoid structure) and a monoid. Makes me wonder about your
question.

Cheers, -- Fred



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categories of models of cartesian PROPs

[John Baez]

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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Re: categories of models of cartesian PROPs

[Richard Garner]

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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Re: categories of models of cartesian PROPs

[John Baez]

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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Re: categories of models of cartesian PROPs

[John Baez]

Hi -

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 that italicized text gets transmogrified here.  I meant:

any one-variable identity (like x^2 = x) obeyed by either C or T will be
> inherited by CommMon[T,C].
>

Best,
jb


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Re: categories of models of cartesian PROPs

[Fabio Gadducci]

Dear John,

do not know if it may be useful, but you can say a few interesting things wrt. models if each object of the prop T has just a co-monoid structure (w/o being natural). In fact, you may e.g. capture partial functions and relational algebras.

Some details in Corradini-Gadducci, A functorial semantics for multi-algebras and partial algebras. TCS 286(2): 293-322 (2002).

Best, Fabio

> On 22/ago/2015, at 03:49, John Baez <baez@math.ucr.edu> wrote:
> 
> Hi -
> 
> 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 that italicized text gets transmogrified here.  I meant:
> 
> any one-variable identity (like x^2 = x) obeyed by either C or T will be
>> inherited by CommMon[T,C].
>> 
> 
> Best,
> jb



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Re: categories of models of cartesian PROPs

[Aleks Kissinger]

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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