Re: colimits of polynomial functors

Re: colimits of polynomial functors

[Joachim Kock]

Hi Ondrej,

> Does the category of (dependent) polynomial functors and strong
> natural transformation have all/some colimits ?
> In general, what is known about them ?

At the risk of being off the point, I think that the colimits
that exist might not have been studied much because they are often
not the 'right' ones, in a sense.  As an example, the polynomial
functor Set -> Set, X \mapsto X^2 (represented by 1 <- 2 -> 1 -> 1)
has two automorphisms (the identity and the twist), and if I am not
mistaken the identity functor X \mapsto X is the coequaliser of those
two in the category of polynomial functors and their strong natural
transformations (just because 1 is the equaliser of the two set auts
2 -> 2).  The functor that 'ought' to be the coequaliser is of course
X \mapsto X^2/2, which is not polynomial.  (For example it does not
preserve pullbacks.)

(I should add that I understand the question as concerning
polynomial functors and those natural transformations compatible
with the canonical tensorial strengths.  If instead, according to
Paul Taylor's interpretation of the question, only cartesian
natural transformations are allowed, then it is easy to see
that the above pair of (cartesian) natural transformations
does not have a coequaliser.)

Cheers,
Joachim.

PS: allow me to advertise a reference:
[Gambino-Kock, Polynomial functors and polynomial monads, arXiv 2009].
It does not have anything about colimits, but it does say a lot about
strong natural transformations, and in particular characterise them
in diagrammatic terms.


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Re: colimits of polynomial functors

[Joachim Kock]

I just wrote, much too quickly:

> At the risk of being off the point, I think that the colimits
> that exist might not have been studied much because they are often
> not the 'right' ones, in a sense.  As an example, the polynomial
> functor Set -> Set, X \mapsto X^2 (represented by 1 <- 2 -> 1 -> 1)
> has two automorphisms (the identity and the twist), and if I am not
> mistaken the identity functor X \mapsto X is the coequaliser of those
> two in the category of polynomial functors and their strong natural
> transformations (just because 1 is the equaliser of the two set auts
> 2 -> 2).

But the last sentence is of course pure nonsense.  The equaliser of the
two set auts 2 -> 2 is 0, and the conclusion is then that the constant
polynomial functor X \mapsto 1 (represented by 1 <- 0 -> 1 -> 1) is
the coequaliser.

Sorry for the nonsense.  I don't know where I had my head.
I can hardly trust myself anymore, but if this second version is
correct, it still illustrates the point I wanted to make, namely that
the colimit is not the 'right' one.

> The functor that 'ought' to be the coequaliser is of course
> X \mapsto X^2/2, which is not polynomial.  (For example it does not
> preserve pullbacks.)

Cheers,
Joachim.





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colimits of polynomial functors

[Ondrej Rypacek]

Dear all

Does the category of (dependent) polynomial functors and strong
natural transformation have all/some colimits ?
In general, what is known about them ?

Many thanks!
Ondrej


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Re: colimits of polynomial functors

[Paul Taylor]

Ondrej Rypacek asked,

> Does the category of (dependent) polynomial functors and strong
> natural transformation have all/some colimits ?
> In general, what is known about them ?

I studied polynomial functors under the name of "stable" functors
in categorical domain theory between approx 1987 and 1993:
     www.PaulTaylor.EU/stable/

I am guessing that, by "strong" natural transformations you mean
those for which the naturality squares are pullbacks, which I
called "cartesian".

I studied cartesian closed 2-categories whose 1- and 2-cells
are stable functors and cartesian natural transformations.

Yes, there are interesting colimits here, although they are multi-
or poly-valued.  Multi-colimits had been introduced by Yves Diers.
I don't remember who introduced poly-colimits, but these are ones
indexed by groupoids instead of sets.

"Quantitative Domains, Groupoids and Linear Logic" was probably
my most readable paper on this topic.

This kind of domain theory was begun by Gerard Berry and popularised
by Jean-Yves Girard.   In its categorical form, Francois Lamarche
also did work of the same kind as mine, except with a weaker
notion of "cartesian" that had been introduced by Andre Joyal.

Paul Taylor





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Re: colimits of polynomial functors

[Thorsten Altenkirch]

On 31 Jan 2011, at 15:13, Ondrej Rypacek wrote:

> Dear all
> 
> Does the category of (dependent) polynomial functors and strong
> natural transformation have all/some colimits ?
> In general, what is known about them ?
> 
> Many thanks!
> Ondrej
> 
> 
> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]


There are not closed under coequalizers, eg. we can obtained unordered pairs as the coequalizer of id,swap : AxA -> AxA where swao (x,y) = (y,x). To include those one has to move to (what we called) "quotient containers" generalizing analytical functors. 

Thorsten



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Re: colimits of polynomial functors

[Marek Zawadowski]

W dniu 2011-01-31 16:13, Ondrej Rypacek pisze:
> Dear all
>
> Does the category of (dependent) polynomial functors and strong
> natural transformation have all/some colimits ?
> In general, what is known about them ?
>
> Many thanks!
> Ondrej
>

In order to make life simpler, I will assume in this note
that polynomial functors are finitary wide
pullback preserving functors on slices of Set.
There are different ways one might organize polynomial functors.
I like to think that they form a fibration over Set
(see Section 6 of  LMF
http://www.mimuw.edu.pl/~zawado/Papers/MonFib.pdf for details).
Then the fiber over 1 is the category of finitary  wide
pullback preserving endofunctors on Set with cartesian
natural transformations as morphisms. If there are any
limits or colimits of polynomial functors any sense this
this category should have them, as well. But this category
is a Kleisli category and one should not expect much from it
in terms of having limits or colimits.

It goes as follows. The category of (algebraic) signatures
(i.e. just operations, no relations) is equivalent to Set/N.
There is a symmetrizations monad S on it. It takes a signature
A-->N and returns a signature S(A)-->N. For each operation a\in A
over n\in N, S(A)  has operation (a,\sigma) for each
permutations \sigma of {1,..,n}. The Kleisli algebras
for this monad form the category of signatures with non-standard
amalgamations considered by Hermida-Makkai-Power.
This category is equivalent to the category of polynomial functors
described above (see LMF). The Eilenberg-Moore category for this monad
is the category of symmetic (non-colored) signatures considered by
Baez-Dolan.
It is equivalent to the category of analytic functors
(by which I mean here the category of finitary  endofunctors on Set
weakly preserving wide pullbacks with wealky cartesian
natural transformations as morphisms c.f.  A. Joyal,
Foncteurs analytiques et especes de structures, Lecture Notes Math.
1234, Springer 1986, 126-159., see also section 7 of LMF for the colored
version).

Thus if one take (co)limits of polynomial functors one takes
(co)limits of free S-algebras and expect to have as a result
an S-algebra i.e. an analytic functor. Not surpisingly,
most of the time this functor is not polynomial. A particular example
of a coequalizer that is analytic but not polynomial was given
by Torsten and commented by Joachim.


NB. I have been talking about the symmetrization monad in Genova
and last two PSSL's reporting joint work with my student S. Szawiel.
Note that here and in many different places it is important that
this monad S is acting directly on signature not on non-symetric operad.
Some people missed this point in Genova, but it is very important
in the above and in many other places.

Best regards,
Marek


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