Re: colimits of polynomial functors

[Marek Zawadowski <zawado@mimuw.edu.pl>]

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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