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

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