Re: [Axiom-developer] Re: Combinat

[Ralf Hemmecke]

> > if we are here dealing already with a sort of pointers (or function
> > arrays) is there already in Aldor/Axiom a 'functor' domain?

Not that I know.

> > I think of a Functor domain in the following sense, it should implement a
> > 'categoy' (in the mathematical sense) of morphisms which opperate on
> > another (pair of) categor(y/ies). The functionality of such a category
> > would be to implement naturality of maps, coherence, composability of
> > morphisms, etc.

> You might be interested in this paper that deals at least in part with
> these issues
>
> Domains of data and domains of terms in AXIOM
> by BROWN, R. & DRECKMANN, W
>
> http://portal.axiom-developer.org/refs/articles/brown-free-c-g.pdf/file_view 
>
> from
>
> http://www.informatics.bangor.ac.uk/public/math/research/preprints/95/algtop95.html 

I am quite disappointed from that paper. It is totally uninteresting 
from my species project point of view. Maybe the other way round, that 
species can be used to construct the directed graphs data structure. But 
that sounds a bit odd if I use functors (species) in order to implement 
categories.

In fact, it isn't. Axiom/Aldor cannot deal with categories and the 
keyword "Category" is a big misnomer. Aldor is better at describing 
universal algebras (or rather the multisorted case). And this is exactly 
what the above paper does. It describes a category by its algebraic 
properties. Unfortunately, all the categories from that paper have only 
finitely many objects. To me that is nothing really useful. For the 
species project I need the category of finite sets and bijections. 
Clearly, that category has infinitely many objects. It even has more 
than uncountably many objects. Take r any real. Then {r} is an object in 
that category.

I must say, I really doubt that any good implementation of a category is 
ever possible. Collecting all the arrows of a category explicitly, makes 
no sense for me. Same with the objects. If you want to compute in a 
category you probably have to do this, but then you are restricted to 
"finitely many".

As you can see in the ToDo of
http://www.risc.uni-linz.ac.at/people/hemmecke/AldorCombinat/combinatsu14.html#x27-320008.1
I did not yet take the time to work out the details of how the 
implementation of a species (i.e. a functor of a special kind) can be 
seen categorially. Note that I have a LabelType. That is a restriction 
that is imposed by the programming language Aldor. I cannot have a List 
of everything but only a List over objects of the same type. Same for a 
species. The labels must all be of the same type. Of course I could form 
unions, but again all objects/labels have to be of the same type.

There is no equivalent in the species theory. A species is a functor F 
that takes a finite set U and returns a finite set F[U]. There is no 
type attached.

> > IFF such a thing would work, species would be an instance, as also lambda
> > rings and symmetric functions ;)

Yes, yes, but don't forget, we have to deal with infinite things. I will 
not even be able to get the full category of natural numbers on my 
computer... ;-)

> I think that is very interesting.

Oh yes, I agree. But that would call for a workshop similar to the one 
we had on symmetric functions. Since I don't work so much with 
categories, it would be interesting to hear what people would like to do 
when the want categories/functors/natural transformations being 
implemented on a computer. To me that sounds much more complicated than 
the symmetric function business. Actually, categories are so simple as 
algebraic structures. But that is not everything.

Ralf