Comments on A Neighborhood of Infinity: Fun with Derivatives of Containers

Duncan, You're right. Not sure how I missed t...

2009-11-09T18:28:40.512-08:00

Duncan,

You're right. Not sure how I missed that.

I just wanted to check my understanding of this st...

2009-11-09T17:50:27.350-08:00

I just wanted to check my understanding of this stuff; but shouldn't the type signature of run be:

run :: (X.F'[X] -> Y) -> F[X] -> F[Y]

?

(i.e. isn't -> right associative?)

cheers,
Duncan

I had my last comment in my head for quite a while...

2007-05-09T01:53:00.000-07:00

I had my last comment in my head for quite a while before I wrote it down. But once again, writing down something helped rethinking it, I see it much more clearly now:

If X.F'[X] is an F[X] with an address, and we only want the address, let's forget the "content" via the map X->1, which induces a map X.F'[X]->F'[1].
A value of type F'[1] represents a position in an F[X] including the shape. For example, if F[X]=X^2+X^3 (pairs and triples), it doesn't just say something like "the first component", but also if it belongs to a pair or triple.

Not sure if that is exactly the "address" you wanted, but it seems to be something a category theorist should be happy with.

You wrote:
(Pity we can't write X.F'[X]-F[X]...

2007-05-08T12:15:00.000-07:00
You wrote:
(Pity we can't write X.F'[X]-F[X] to mean just the address, but '-' makes no sense in this context.)

Since having two information is represented by multiplication (product type), one would rather expect to divide than subtract, giving X.F'[X]/F[X]. If we don't mind just canceling those Xs, that's even useful for tuples. For sum types, we need the fact that the terms are related and have to 'divide' like this: (a+b)/(c+d):=a/c+b/d.
While I have no good justification for this, it works for sums of tuples like Maybe, i.e. 1+X: X.(0+1)/(1+X)=(0+X)/(1+X)=0/1+X/X=0+1=1.

Note also that the function you call projection is not generally onto, its image is what might be called F[X]-F[0].
(Subtraction is useful if it somehow is of the form A+B-A.)

I don't mean Joyal's book, I mean this.

Yeah, Joyal found a nice way to interpret generati...

2006-06-14T16:38:00.001-07:00
Yeah, Joyal found a nice way to interpret generating functions, not just as generating functions, which merely count the size of various structures, but as a functorial representation of the combinatorial objects themselves. Unfortunately Joyal's paper isn't available on the web (and my French isn't too good anyway) and his book is way too expensive (even on ebay) and I don't have easy access to a mathematics library. So most of what I know on this is second hand from John Baez.

There's this paper but it's over my head. (Need to read up on linear logic...)

Much of this stuff originates with Andres Joyal, w...

2006-06-14T16:19:00.000-07:00
Much of this stuff originates with Andres Joyal, when he invented the concept of a species of structures, and showed how you can give a combinatorial interpretation of the calculus of formal power series. It's basically the theory of the category of (objects) finite sets and (arrows) isomorphisms between them.

I desperately want to learn more about this.

Comments on A Neighborhood of Infinity: Fun with Derivatives of Containers

Duncan, You're right. Not sure how I missed t...

I just wanted to check my understanding of this st...

I had my last comment in my head for quite a while...

You wrote:(Pity we can't write X.F'[X]-F[X]...

I don't mean Joyal's book, I mean this.

Yeah, Joyal found a nice way to interpret generati...

Much of this stuff originates with Andres Joyal, w...

You wrote:
(Pity we can't write X.F'[X]-F[X]...