If you look at some Web sites (Mathworld, the site of John Baez - a known spec. in algebraic methods in physics), or into some books on differential geometry, you might easily find something which is called pullback or pull-back.
Actually, it is the construction of a dual, whose meaning can be distilled and implemented in Haskell as follows. The stuff is very old, and very well known.
Suppose you have two domains X and Y. A function F : X -> Y. The form (F x) gives some y.
You have also a functor which constructs the dual spaces, X* and Y* - spaces of functionals over X or Y. A function g belongs to Y* if g : Y -> Z (some Z, let's keep one space like this).
Now, I can easily construct a dual to F, the function F* : Y* -> X* by
(F* g) x = g (F x)
and this mapping is called pullback...
While there is nothing wrong with that, and in Haskell one may easily write the 'star' generator
(star f) g x = g (f x)
or
star = flip (.)
... I have absolutely no clue why this is called a pullback. Moreover, in the incriminated diff. geom. books, its inverse is *not* called pushout, but push-forward. Anyway, I cannot draw any pullback diagram from that.
The closest thing I found is the construction in Asperti & Longo, where a F in C[A,B] induces F* : C!B -> C!A where the exclam. sign is \downarrow, the "category over ...".
The diagram is there, a 9-edge prism, but - in my eyes - is quite different from what one can get from this "contravariant composition" above. But my eyes are not much better than my ears, so...
I sent this question to a few gurus, and the answers are not conclusive, although it seems that this *is* a terminologic confusion.
Vincent Danos <[EMAIL PROTECTED]> wrote:
> it really doesn't look like a categorical pullback > and it might well be a "pull-back" only in the sense > that if if F:A->B is a linear map say and f is a linear form on B, then F*(f) > is a linear form on A > defined as F*(f)(a)=f(b=F(a)) so one can "pull back" (linearly of course!) > linear forms on B to linear forms on A > "back" refers to the direction of F, i'd say.
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Does anybody have a different (or any!) idea about that?
Thank you in advance for helping me to solve my homework.
Jerzy Karczmarczuk Caen, France
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