[[Sent to Haskell-café, and to comp.lang.fun]] 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 <Vincent.Danos@pps.jussieu.fr> 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.
================================== 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