[Haskell-cafe] Very freaky
Andrew Coppin
andrewcoppin at btinternet.com
Thu Jul 12 15:36:47 EDT 2007
> AC> I'm still puzzled as to what makes the other categories so magical
> AC> that they cannot be considered sets.
>
> They are just too big. The set of all sets can't exist, you know.
>
That's news.
How come the set of all sets doesn't exist?
> Well, you mentioned that you have some knowledge of group theory, so
> let me give you three examples of adjoint functors (don't worry, it
> won't hurt) - two from the group theory and one related to Haskell.
>
> 1) Consider the category Set of all sets - it's objects are sets and
> it's morphisms (=arrows) are functions between sets. Also, there is a
> category Grp of groups - with groups as objects and homomorphisms as
> morphisms. Then, the trivial mapping G: Grp -> Set, which maps each
> group to it's base set (and each homomorphisms to itself - as a
> function from one base set to another) is a functor (it is called
> "forgetting functor", I guess, since it "forgets" the group
> structure).
>
> There is a very natural functor F: Set -> Grp. Namely, F maps each set
> X to the free group, with generators corresponding to elements of X.
> By definition, each function f:X->H, where H is a group, can be
> extended to the homomorphism f*:F(X)->H. That means that there is a
> natural bijection between functions X->H (more precisely, from X to
> G(H), since these functions aren't related to the group structure on
> H) and homomorphisms F(X)->H:
>
> Set(X,G(H)) ~ Grp(F(X),H)
>
> Here by Grp(H1,H2) I denote the set of morphisms (=homomorphisms) from
> H1 to H2; the same notation is used for Set.
>
> That means exactly that F is LEFT ADJOINT to G (or, equivalently, G is
> right adjoint to F).
>
> Their composition GF is a monad on the category Set. GF(X) is the set
> of all elements of the free group, generated by X. For a in X,
> return(x) is an element of the free group, corresponding to x. And if
> we have a map X->GF(Y) and an element of GF(X), we can remember that
> GF(Y) carries some group structure, so our map is in fact a map from X
> to some group, which extends to a homomorphism from F(X) to F(Y),
> which is a map from GF(X) to GF(Y), which maps our chosen element of
> GF(X) to an element of GF(Y) - that gives us (>>=).
>
That almost made sense most of the way through... but... ooouch... x_x
> 2) There is a category Ab of abelian groups (and homomorphisms). Of
> course, there is a trivial functor G: Ab -> Grp, which maps each
> abelian group to itself. There is also a functor F: Grp -> Ab; it maps
> each group H to it's "abelianization": F(H) = H/[H,H]. F is also left
> adjoint to G: there is a bijection between homomorphisms H -> A, where
> A is abelian, and homomorphisms H/[H,H] -> A. Again, there composition
> GF: Grp -> Grp is a monad (on Grp this time).
>
> There are many other constructions that happen to be adjoint functors;
> and that is a kind of generalization that makes mathematics so useful
> and exciting. These constructions include all kinds of "free"
> structures - free modules, algebras etc.; discrete and codiscrete
> topological spaces, and many others.
>
> 3) Let X be a set. I'll denote here the set of all functions from X to
> Y by X->Y, and the product XxY (small x stands here for the "times"
> sign) by (X,Y), sticking to the Haskell notation. Then the functor
> (X ->): Set -> Set which maps each set Y to the set X->Y, is right
> adjoint to the functor (,X), mapping each set Y to (Y,X): there is a
> bijection between functions from Z to (X->Y) and functions from (Z,X)
> to Y. This bijection is called "currying". The composition of this
> functors is - as always - monad: it maps Y to X->(Y,X). And this is a
> kind of monad we are familiar with: it's the State monad. Summarizing,
> we have the following: State monad exists because of currying.
>
Again... that almost makes some sort of sense... but this is REALLY
making my head hurt!
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