Talk:Natural transformation

From Wikipedia, the free encyclopedia

Mathematics Portal This article is within the scope of WikiProject Mathematics, which collaborates on articles related to mathematics.
Mathematics rating:	Start Class	High Priority	Field: Foundations, logic, and set theory
Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. Please add useful comments here--Cronholm144 08:23, 28 May 2007 (UTC)

Would you give an example of a non-natural isomorphism to the article? -- Taku 00:26, 10 November 2005 (UTC)

It looks like it has been added... Marc Harper 15:08, 26 March 2006 (UTC)

An isomorphism between a vector space and its dual V* is perhaps the commonest example of a non-natural isomorphism. Is there a sense in which it is a theorem that there is no natural isomorphism between V and V* ?Daqu 04:46, 21 February 2006 (UTC)

I removed the bit saying it was a 'counterexample'.

metterklume

It's probably not too hard to show that there can be no transformations between functors of different variance. -lethe ^{talk +} 05:25, 21 February 2006 (UTC)

[edit] example for fun

I wrote this example natural transformation for the benefit of a friend in my user space. He suggested that it was well-enough presented to warrant inclusion in this article, or else perhaps in its own article. I don't really think so; it seems to me to be too tutorial-ish to be encyclopedic. I post it on the talk page, and you can decide whether it has a place in this article (or any other).

We want to verify the equation

where τ_C: P(C) → 2^C is the map which sends any subset of the set C to the characteristic function on that subset, i.e.

τ_C(U) = χ_U,

where χ_U is given by

for any subset U ⊆ C and any element c ∈ C. To verify the equation, let both sides act on some subset S ⊆ B. We have

by the definition of the powerset functor, and so

On the right-hand side of the equation, we have

τ_B(S) = χ_S

and recall that f* is the pullback by f induced by the contravariant hom-functor; it acts on maps by multiplication on the right:

So it remains to check the equality

To verify this equation, act both maps in 2^A on an arbitrary element a ∈ A.

Since a ∈ f^–1(S) iff f(a) ∈ S, these maps are equal.

It's better than the "every group is naturally isomorphic to its opposite group"... —Preceding unsigned comment added by 130.126.108.212 (talk) 19:12, 9 November 2007 (UTC)

[edit] natural isomorphism

I removed this paragraph from the article. I think it's wrong, but I don't know enough math to be sure of it. Paisa (talk) 02:23, 21 December 2007 (UTC)

A natural transformation η : F → G is a natural isomorphism if and only if there exists a natural transformation ε : G → F such that ηε = 1_G and εη = 1_F (where 1_F : F → F is the natural transformation assigning to every object X the identity morphism on F(X)).

I don't have CWM handy, but Mac Lane and Moerdijk in Sheaves and Geometry in Logic on page 13 call a natural transformation η : F → G a natural isomorphism if it is componentwise an isomophism, that is, if η_X is an isomorphism for each object X. (This is the definition currently in the article.) The Mac Lane–Moerdijk definition is equivalent to the one above. If there exists a natural transformation ε as above, then the compositional identities imply that η_X and ε_X are inverse morphisms. Thus η is componentwise an isomorphism, hence a natural isomorphism. Conversely, given a natural isomorphism η, define ε by the rule ε_X := η_X^–1 for each object X. One can verify that this definition yields a natural transformation satisfying the compositional identities above. I did not find an explicit comment in Mac Lane–Moerdijk acknowledging this equivalent definition. Michael Slone (talk) 04:09, 21 December 2007 (UTC)