Talk:Natural transformation

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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.