Talk:Natural transformation

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Comment by Taku[edit]

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)

example for fun[edit]

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).

Powersetiso.png


We want to verify the equation

\tau_A\circ \mathcal{P}(f) = f^*\circ\tau_B

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

\tau_C(U)=\chi_U,

where χU is given by


\chi_U(c) =
\begin{cases}
1 & c \in U\\
0 & c \not\in U
\end{cases}

for any subset UC and any element cC. To verify the equation, let both sides act on some subset SB. We have

\mathcal{P}(f)(S) = f^{-1}(S)

by the definition of the powerset functor, and so

\tau_A(\mathcal{P}(f)(S)) =\chi_{f^{-1}(S)}.

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

\tau_B(S) = \chi_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:

f^*(\chi_S)=\chi_S\circ f.

So it remains to check the equality

\chi_{f^{-1}(S)} = \chi_S\circ f.

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


\chi_{f^{-1}(S)}(a) =
\begin{cases}
1 & a\in f^{-1}(S)\\
0 & a \not\in f^{-1}(S)
\end{cases}


(\chi_S\circ f)(a)=
\begin{cases}
1 & f(a)\in S\\
0 & f(a) \not\in S
\end{cases}

Since af–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)

natural isomorphism[edit]

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 : FG is a natural isomorphism if and only if there exists a natural transformation ε : GF such that ηε = 1G and εη = 1F (where 1F : FF 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 : FG a natural isomorphism if it is componentwise an isomophism, that is, if ηX 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 ηX and εX are inverse morphisms. Thus η is componentwise an isomorphism, hence a natural isomorphism. Conversely, given a natural isomorphism η, define ε by the rule εX := ηX–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)

Actually, I think that the definition

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

is better than the one that requires each ηX to be an isomorphism, as it gives the right generalisations to terms such as natural retraction, for which the two mechanisms of definition give non-equivalent concepts. I.e. a natural retraction should be a natural transformation with a right-inverse natural transformation, not a natural transformation η for which each ηX is a retraction (and which retractions may not necessarily fit together to make a right inverse natural transformation, unlike the case with isomorphisms). If nobody objects I may therefore update the article with this definition, pointing out the equivalence with the other definition for isomorphisms, and indicating the generalisations to retraction, coretraction, etc. Rfs2 (talk) 09:38, 7 October 2011 (UTC)

Natural map vs transformation[edit]

I think there's an informal notion of natural map that's not a natural transformation. I posted a section in Talk:Eilenberg-Steenrod axioms. Can someone explain how exactly the connecting homomorphism is a transformation? Money is tight (talk) 05:50, 12 January 2011 (UTC)

opposite group of abelian group[edit]

For all abelian groups X, Y and Z we have a group isomorphism
Hom(X Y, Z) → Hom(X, Hom(Y, Z)). These isomorphisms are "natural" in the sense that they define a natural transformation between the two involved functors Ab × Abop × AbopAb.

What is the opposite group of an abelian group? In line with the first example, it should be the group itself with "reversed order" operation, i.e., the abelian group itself without modification to its group operation. But then the "op" markings are unecessary and misleading. --217.253.231.134 (talk) 09:14, 23 January 2012 (UTC)

Is dual really a counterexample?[edit]

Admittedly, this is far from my expertise; I am hoping someone will enlighten me. I looked at the MacLane and Birkhoff book, secion 6.4, and didn't see any detailed discussion of the assertion that V and V* are not naturally isomorphic.

I thought the reason we say that V and V** are naturally isomorphic is that if C is the category of finite-dimensional vector spaces over K, then there's a functor F: C --> C satisfying two properties: (1) F(V) = V** for all V in C (2) F is naturally isomorphic to the identity functor.

Why does the following not work: define, for each V, an isomorphism phi_V: V --> V*. Now define a functor F: C --> C by F(V) = V*, and for a linear map psi: V --> W, F(psi) = phi_W o psi o phi_V^-1. I think this is a functor, and (1) F(V) = V* for all V in C (2) F is naturally isomorphic to the identity functor, via the system of isomorphisms that we've chosen.

So why do we not say V and V* are naturally isomorphic? My procedure "requires choices," so it may be aesthetically unappealing, but what's the rigorous mathematical reason it's not allowed? Kier07 (talk) 22:14, 23 April 2012 (UTC)

  • What you're missing is that the functor F isn't the dual-space functor; if psi: V --> W, then F(psi): V* --> W*; but the dual-space functor instead gives a psi*: W* --> V*. By contrast, the double-dual-space functor G has G(psi) = psi**: V** --> W**.

Obscure wording in definition[edit]

a natural transformation η from F to G associates to every object X in C a morphism ηX : F(X) → G(X) between objects of D, called the component of η at X

Who or what is called "component"? Are objects called the "component", or are all morphisms called "component"? Perhaps it is a synonym of "natural transformation" itself? Sorry, I'm not familiar with the term, so I just cannot make it up from the description. A general request (if you will edit the description). Can you please avoid sentences with long chains of descriptive clauses like this? These make not the easiest to understand content particularly unintelligible. 109.65.104.130 (talk) 14:07, 30 September 2013 (UTC)

The components are morphisms. I agree that the wording is ambiguous, though it's not obvious how to best fix it. I've updated it in https://en.wikipedia.org/w/index.php?title=Natural_transformation&diff=610072841&oldid=609455728 at the cost of more verbosity, but I think it helps clarify matters (see "Style — Towards clarity and grace" for an analysis of the tradeoffs). --Blaisorblade (talk) 14:02, 25 May 2014 (UTC)

Definitions of horizontal composition[edit]

The article contains no definition of horizontal composition, nor does it state the interchange law. I think there are two definitions — for instance they're given in this blog post, though in form of Haskell code and one of them is incorrectly called vert, for vertical composition. I think they're equivalent by naturality, but I've not seen a proof. In more category-theoretic term, the two definition of a \circ_0 b : G'F' \to GF (with a: G' \to G, b: F' \to F natural transformations) would be

a \circ_0 b : G'F' \to GF
(a \circ_0 b)_X = a_{FX} \circ G'(b_X) = G(b_X) \circ a_{F'X}

And the interchange law (with \circ_0, \circ_1 being respectively horizontal and vertical composition, as in 2-category) — see the diagram in the linked post:

(a \circ_1 b) \circ_0 (a' \circ_1 b') = (a \circ_0 a') \circ_1 (b \circ_0 b')

But I am not entirely sure whether the above are correct, much less a source for them. (I've typechecked them in Haskell, https://github.com/Blaisorblade/CategoryTheory/blob/master/code.hs#L22, and Haskell's type system tends to guarantee that a function with the right type has the right behavior — the type of natural transformation between f and g is forall a. f a -> g a, and it is all is needed to enforce naturality, for instance). --Blaisorblade (talk) 14:36, 25 May 2014 (UTC)