How Natural Isomorphisms Work: Understanding the Basics

[Hurkyl]

Still working on wrapping my head around these things in general...

It strikes me that, unless I'm really flubbing something up, a natural isomorphism T between two functors F,G:A-->B can be regarded itself as a functor B-->B.

For the objects, we just send each object X to the target of the morphism T_X.

For a morphism f:X-->Y, we have:

<br /> Tf := T(X) \stackrel{T_X^{-1}}{\longrightarrow}<br /> X \stackrel{f}{\longrightarrow}<br /> Y \stackrel{T_Y}{\longrightarrow}<br /> T(Y)<br />

So this appears to be a functor... then, of course we have that the evaluation T(F) is simply given by T o F as functors.


As I write this I realize that this can't be a useful interpretation of natural transformations in general, because I had to use the fact the isomorphisms have inverses...

But is it a useful way to think of natural isomorphisms? Or am I just going to mislead and confuse myself and should stop thinking about this immediately?

 

[mathwonk]

let me copy the discussion from my grad algebra book available as a large pdf file to the interested. oops it doesn't copy well here:

Definition: Two (covariant) functors F,G are called "equivalent",
or "naturally equivalent" if there exist isomorphisms ƒM:F(M)-->G(M), one for each module M, such that for each map a:M-->N, the induced maps F(a):F(M)-->F(N) and G(a):G(M)-->G(N) are identified by the isomorphisms ƒM and ƒN. I.e. the following two compositions are equal: (G(a)oƒM) = (ƒNoF(a)):F(M)-->G(N).

Remarks: (a) Equivalence of functors is an equivalence relation.
(b) Equivalence of contravariant functors is defined analogously.
(c) If we drop the requirement that the maps ƒM be isomorphisms, then the resulting family of ƒM is called a "natural transformation" of the functors F,G.
(d) Peter Freyd emphasized that natural transformations are an essential concept, asserting that the only reason we define categories is to define functors, and the only reason we define functors is to be able to define natural transformations.

Example: If a:M-->N is an isomorphism, then the two functors Hom(N,.) and Hom(M,.) are equivalent. I.e. for each X, the map a*om(N,X)¨Hom(M,X), "preceding by a", is an isomorphism, since it has as inverse (a*)-1 = (a-1)*. Secondly, if f:X-->Y is a map, then the associated maps f*om(N,X)-->Hom(N,Y), and f*om(M,X)-->Hom(M,Y) are identified by a*. That is, both compositions (f*oa*)om(N,X)-->Hom(M,X)-->Hom(M,Y), and
(a*of*)om(N,X)-->Hom(N,Y)-->Hom(M,Y), are equal. This is true by associativity of composition, since for g:N-->X, (f*oa*)(g) = fo(goa) = (fog)oa =(a*of*)(g).

Our goal is the following converse assertion:
Theorem: Suppose the functors Hom(N,.) and Hom(M,.) are equivalent, via the isomorphisms ƒXom(N,X)-->Hom(M,X), for all modules X. Then N and M are isomorphic via a unique map a:M-->N such that for all X, ƒX = a*.


I hope I filled in all the missing symbols consistently.

And, oh, a natural transformation is not a functor, it is a map of functors, i.e. a morphism in the category of functors.

The basic example is the family of maps from each vector space V to its double dual V**. this is a transformation from the identity functor to the double dual functor.

 

[Hurkyl]

Yah, I know I'm breaking the type system. But your example (and one that I considered) is the kind of thing I was talking about. The natural transformation between Id and _** gives rise to a functor on vector spaces. (In this case, _** itself) -- in general, it seems any natural isomorphism between two functors F and G with target C gives rise to a functor T on C such that ToF = G.

If I was proficient with this stuff, I'd probably make a claim that sounds like "there's a functor from the subcategory of Funct with natural isomorphisms as arrows to the subcategory of Cat with endofunctors as arrows".

I guess, even if I haven't made a horrible mistake, that I should take away from your response that this isn't a profitable way of thinking of things.

 

[Don Aman]

And, oh, a natural transformation is not a functor, it is a map of functors, i.e. a morphism in the category of functors.

Also, a natural transformation is a 2-morphism in the 2-category of all categories. something I'm still struggling with.

 

[mathwonk]

well Hurkyl what evidence is there for your statement? I.e. F and G may not have as values all objects in C, so why should there be a functor T on all of C that connects them?

it seems to mem also that the natural transformation muses more of the structure of F(X) than just its nature as an object of C.

e.g. take the functors Bil(AxB,.) and Hom(A(tens)B,.).

Now every bilinear map from AxB to M indices a linear map from A(tens)B to M, so there is a natural transformation between these two functors from modules to sets, or even from modules to modules.

But what functor from modules to modules takes every module of bilinear maps, to the corresponding module of linear maps?

as for 2 - morphisms, I have never heard of them, and am not troubled by that fact. that just sounds like excess terminology to me.

i.e. the natural transformations are indexed by X, not F(X).

 

[Hurkyl]

Hrm, I thought the natural transformation was supposed to define a morphism for every object of the target category, whether or not it's in the image of the functors? In any case, you can just pick the identity morphism for all the extra objects.

My functor only works with isomorphism (or things with inverses on the correct side), because I need to invert one of the object morphisms defined by the natural transformation to get the functor...

Saunders-MacLane talks about 2-morphisms and 2-categories... 2-morphisms have two composition rules that satisfy a composition law. Cat is the canonical example, whose 1-morphisms are the functors and 2-morphisms are the natural transformations. They also point out that Adj, is also a 2-category, for what it's worth. John Baez goes on all the time about n-categories in theoretical physics, so at least they have a use somewhere.

 

[mathwonk]

Thank you for the explanation. The reason i am not troubled by not knowing the word, is borne out by the fact that since CAT is the canonical example, it means I already understand the concept, just not the terminology. To me this is a canonical example of BS.

math and science are full of secret handshakes to mystify the uninitiated. I have noticed every concept has a different term in every subject, but there are not that many ideas. students should take heart.

 

[mathwonk]

let me just say that very clever people like you hurkyl, can sometimes figure out ways to force something artificially to be a functor, as perhaps you do above, but the word "natural" in the name should suggest to you that this is not likely to be helpful as a point of view.

I admit that the first other example i thought of, looking at loops in a topological space, first up to homotopy, and then up to homology, (i.e passage from <pi>(1) = "Pi(1)" to H1), is related by the operation of "making a group abelian", which in fact is a functor defined on all groups, not just those that are fundamental groups of top spaces.

In the example of bilinear maps and linear maps on the tensor product I am not so persuaded that it is helpful.

But who knows? see if you can make anything of it. or maybe find more examples where there is a natural extension of the functor to all objects in B, and better yet, an interpretation of how the morphisms behave when things are not isomorphisms.

i.e. in the fundamental group case, the construction was purely group theoretic, but in the bilinear map case it depended on knowing not the module structure of the object but the fact that the object was a module of bilinear maps, I believe. i.e. the construction came really from the original category A, and was not just a construction on B.

 

[matt grime]

Interesting you should bring up homotopy in the same area s 2-categories. This is essentially where 2-cats and n-cats came from.

Just as maps between topological spaces may not be equal, they may be the same up to homotopy.

So there is a category where the objects are spaces, and the maps are toplogical maps, and there are maps of maps, the homotopies.

A 2-cat is a category with objects, maps, and maps of the maps. n-category is that repeated *in some manner*. I emphasize that because it isn't clear at this stage what the *in some manner ought to be*. The reason for this is that when we take maps of maps we automatically think of natural transformations and commuting squares. However, in the higher dimensions (as the n-cat people like to say), it isn't obvious what diagrams (of polytopes, I seem to recall) they ought to look at and what commutativity rules they need. For instance must they commute "on the nose" or "only up to some 'higher homotopy'". The way forward seems to be to define things in succession: cats, then 2-cats, then 3-cats and so on.



There is a nice topological analogy I like, can't remember who showed it to me, that shows this is actually a delicate question: up to what do we require commuting diagrams.

Consider the homotopy category of topolgical spaces with maps homotopy equivalence classes. Then when taking the functor hocolim (just a lim in a triagulated category, this is a fancy way of saying: take a set of objects and maps between them, build a bigger object by taking the union of all the objects and identifying the images of all the maps), different choices of objects and maps in the equivalence classes create different hocolims.

For instance: take the diagram

X <-- Y --> Z

and let X and Z be 2-discs, and Y a circle (1-sphere), and the maps identify Y with the boundary of X and Z. Its hocolim is the 2-sphere.

If we change the diagram up to homotopy we can let X and Z be points, thus the hocolim is also a point, which isn't homotopic to the 2-sphere.