Interpretation of categorical construction

[Hurkyl]

The other day, I (once again) decided that I simply don't understand natural transformations. (Or functor valued functors, for that matter... which greatly disturbs me because I'm usually quite comfortable with function valued functions)

So, I sat down to try and figure them out, and I stumbled across this...

Suppose I have functors F, G:A→B.

The defining charactersitic is that if η:F→G is a natural transformation, and f:A→B is a morphism of A, then:

      η(A)
F(A) ------> G(A)
 |            |
 |F(f)        |G(f)
 |            |
 V    η(B)    V
F(B) ------> G(B)

Eventually it struck me to write everything in terms of the morphism f and natural transformation η. That is, replace the functors and objects with the appropriate "source" and "target" operations:

                η(src f)
(src η)(src f) ----------> (tgt η)(src f)
     |                      |
     | (src η)(f)           | (tgt η)(f)
     |          η(tgt f)    |
(srg η)(tgt f) ----------> (tgt η)(tgt f)

Where, for example, src f = A, and tgt η = G.

This is a wonderfully symmetric diagram, which eventually made me realize that the objects of A can be regarded as functors from the functor category Funct(A, B) to B, and the morphisms of A would be the corresponding natural transformations.

I.E., there's a functor A→Funct(Funct(A, B), B).

But more interestingly, it seems to suggest that it should make sense to define the product ηf as the above commutative diagram. Then if η is an identity natural transformation (and thus a functor), ηf is simply the evaluation of the functor. Similarly, if f is an identity morphism (and thus an object), then ηf is simply the component of the natural transformation at that object.

Generalizing, it seems to make sense to talk about the "outer product" of two categories A and B.

Such a "category" (call it C) would be composed of the pairs (a, b) where a is a morphism of A and b is a morphism of B. We have two compositions that could possibly be defined: (a, b)(c, b) = (ac, b) and (a, b)(a, c) = (a, bc)

(I'll start writing capital letters for objects, aka identity morphisms)

(A, B) is an object of C.
(a, B) and (A, b) are "morphisms" of this category, the left and the right morphisms. Unless I've made a silly mistake, either on their own would give a category.

But there's a third type of thing in the "category", we have the things of the form (a, b) which are the commutative squares, and they can be composed if they share a side.


It seems to me that this should be something useful, but I've been having trouble coming up with any actual examples that correspond to ordinary, everyday things like groups or vector spaces...

So I'm wondering if anyone else can offer up a useful interpretation of this sort of thing... or maybe give a better definition of what sort of thing the product of a natural transformation and a morphism ought to be.

 

[fresh_42]

I got lost when source and target were given. For example is not a good definition.
A natural transformation of functors is given if it transforms morphisms into morphisms, i.e. if it extends naturally - will say without extra instructions - the function between objects of categories to morphisms of these categories.

It is the same as on category level, only on the first meta-level. As soon as one operates on different levels, it is extremely important to distinguish those levels, preferably by notation, i.e. different alphabets, not source or target.