There is a spoken advice to capture with the pawns always towards the center. This situation let me think for a while. Is undoubling here not better?
In this particular position both ...ab6 and ...cb6 are reasonable.
Generally speaking I would prefer to take toward the center because the doubled pawns are not weak at all when they have neighbours and the enhanced control of the center might prove useful.
Here however ab6 has the minus that it allows White counterplay thanks to the outside passed pawn after a4-a5.
The evaluation of this position will depend on Black breaking through or not on the kingside, with ...Rf3 or ...g4. My intuition is that White should be able to hold. If that's correct, the best choice might very well be to limit counterplay with 1...cb6. But once again, this is merely due to the specifics of this particular position, and not because undoubling would be better than taking toward the center in general.
Here all is about whether white a-pawn will be strong passed pawn or weakness on semi open file. Possible undoubling is just a side effect of more important consideration. If you can blow the kingside, make your center as solid as possible and wait a3 will become a serious weakness with white pieces tied to defense. If you can't, you will have to deal with huge distraction problem on the a-file, forcing you to leave attacking position. My first idea was clearly cxb6 but it's better to ask the engine :-)
I think Olivier is right to prefer axb6. The drawback of allowing White a potential passed pawn is not serious, unless it occurs at some late stage of an endgame. Until then it is just a backward pawn on an open file. But Black also has a great position on the other wing. A Fischer block with Rf3 must be taken seriously.
In the general situation, you should ask, why are doubled pawns weak? Sometimes because they cannot protect each other, sometimes because the square ahead is open to blockade. Neither applies here, but they would exert great control over the center. Whether you are playing with doubled pawns or against them, you have to "accentuate the positive". Remember that anything that cannot be attacked is not weak.