This question refers to the discussion here: https://math.stackexchange.com/questions/1084883/chomsky-feynman-thom
I will try the divide-and-conquer strategy to try to make some progress in the problem dealt with in the above-mentioned link.
My question is then a very straightforward one. Let us consider a double potential well such as the first figure in the following link (it is crucial that the depth of both wells be the same) http://inspirehep.net/record/862281/plots
We assume an object, particle, whatever, to be in one of those two stable positions. We want to get to a state where the particle ends up in the other stable position and stays there. From a geometrical or topological point of view it is quite obvious how we get there. The question is: How would you formulate the whole thing algebraically, be it with abstract algebra, with category theory, whatever?