@madnight - Actually I was staying well away from the weeds, giving the view from miles up.
There are categories like what you said, but formalizing them requires formalizing some particular string theory well enough to know what these 'states of open strings' are: there should be a vector space of such states. In mirror symmetry people are studying two different string theories, the "A-model" and the "B-model", both of which are quite technical, and claiming that sometimes the A-model on one space is equivalent to the B-model in some other space. People got excited because some physicists used this conjecture to do some astounding computations that mathematicians had been struggling to do for decades.
If you want something much more easy to get into, I immodestly recommend my paper "A prehistory of n-categorical physics":