John Baez has written various things about this. Briefly, cobordisms should be thought of in terms of time evolution: you have two manifolds which represent space, and a cobordism between them represents time evolution. Of course to be more physically realistic one should put a Lorentzian structure on the cobordism and make the two manifolds spacelike slices, but I guess the point of the adjective "topological" is to ignore these extra details for the sake of mathematical simplicity.
Then the functor to Vect is supposed to be a simple version of a functor to Hilb (the category of Hilbert spaces) assigning to a manifold the Hilbert space of states on it, and assigning to a cobordism a linear operator representing time evolution. Again, to be more physically realistic one should demand that the operator be unitary and indeed there is a notion of unitary TQFT (but many TQFTs of interest to mathematicians are not unitary).
31 Dec. 2012 確率振幅のみに話を限定してトークをしたことは良くなかったと反省している．他に
返信削除１、境界値
２、内積
３、（外）微分形式
４、Category化との関係(TFTはSymmetric Monoidal Category)
の基本的な話をすべきでした．
4 Aug 2013
返信削除板書の中では、時間発展の定義は正しいが、Riemann構造が入っていないと、おそらく確率振幅が定義できない．位相だけだとrigidであり、局所自由度がない．
John Baez has written various things about this. Briefly, cobordisms should be thought of in terms of time evolution: you have two manifolds which represent space, and a cobordism between them represents time evolution. Of course to be more physically realistic one should put a Lorentzian structure on the cobordism and make the two manifolds spacelike slices, but I guess the point of the adjective "topological" is to ignore these extra details for the sake of mathematical simplicity.
Then the functor to Vect is supposed to be a simple version of a functor to Hilb (the category of Hilbert spaces) assigning to a manifold the Hilbert space of states on it, and assigning to a cobordism a linear operator representing time evolution. Again, to be more physically realistic one should demand that the operator be unitary and indeed there is a notion of unitary TQFT (but many TQFTs of interest to mathematicians are not unitary).