A two dimensional TQFT can be described as a commutative Frobenius algebra by breaking up bordisms along submanifolds of codimension one into a set of generating bordisms. I was wondering if there is a similar way to break up bordisms of any dimension along codimension one submanifolds? 



Generating bordims in TQFTs
András Juhász recently posted a [paper](http://arxiv.org/abs/1408.0668) to the arxiv with a result in this direction. Of course the answer is much much more complicated than the 2-dimensional case, even just in 3-dimensions.


Somewhat nicer answers can be found if you allow "extended TFTs" where you are allowed to continue cutting in lower dimensions. The cobordism hypothesis (see [here](http://www-math.mit.edu/~lurie/papers/cobordism.pdf) and [here](https://www.ma.utexas.edu/users/dafr/cobordism.pdf)) is a particularly nice statement in this setting for TFTs extended all the way down to points. An intermediate example, which is more in the spirit of the Frobenius algebra classification in 2-dimensions, is the Bartlett--Douglas--Schommer-Pries--Vicary classification of 321 TFTs in terms of modular tensor categories ([two](http://arxiv.org/abs/1411.0945) [of](http://arxiv.org/abs/1509.06811) the four papers are posted).


