Linear maps in TQFTs In the axiomatic formulation of TQFTs we assign linear maps between vector spaces attached to smooth manifolds. When the smooth manifolds are inequivalent (i.e. topology changes) we get linear maps between different vector spaces. Are there examples in quantum mechanics where we have linear maps between different Hilbert spaces? Is there a way to append all the Hilbert spaces of a TQFT into  a large, encompassing space and then view the maps as mapping of this space to itself?
 A: Axiomatically, a TQFT is a functor $F$ from a category of corbordisms, $Cob$ to that of hilbert spaces, $Hilb$ (this encapsulates the definition that you gave and more); so what we have is:

$F: Cob \rightarrow Hilb$

Now, the category of corbordism has objects that are closed manifolds - such as the circle or the sphere; and morphisms that are manifolds with two disjoint boundaries, the incoming and the outgoing; for example, a tube, whose incoming boundary is a circle, and whose outgoing manifold is also a circle; its worth noting that there is no topology change here - and in fact it represents an evolution of 'no change'. 
Lets write this morphism as:

$p:S^1 \rightarrow S^1$

then the by the definition of a functor we get:

$Fp: FS^1 \rightarrow FS^1$

Now the functor maps objects of corbordisms to hilbert spaces, thus it maps the circle $S^1$ to some hilbert space, say $H$; so we get: 

$Fp:H \rightarrow H$ 

which is a morphism in $Hilb$, and so is a linear map; and in fact, we can identify it: it represents 'no change', so its simply the identity, $I$.
Another example of a morphism that does have topology change is the so called 'pair of pants'; the incoming boundary is a single circle (the waist of the trousers) and the outgoing is a pair of circles (where the feet pop out of the trousers); what we see here is a circle split into two.
We can write this morphism as $q: S^1 \rightarrow S1 \sqcup S^1$; and the above considerations means we get:

$Fq: F(S1) \rightarrow F(S^1 \sqcup S^1)$

which, because $F$ is a monoidal functor between monoidal categories simplifies to 

$Fq: F(S1) \rightarrow F(S^1) \otimes F(S^1)$

As you can see, the latter part is tensoring two hilbert spaces together; so they're, in a sense, being joined into a larger space.
I don't know whether there are QM systems where one has maps from one hilbert space of states to another - but it wouldn't surprise me if there were; since TQFTs are QFTs, they are also quantum - they arise as a description of a quantisation of 2+1 gravity - so you could take them as a an example of such.
