Is there a definition of an infinite dimensional 2-Hilbert space?

Finite dimensional 2-Hilbert spaces have been discussed by Baez in http://arxiv.org/abs/q-alg/9609018 In the more recent paper by Baez, Baratin, Freidel and Wise http://arxiv.org/abs/0812.4969 a notion of infinite dimensional 2-vector space is discussed, building on work by Crane, Sheppeard and Yetter. Is that the current state of knowledge?

Another way of asking this question is the following. If I want to see a field theory that has an infinite dimensional Hilbert space of states as an extended field theory, what kind of objects should I assign to codimension 2 manifolds?

Is there a definition of an infinite dimensional 2-Hilbert space?

Finite dimensional 2-Hilbert spaces have been discussed by Baez in http://arxiv.org/abs/q-alg/9609018 In the more recent paper by Baez, Baratin, Freidel and Wise http://arxiv.org/abs/0812.4969 a notion of infinite dimensional 2-vector space is discussed, building on work by Crane, Sheppeard and Yetter. They also have a few proposals for what an infinite dimensional 2-Hilbert space should be, but "the details still need to be worked out". Is that the current state of knowledge?

Another way of asking this question is the following. If I want to see a field theory that has an infinite dimensional Hilbert space of states as an extended field theory, what kind of objects should I assign to codimension 2 manifolds?