nLab 2-Hilbert space

Idea

A concept of $2$-Hilbert spaces is supposed to be a categorification of that of Hilbert spaces, or at least of finite dimensional such: inner product spaces.

One way to define this is as a Kapranov-Voevodsky 2-vector space where the hom-functor plays the role of the categorified inner product (Baez 97).

In other words, a $2$-Hilbert space is an abelian $H^\ast$-category.

More generally this works for semisimple categories (…)

Related concepts

• Schur's lemma

• 2-vector space

References

One approach:

• John Baez, Higher-dimensional algebra II: 2-Hilbert spaces, Adv. Math. 127 (1997) 125-189 [arXiv:q-alg/9609018]

• Bruce Bartlett, On unitary 2-representations of finite groups and topological quantum field theory, PhD thesis, University of Sheffield (2008) [arXiv:0901.3975]

Understanding complete $W^\ast$-categories as 2-Hilbert spaces:

• André Henriques, Nivedita, David Penneys: Complete $W^\ast$-categories [arXiv2411.01678]

• Nivedita: Towards Models for 2-Hilb and 3-Hilb as targets for functorial field theories, talk notes (2024) [pdf]

• Giovanni Ferrer, Lukas Müller, David Penneys, Luuk Stehouwer: The many faces of higher Hilbert spaces [arXiv:2606.11334]

A proposal for further categorification to 3-Hilbert spaces:

• Quan Chen, Giovanni Ferrer, Brett Hungar, David Penneys, Sean Sanford: Manifestly unitary higher Hilbert spaces [arXiv:2410.05120]