nLab 2-Hilbert space

Idea

The concept of $2$-Hilbert space is supposed to be a categorification of that of Hilbert space, 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 96).

More generally this works for semisimple categories (…)

Related concepts

• Schur's lemma

• 2-vector space

References

See

• John Baez, Higher-dimensional algebra II: 2-Hilbert spaces, Adv. Math. 127 (1997), 125-189. (arXiv)

for one approach, and

• Bruce Bartlett, On unitary 2-representations of finite groups and topological quantum field theory, PhD thesis, University of Sheffield, submitted Oct 2008. (arXiv)

for a refinement of this approach.

A treatment of 3-Hilbert spaces, the further categorification of 2-Hilbert spaces, is in

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