nLab Kapranov-Voevodsky 2-vector space

A Kapranov–Voevodsky 2-vector space is a kind of 2-vector space, in this case a category equivalent to Vect${}^n$ for some finite $n$. For details, see:

• Mikhail Kapranov and Vladimir Voevodsky, 2-categories and Zamolodchikov tetrahedra equations, in Algebraic Groups and Their Generalizations: Quantum and Infinite-Dimensional Methods, Proc. Sympos. Pure Math. 56, Part 2, AMS, Providence, RI, 1994, pp. 177–259. (pdf)

• Josep Elgueta, A strict totally coordinatized version of Kapranov and Voevodsky’s 2-category $2Vect$. (arXiv)

There is also a more abstract characterization of Kapranov–Voevodsky 2-vector spaces, described here:

• Martin Neuchl, Representation Theory of Hopf Categories, Ph.D. dissertation, University of Munich, 1997.

• David Yetter, Categorical linear algebra—a setting for questions from physics and low-dimensional topology, Kansas State University preprint.

Namely, they are semisimple $k$-linear abelian categories with finitely many simple objects. We may also drop the finiteness condition here to define a class of ‘infinite-dimensional’ Kapranov–Voevodsky 2-vector spaces. For further discussion and more references, see:

• John Baez, Aristide Baratin, Laurent Freidel and Derek Wise, Infinite-dimensional representations of 2-groups. (arXiv)

For representations of 2-groups on Kapranov–Voevodsky 2-vector spaces see:

• Josep Elgueta, Representation theory of 2-groups on finite dimensional 2-vector spaces. (arXiv)