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.
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: