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Added reference
Many years ago at the $n$Café I remember discussions on varieties of 2-vector space. The author here writes:
There are other two types of 2-vector spaces, that people might be familiar with. One is the 2-vector space via categorification [BDR04] by Baez, Dundas and Rognes, and the other is Kapranov-Voevodsky’s 2-vector space [Kap99]. Philosophically [2ve13] our 2-vector spaces may be viewed as a sort of unification of these two.
The reference [2ve13] is to 2-vector bundle.
I have added more references to the actual References-section, such as to Kapranov & Voevodsky and to Baez & Crans, but also for instance to the review in BDSPV15.
Then I took the liberty of making the following explicit (now here):
The notion of 2-vector spaces with 2-linear maps between them as algebras with bimodules between them (subsuming the definition in Kapranov & Voevodsky 1991 as the special case of algebras that are direct sums of the ground field) is due to
following earlier discussion in
Urs Schreiber, 2-vectors in Trondheim (2006)
Urs Schreiber, Topology in Trondheim and Kro, Baas & Bökstedt on 2-vector bundles (2007)
which is picked up in
and further developed into a theory of 2-vector bundles (via algebra bundles with bundles of bimodules between them) in:
Essentially the same notion also appears in:
The notion is reviewed in a list of “standard” definitions in BDSPV15, without however referencing it.
When BDSPV15 came out I expressed my surprise to Bruce B. who had been around when I promoted the notion and and knew that people certainly did not regard it as standard for a long time to come. I seem to remember that Bruce agreed to fix this in a revision, but it seems this article was never revised or published.
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