That was one of the original points of intersection at the $n$Café, your interest in 2-vector spaces and mine in Klein 2-geometry, as here.

I see there’s me a little earlier chipping in here at The String Coffee Table to your post.

]]>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

- Urs Schreiber, §A of:
*AQFT from n-functorial QFT*, Commun. Math. Phys.**291**(2009) 357-401 [arXiv:0806.1079, doi:10.1007/s00220-009-0840-2]

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

- Urs Schreiber, Konrad Waldorf, §4.4
*Connections on non-abelian Gerbes and their Holonomy*, Theory Appl. Categ.,**28**17 (2013) 476-540 [arXiv:0808.1923, tac:28-17]

and further developed into a theory of 2-vector bundles (via algebra bundles with bundles of bimodules between them) in:

- Peter Kristel, Matthias Ludewig, Konrad Waldorf,
*The insidious bicategory of algebra bundles*[arXiv:2204.03900]

Essentially the same notion also appears in:

- Daniel S. Freed, Michael J. Hopkins, Jacob Lurie, Constantin Teleman, §7.1 with Ex. 2.13 in:
*Topological Quantum Field Theories from Compact Lie Groups*[arXiv:0905.0731]

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.

]]>Added reference

- Zhen Huan,
*2-Representations of Lie 2-groups and 2-Vector Bundles*(arXiv:2208.10042)

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.

]]>Removed the linking to Forrester-Barker as he is no longer active in mathematics and the grey link was not useful. The link to his PhD thesis remains.

]]>Correction of non-mathematical typo

Anonymous

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