Baez and Crans defined a 2-vector space to be a category internal to the category of vector spaces (say over the reals). I am interested in categories that are equivalent to Baez-Crans vector spaces but are not necessarily isomorphic. Do such categories have a name? Do they have a nice description? For example $+$ seem to make such a category into a symmetric monoidal category. What about multiplication by scalars?
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Let T be the algebraic theory of vector spaces (in the sense of Lawvere). Then a T-algebra in the category of small categories is a Baez-Crans 2-vector space. You seem to be asking about pseudo-T-algebras? See arXiv:math/0408298.
I am not sure what you could call these. "pseudo vector space"? "pseudo 2-vector space"?.
A vector space has an underlying commutative monoid, so a pseudo T-algebra gives a pseudo commutative monoid, which is nearly the same as a symmetric monoidal category. However this commutative monoid is not just a monoid but a group. So you will get an Abelian 2-group (which is a symmetric monoidal category in which everything is invertible (objects and morphisms)).
answered Aug 5, 2014 at 14:09
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$\begingroup$ Thanks. I appreciate the suggestions. What do you do with scalar the multiplication? Do you categorify the ground field? and weaken the notion of a module? $\endgroup$ Commented Aug 5, 2014 at 19:20