Generalized 2vector spaces and general linear 2groups
Abstract
In this paper a notion of {\it generalized 2vector space} is introduced which includes Kapranov and Voevodsky 2vector spaces. Various kinds of generalized 2vector spaces are considered and examples are given. The existence of non free generalized 2vector spaces and of generalized 2vector spaces which are non Karoubian (hence, non abelian) categories is discussed, and it is shown how any generalized 2vector space can be identified with a full subcategory of an (abelian) functor category with values in the category ${\bf VECT}_K$ of (possibly infinite dimensional) vector spaces. The corresponding general linear 2groups $\mathbb{G}\mathbb{L}({\bf Vect}_K[\mathcal{C}])$ are considered. Specifically, it is shown that $\mathbb{G}\mathbb{L}({\bf Vect}_K[\mathcal{C}])$ always contains as a (non full) sub2group the 2group ${\sf Equiv}_{Cat}(\mathcal{C})$ (hence, for finite categories $\mathcal{C}$, they contain {\sl Weyl sub2groups} analogous to usual Weyl subgroups of the general linear groups), and $\mathbb{G}\mathbb{L}({\bf Vect}_K[\mathcal{C}])$ is explicitly computed (up to equivalence) in a special case of generalized 2vector spaces which include those of Kapranov and Voevodsky. Finally, other important drawbacks of the notion of generalized 2vector space, besides the fact that it is in general a non Karoubian category, are also mentioned at the end of the paper.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 2006
 arXiv:
 arXiv:math/0606472
 Bibcode:
 2006math......6472E
 Keywords:

 Mathematics  Category Theory
 EPrint:
 35 pages