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 Citació: Elgueta, J. On the regular representation of an (essentially) finite 2-group. "Advances in mathematics", 01 Maig 2011, vol. 227, núm. 1, p. 170-209. Títol: On the regular representation of an (essentially) finite 2-group Autor: Elgueta Montó, Josep Data: 1-mai-2011 Tipus de document: Article Resum: The regular representation of an essentially finite 2-group $\mathbb{G}$ in the 2-category $\mathbf{2Vect}_k$ of (Kapranov and Voevodsky) 2-vector spaces is defined and cohomology invariants classifying it computed. It is next shown that all hom-categories in $\mathbf{Rep}_{\mathbf{2Vect}_k}(\mathbb{G})$ are 2-vector spaces under quite standard assumptions on the field $k$, and a formula giving the corresponding "intertwining numbers" is obtained which proves they are symmetric. Finally, it is shown that the forgetful 2-functor ${\boldmath$\omega$}:\mathbf{Rep}_{\mathbf{2Vect}_k}(\mathbb{G})\To\mathbf{2Vect}_k$ is representable with the regular representation as representing object. As a consequence we obtain a $k$-linear equivalence between the 2-vector space $\mathbf{Vect}_k^{\mathcal{G}}$ of functors from the underlying groupoid of $\mathbb{G}$ to $\mathbf{Vect}_k$, on the one hand, and the $k$-linear category $\mathcal{E} nd({\boldmath$\omega$})$ of pseudonatural endomorphisms of ${\boldmath$\omega$}$, on the other hand. We conclude that $\mathcal{E} nd({\boldmath$\omega$})$ is a 2-vector space, and we (partially) describe a basis of it. ISSN: 0001-8708 URI: http://hdl.handle.net/2117/12492 DOI: 10.1016/j.aim.2011.01.016 Apareix a les col·leccions: Altres. Enviament des de DRACEGSA - Equacions Diferencials, Geometria, Sistemes Dinàmics i de Control, i Aplicacions. Articles de revistaDepartaments de Matemàtica Aplicada. Articles de revista Comparteix: