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On the regular representation of an (essentially) finite 2-group
dc.contributor.author | Elgueta Montó, Josep |
dc.contributor.other | Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada II |
dc.date.accessioned | 2011-04-19T09:39:19Z |
dc.date.available | 2011-04-19T09:39:19Z |
dc.date.issued | 2010-12-01 |
dc.identifier.uri | http://hdl.handle.net/2117/12409 |
dc.description.abstract | 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. |
dc.format.extent | 30 p. |
dc.language.iso | eng |
dc.relation.ispartofseries | MAII-IR-10-00002 |
dc.rights | Attribution-NonCommercial-NoDerivs 3.0 Spain |
dc.rights.uri | http://creativecommons.org/licenses/by-nc-nd/3.0/es/ |
dc.subject | Àrees temàtiques de la UPC::Matemàtiques i estadística::Àlgebra::Teoria de categories; àlgebra homològica |
dc.subject.lcsh | Categories (Mathematics) |
dc.subject.lcsh | Vector spaces |
dc.title | On the regular representation of an (essentially) finite 2-group |
dc.type | External research report |
dc.subject.lemac | Categories (Matemàtica) |
dc.subject.lemac | Espais vectorials |
dc.contributor.group | Universitat Politècnica de Catalunya. EGSA - Equacions Diferencials, Geometria, Sistemes Dinàmics i de Control, i Aplicacions |
dc.rights.access | Open Access |
local.identifier.drac | 5479939 |
dc.description.version | Preprint |
local.personalitzacitacio | true |
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