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http://dx.doi.org/10.18419/opus-11536
Autor(en): | Truong, Monika |
Titel: | Basic representation theory of crossed modules |
Erscheinungsdatum: | 2018 |
Dokumentart: | Abschlussarbeit (Master) |
Seiten: | xvi, 196 |
URI: | http://nbn-resolving.de/urn:nbn:de:bsz:93-opus-ds-115535 http://elib.uni-stuttgart.de/handle/11682/11553 http://dx.doi.org/10.18419/opus-11536 |
Zusammenfassung: | A group corresponds to a topological space with one nontrivial homotopy group. A crossed module corresponds to a topological space with two nontrivial homotopy groups. In classical group theory, Cayley's Theorem constructs for every group G an injective group morphism to the symmetric group S_G. For a crossed module V, we have a similar statement. For every category C, we have the symmetric crossed module S_C. For every crossed module V, we construct an injective crossed module morphism to the symmetric crossed module S_VCat. Suppose given an R-linear category M. On the one hand, we obtain the invertible monoidal category Aut_R(M) by means of category theory. On the other hand, we have the symmetric crossed module S_M as in the Cayley context. In S_M, we have the crossed submodule Aut^CM_R(M) containing only the R-linear elements of S_M. We consider the corresponding invertible monoidal category (Aut^CM_R(M))Cat. We show that there exists a monoidal isofunctor Real_M : (Aut^CM_R(M))Cat -~-> Aut_R(M). This means that starting with M, we obtain essentially the same object via crossed module theory as via category theory. A representation of a group G on an R-module N is given by a group morphism G -> Aut_R(N). Analogously, a representation of a crossed module V on an R-linear category M is given by a crossed module morphism V -> Aut^CM_R(M). We begin to study the representation theory of crossed modules. |
Enthalten in den Sammlungen: | 08 Fakultät Mathematik und Physik |
Dateien zu dieser Ressource:
Datei | Beschreibung | Größe | Format | |
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master_truong_monika_pdfa.pdf | 1,28 MB | Adobe PDF | Öffnen/Anzeigen |
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