G-kernels and Crossed Modules
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arXiv
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| Format: | Preprint |
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2025
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| author | Pacheco, Sergio Girón Izumi, Masaki Pennig, Ulrich |
| author_facet | Pacheco, Sergio Girón Izumi, Masaki Pennig, Ulrich |
| contents | We develop a unified framework based on topological crossed modules for various lifting obstructions for $Γ$-kernels. It allows us to identify actions, cocycle actions and $Γ$-kernels up to their natural equivalence relations with cohomology sets. The obstructions then appear as boundary maps in corresponding exact sequences. Since topological crossed modules are topological $2$-groups (in the categorical sense), they have classifying spaces, which come with a natural transformation from the cohomology to a homotopy set. For the crossed module that gives cocycle actions we prove a weak equivalence of the classifying space of the crossed module with one from bundle theory. In case the algebra is strongly self-absorbing we show that the homotopy set is a group and that the above natural transformation is a group isomorphism on an appropriate restriction of the cohomology set. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_04134 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | G-kernels and Crossed Modules Pacheco, Sergio Girón Izumi, Masaki Pennig, Ulrich Operator Algebras Algebraic Topology 46L35, 46L35, 46L80, 55N20 We develop a unified framework based on topological crossed modules for various lifting obstructions for $Γ$-kernels. It allows us to identify actions, cocycle actions and $Γ$-kernels up to their natural equivalence relations with cohomology sets. The obstructions then appear as boundary maps in corresponding exact sequences. Since topological crossed modules are topological $2$-groups (in the categorical sense), they have classifying spaces, which come with a natural transformation from the cohomology to a homotopy set. For the crossed module that gives cocycle actions we prove a weak equivalence of the classifying space of the crossed module with one from bundle theory. In case the algebra is strongly self-absorbing we show that the homotopy set is a group and that the above natural transformation is a group isomorphism on an appropriate restriction of the cohomology set. |
| title | G-kernels and Crossed Modules |
| topic | Operator Algebras Algebraic Topology 46L35, 46L35, 46L80, 55N20 |
| url | https://arxiv.org/abs/2509.04134 |