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| Format: | Preprint |
| Published: |
2023
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| Online Access: | https://arxiv.org/abs/2305.01630 |
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| _version_ | 1866914914356953088 |
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| author | Wolfson, Jesse |
| author_facet | Wolfson, Jesse |
| contents | Simplicial objects $\mathsf{sC}$ in descent categories $\mathsf{C}$, as introduced by Behrend and Getzler, provide a context in which to study higher stacks. In this note, we extend the construction of the canonical cocycle of a smooth principal $G$-bundle to the context of principal $G$-bundles in $\mathsf{sC}_{/X}$. As an application, we show how this specializes to $\mathsf{C}=\mathsf{Sets}$ to give a streamlined construction of $k$-invariants of reduced Kan complexes. We adapt this to give a similarly streamlined construction of minimal Kan complexes, with the goal of clarifying the role of the axiom of choice; Postnikov towers of minimal Kan complexes provide examples of towers of simplicial principal bundles of the type we consider. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_01630 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On Simplicial Principal Bundles in Descent Categories Wolfson, Jesse Algebraic Topology 55R10, 55R65, 55R91, 55U10 Simplicial objects $\mathsf{sC}$ in descent categories $\mathsf{C}$, as introduced by Behrend and Getzler, provide a context in which to study higher stacks. In this note, we extend the construction of the canonical cocycle of a smooth principal $G$-bundle to the context of principal $G$-bundles in $\mathsf{sC}_{/X}$. As an application, we show how this specializes to $\mathsf{C}=\mathsf{Sets}$ to give a streamlined construction of $k$-invariants of reduced Kan complexes. We adapt this to give a similarly streamlined construction of minimal Kan complexes, with the goal of clarifying the role of the axiom of choice; Postnikov towers of minimal Kan complexes provide examples of towers of simplicial principal bundles of the type we consider. |
| title | On Simplicial Principal Bundles in Descent Categories |
| topic | Algebraic Topology 55R10, 55R65, 55R91, 55U10 |
| url | https://arxiv.org/abs/2305.01630 |