A stable $\infty$-category for equivariant $KK$-theory
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866917116558442496 |
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| author | Bunke, Ulrich Engel, Alexander Land, Markus |
| author_facet | Bunke, Ulrich Engel, Alexander Land, Markus |
| contents | For a countable group $G$ we construct a small, idempotent complete, symmetric monoidal, stable $\infty$-category $\mathrm{KK}^{G}_{\mathrm{sep}}$ whose homotopy category recovers the triangulated equivariant Kasparov category of separable $G$-$C^*$-algebras, and exhibit its universal property. Likewise, we consider an associated presentably symmetric monoidal, stable $\infty$-category $\mathrm{KK}^{G}$ which receives a symmetric monoidal functor $\mathrm{kk}^{G}$ from possibly non-separable $G$-$C^*$-algebras and discuss its universal property. In addition to the symmetric monoidal structures, we construct various change-of-group functors relating these KK-categories for varying $G$. We use this to define and establish key properties of a (spectrum valued) equivariant, locally finite $K$-homology theory on proper and locally compact $G$-topological spaces, allowing for coefficients in arbitrary $G$-$C^*$-algebras. Finally, we extend the functor $\mathrm{kk}^{G}$ from $G$-$C^*$-algebras to $G$-$C^*$-categories. These constructions are key in a companion paper about a form of equivariant Paschke duality and assembly maps. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2102_13372 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | A stable $\infty$-category for equivariant $KK$-theory Bunke, Ulrich Engel, Alexander Land, Markus Operator Algebras Algebraic Topology K-Theory and Homology For a countable group $G$ we construct a small, idempotent complete, symmetric monoidal, stable $\infty$-category $\mathrm{KK}^{G}_{\mathrm{sep}}$ whose homotopy category recovers the triangulated equivariant Kasparov category of separable $G$-$C^*$-algebras, and exhibit its universal property. Likewise, we consider an associated presentably symmetric monoidal, stable $\infty$-category $\mathrm{KK}^{G}$ which receives a symmetric monoidal functor $\mathrm{kk}^{G}$ from possibly non-separable $G$-$C^*$-algebras and discuss its universal property. In addition to the symmetric monoidal structures, we construct various change-of-group functors relating these KK-categories for varying $G$. We use this to define and establish key properties of a (spectrum valued) equivariant, locally finite $K$-homology theory on proper and locally compact $G$-topological spaces, allowing for coefficients in arbitrary $G$-$C^*$-algebras. Finally, we extend the functor $\mathrm{kk}^{G}$ from $G$-$C^*$-algebras to $G$-$C^*$-categories. These constructions are key in a companion paper about a form of equivariant Paschke duality and assembly maps. |
| title | A stable $\infty$-category for equivariant $KK$-theory |
| topic | Operator Algebras Algebraic Topology K-Theory and Homology |
| url | https://arxiv.org/abs/2102.13372 |