G-Global Homotopy Theory and Algebraic K-Theory
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2020
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| _version_ | 1866912236412338176 |
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| author | Lenz, Tobias |
| author_facet | Lenz, Tobias |
| contents | We develop the foundations of $G$-global homotopy theory as a synthesis of classical equivariant homotopy theory on the one hand and global homotopy theory in the sense of Schwede on the other hand. Using this framework, we then introduce the $G$-global algebraic $K$-theory of small symmetric monoidal categories with $G$-action, unifying $G$-equivariant algebraic $K$-theory, as considered for example by Shimakawa, and Schwede's global algebraic $K$-theory.
As an application of the theory, we prove that the $G$-global algebraic $K$-theory functor exhibits the category of small symmetric monoidal categories with $G$-action as a model of connective $G$-global stable homotopy theory, generalizing and strengthening a classical non-equivariant result due to Thomason. This in particular allows us to deduce the corresponding statements for global and equivariant algebraic $K$-theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2012_12676 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | G-Global Homotopy Theory and Algebraic K-Theory Lenz, Tobias Algebraic Topology K-Theory and Homology 55P91, 19D23, 18F25(Primary), 55P48(Secondary) We develop the foundations of $G$-global homotopy theory as a synthesis of classical equivariant homotopy theory on the one hand and global homotopy theory in the sense of Schwede on the other hand. Using this framework, we then introduce the $G$-global algebraic $K$-theory of small symmetric monoidal categories with $G$-action, unifying $G$-equivariant algebraic $K$-theory, as considered for example by Shimakawa, and Schwede's global algebraic $K$-theory. As an application of the theory, we prove that the $G$-global algebraic $K$-theory functor exhibits the category of small symmetric monoidal categories with $G$-action as a model of connective $G$-global stable homotopy theory, generalizing and strengthening a classical non-equivariant result due to Thomason. This in particular allows us to deduce the corresponding statements for global and equivariant algebraic $K$-theory. |
| title | G-Global Homotopy Theory and Algebraic K-Theory |
| topic | Algebraic Topology K-Theory and Homology 55P91, 19D23, 18F25(Primary), 55P48(Secondary) |
| url | https://arxiv.org/abs/2012.12676 |