The presentable stable envelope of an exact category
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908902434537472 |
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| author | Nielsen, Marius Winges, Christoph |
| author_facet | Nielsen, Marius Winges, Christoph |
| contents | We prove an analogue of the Gabriel--Quillen embedding theorem for exact $\infty$-categories, giving rise to a presentable version of Klemenc's stable envelope of an exact $\infty$-category. Moreover, we construct a symmetric monoidal structure on the $\infty$-category of small exact $\infty$-categories and discuss the multiplicative properties of the Gabriel--Quillen embedding. For $E$ an Adams-type homotopy associative ring spectrum, this allows us to identify the symmetric monoidal $\infty$-category of $E$-based synthetic spectra with the presentable stable envelope of the exact $\infty$-category of compact spectra with finite projective $E$-homology.
In addition, we show that algebraic K-theory, considered as a functor on exact $\infty$-categories, admits a unique delooping as a localising invariant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_02598 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The presentable stable envelope of an exact category Nielsen, Marius Winges, Christoph Algebraic Topology Category Theory K-Theory and Homology We prove an analogue of the Gabriel--Quillen embedding theorem for exact $\infty$-categories, giving rise to a presentable version of Klemenc's stable envelope of an exact $\infty$-category. Moreover, we construct a symmetric monoidal structure on the $\infty$-category of small exact $\infty$-categories and discuss the multiplicative properties of the Gabriel--Quillen embedding. For $E$ an Adams-type homotopy associative ring spectrum, this allows us to identify the symmetric monoidal $\infty$-category of $E$-based synthetic spectra with the presentable stable envelope of the exact $\infty$-category of compact spectra with finite projective $E$-homology. In addition, we show that algebraic K-theory, considered as a functor on exact $\infty$-categories, admits a unique delooping as a localising invariant. |
| title | The presentable stable envelope of an exact category |
| topic | Algebraic Topology Category Theory K-Theory and Homology |
| url | https://arxiv.org/abs/2506.02598 |