Spectral Mackey functors and equivariant algebraic K-theory (I)
Spectral Mackey functors are homotopy-coherent versions of ordinary Mackey functors as defined by Dress. We show that they can be described as excisive functors on a suitable ∞-category, and we use this to show that universal examples of these objects are given by algebraic K-theory. More importantl...
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| Format: | Article |
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Elsevier BV
2019
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| Online Access: | http://hdl.handle.net/1721.1/120914 |
| _version_ | 1826190647711760384 |
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| author | Barwick, Clark Edward |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Barwick, Clark Edward |
| author_sort | Barwick, Clark Edward |
| collection | MIT |
| description | Spectral Mackey functors are homotopy-coherent versions of ordinary Mackey functors as defined by Dress. We show that they can be described as excisive functors on a suitable ∞-category, and we use this to show that universal examples of these objects are given by algebraic K-theory. More importantly, we introduce the unfurling of certain families of Waldhausen ∞-categories bound together with suitable adjoint pairs of functors; this construction completely solves the homotopy coherenc e problem that arises when one wishes to study the algebraic K-theory of such objects as spectral Mackey functors. Finally, we employ this technology to lay the foundations of equivariant stable homotopy theory for profinite groups; the lack of such foundations has been a serious impediment to progress on the conjectures of Gunnar Carlsson. We also study fully functorial versions of A-theory, upside-down A -theory, and the algebraic K-theory of derived stacks. |
| first_indexed | 2024-09-23T08:43:33Z |
| format | Article |
| id | mit-1721.1/120914 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T08:43:33Z |
| publishDate | 2019 |
| publisher | Elsevier BV |
| record_format | dspace |
| spelling | mit-1721.1/1209142022-09-23T14:09:02Z Spectral Mackey functors and equivariant algebraic K-theory (I) Barwick, Clark Edward Massachusetts Institute of Technology. Department of Mathematics Barwick, Clark Edward Spectral Mackey functors are homotopy-coherent versions of ordinary Mackey functors as defined by Dress. We show that they can be described as excisive functors on a suitable ∞-category, and we use this to show that universal examples of these objects are given by algebraic K-theory. More importantly, we introduce the unfurling of certain families of Waldhausen ∞-categories bound together with suitable adjoint pairs of functors; this construction completely solves the homotopy coherenc e problem that arises when one wishes to study the algebraic K-theory of such objects as spectral Mackey functors. Finally, we employ this technology to lay the foundations of equivariant stable homotopy theory for profinite groups; the lack of such foundations has been a serious impediment to progress on the conjectures of Gunnar Carlsson. We also study fully functorial versions of A-theory, upside-down A -theory, and the algebraic K-theory of derived stacks. 2019-03-12T13:36:01Z 2019-03-12T13:36:01Z 2017-01 2019-02-11T17:36:32Z Article http://purl.org/eprint/type/JournalArticle 00018708 http://hdl.handle.net/1721.1/120914 Barwick, Clark. “Spectral Mackey Functors and Equivariant Algebraic K-Theory (I).” Advances in Mathematics 304 (January 2017): 646–727. http://dx.doi.org/10.1016/J.AIM.2016.08.043 Advances in Mathematics Creative Commons Attribution-NonCommercial-NoDerivs License http://creativecommons.org/licenses/by-nc-nd/4.0/ application/pdf Elsevier BV arXiv |
| spellingShingle | Barwick, Clark Edward Spectral Mackey functors and equivariant algebraic K-theory (I) |
| title | Spectral Mackey functors and equivariant algebraic K-theory (I) |
| title_full | Spectral Mackey functors and equivariant algebraic K-theory (I) |
| title_fullStr | Spectral Mackey functors and equivariant algebraic K-theory (I) |
| title_full_unstemmed | Spectral Mackey functors and equivariant algebraic K-theory (I) |
| title_short | Spectral Mackey functors and equivariant algebraic K-theory (I) |
| title_sort | spectral mackey functors and equivariant algebraic k theory i |
| url | http://hdl.handle.net/1721.1/120914 |
| work_keys_str_mv | AT barwickclarkedward spectralmackeyfunctorsandequivariantalgebraicktheoryi |