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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Bibliographic Details
Published in:Advances in mathematics (New York. 1965) 2017-01, Vol.304, p.646-727
Main Author: Barwick, Clark
Format: Article
Language:English
Subjects:
Equivariant algebraic K-theory
Equivariant stable homotopy theory
Spectral Mackey functors
Unfurling
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Summary: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 coherence 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 introduce fully functorial versions of A-theory, upside-down A-theory, and the algebraic K-theory of derived stacks. We use this to give what we think is the first general construction of π1ét-equivariant algebraic K-theory for profinite étale fundamental groups. This is key to our approach to the “Mackey functor case” of a sequence of conjectures of Gunnar Carlsson.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2016.08.043