## Abstract

*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 퐾-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 퐾-theory of such objects as spectral Mackey functors. Finally, we employ this technology to introduce fully functorial versions of 퐴-theory, upside-down 퐴-theory, and the algebraic 퐾-theory of derived stacks. We use this to give

what we think is the first general construction of 휋^{ét}* _{1}* -equivariant algebraic 퐾-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.

Original language | English |
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Pages (from-to) | 646-727 |

Number of pages | 60 |

Journal | Advances in Mathematics |

Volume | 304 |

Early online date | 15 Sep 2016 |

DOIs | |

Publication status | Published - 2 Feb 2017 |