Spectral Mackey functors and equivariant algebraic K-theory (I)
By Clark Barwick. 2014
194 citations recorded by the source corpus.
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 infinity-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 infinity-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 lay the foundations of equivariant stable homotopy theory for profinite groups and to study fully functorial versions of A-theory, upside-down A-theory, and the algebraic K-theory of derived stacks.
Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/6e4e6e6b1c3b6b276f3185e46d5b92994abfe3f9
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Citations in the corpus
- Incomplete Tambara functors (2016, background): “Remark 2.13 Proposition 2.12 holds much more generally: if C is a disjunctive category in the sense of Barwick [1], ie one in which any map to the coproduct splits as a coproduct of maps to the summands, then the same argument given goes through.”
- Nilpotence and descent in equivariant stable homotopy theory (2015, methodology): “[11]), we will need to use the existence of models of certain E∞-algebras in SpG (namely, equivariant real and complex K-theory), even though an ∞-categorical treatment (and new construction) is to appear in Lurie’s forthcoming work on elliptic cohomology (see [43] for a survey).”
- Lax colimits and free fibrations in ∞-categories (2017, methodology): “The twisted arrow ∞ -category, which was originally introduced by Joyal, has previously been extensively used by Barwick [Bar13,Bar17] and collaborators [BGN14], and by Lurie [ A natural definition of (co)ends in the ∞ -categorical setting is then the following.”