A convenient setting for equivariant higher algebraic K-theory

Dress, Andreas; Kuku, Aderemi O.

doi:10.1007/bfb0062166

Cited by 9 publications

(11 citation statements)

References 4 publications

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“…Aderemi Kuku, extending work of Andreas Dress [5], has worked for many years on the foundations of equivariant higher algebraic K-theory as a construction that yields ordinary abelian group-valued Mackey functors (partly joint with Dress) [6,7,19,20,21,22,23]. For Mackey functors for finite groups, our work can be understood as lifting the target of Kuku's constructions to the ∞-category of spectrum-valued Mackey functors.…”

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confidence: 99%

Spectral Mackey functors and equivariant algebraic K-theory (I)

Barwick

2017

Advances in Mathematics

100

132

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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 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 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.

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confidence: 99%

Spectral Mackey functors and equivariant algebraic K-theory (I)

Barwick

2017

Advances in Mathematics

100

132

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“…Together with (6.9) and (6.10), this completes the proof of Theorem 6.7, and Theorem 6.5 follows. 7. The E ∞ operads V G , V × G , and W G The operad P G has a privileged conceptual role, but there are other categorical E ∞ G-operads with different good properties.…”

Section: Proofmentioning

confidence: 99%

“…To topologize E U G (X), give U and X disjoint basepoints * . 7 View the set Ob of objects of E U G (X) as the set of based functions p : U + −→ X + such that p −1 ( * ) is the complement of a finite set A ⊂ U . Topologize Ob as a subspace of X U+ + .…”

Section: Multiplicative Actions On E Umentioning

confidence: 99%

“…A recognition principle for naive G-spectra, with G not necessarily finite, is given in §2. 7. An interesting detail there shows how to use the recognition principle to construct change of universe functors on the space level.…”

Section: Introductionmentioning

confidence: 99%

“…It is the equivariant analogue of Quillen's original definition in terms of the plus construction. With essentially the same level of generality, the analogue of Quillen's definition in terms of the Q-construction has been studied by Dress and Kuku [7,19]. Shimada [46] has given an equivariant version of Quillen's "plus = Q" theorem in this context.…”

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confidence: 99%

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Equivariant iterated loop space theory and permutative G–categories

Guillou

May

2017

Algebr. Geom. Topol.

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We set up operadic foundations for equivariant iterated loop space theory. We start by building up from a discussion of the approximation theorem and recognition principle for V -fold loop G-spaces to several avatars of a recognition principle for infinite loop G-spaces. We then explain what genuine permutative G-categories are and, more generally, what E∞ G-categories are, giving examples showing how they arise. As an application, we prove the equivariant Barratt-Priddy-Quillen theorem as a statement about genuine G-spectra and use it to give a new, categorical, proof of the tom Dieck splitting theorem for suspension G-spectra. Other examples are geared towards equivariant algebraic K-theory.

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Mackey structures on equivariant algebraic K-theory

Shimakawa¹

1991

K-Theory

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A convenient setting for equivariant higher algebraic K-theory

Cited by 9 publications

References 4 publications

Spectral Mackey functors and equivariant algebraic K-theory (I)

Spectral Mackey functors and equivariant algebraic K-theory (I)

Equivariant iterated loop space theory and permutative G–categories

Mackey structures on equivariant algebraic K-theory

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