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Abstract
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We study the “higher algebra” of spectral Mackey functors, which
the first named author introduced in Part I of this paper. In
particular, armed with our new theory of symmetric promonoidal
-categories
and a suitable generalization of the second named author’s Day convolution, we endow the
-category
of Mackey functors with a well-behaved symmetric monoidal structure.
This makes it possible to speak of
spectral Green functors for any operad
.
We also answer a question of Mathew, proving that the algebraic
-theory of
group actions is lax symmetric monoidal. We also show that the algebraic
-theory of
derived stacks provides an example. Finally, we give a very short, new proof of the
equivariant Barratt–Priddy–Quillen theorem, which states that the algebraic
-theory of the category
of finite
-sets is simply
the
-equivariant
sphere spectrum.
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Keywords
spectral Mackey functors, spectral Green functors,
equivariant algebraic $K\mkern-2mu$-theory, Day
convolution, symmetric promonoidal infinity-categories,
equivariant Barratt–Priddy–Quillen
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Mathematical Subject Classification 2010
Primary: 19D99, 55P91
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Milestones
Received: 30 July 2018
Revised: 11 December 2018
Accepted: 27 December 2018
Published: 22 March 2019
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