Twisted R-algebra structures on quotients of real ring spectra are built via Thom spectra, enabling real THH computations including for KR/2.
Parametrized higher category theory and higher algebra: Expos\'e I -- Elements of parametrized higher category theory
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abstract
We introduce the basic elements of the theory of parametrized $\infty$-categories and functors between them. These notions are defined as suitable fibrations of $\infty$-categories and functors between them. We give as many examples as we are able at this stage. Simple operations, such as the formation of opposites and the formation of functor $\infty$-categories, become slightly more involved in the parametrized setting, but we explain precisely how to perform these constructions. All of these constructions can be performed explicitly, without resorting to such acts of desperation as straightening. The key results of this Expos\'e are: (1) a universal characterization of the $T$-$\infty$-category of $T$-objects in any $\infty$-category, (2) the existence of an internal Hom for $T$-$\infty$-categories, and (3) a parametrized Yoneda lemma.
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math.AT 2years
2026 2verdicts
UNVERDICTED 2representative citing papers
Normed algebras in rational G-spectra for finite G are equivalent to families of commutative rational algebras with compatible maps indexed by conjugacy classes of subgroups according to an indexing system I.
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Equivariant twisted $R$-algebras via Thom spectra
Twisted R-algebra structures on quotients of real ring spectra are built via Thom spectra, enabling real THH computations including for KR/2.
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A model for normed algebras in rational G-spectra
Normed algebras in rational G-spectra for finite G are equivalent to families of commutative rational algebras with compatible maps indexed by conjugacy classes of subgroups according to an indexing system I.