Parametrized spectra, multiplicative Thom spectra, and the twisted Umkehr map

Abstract

We introduce a general theory of parametrized objects in the setting of infinity categories. Although spaces and spectra parametrized over spaces are the most familiar examples, we establish our theory in the generality of objects of a presentable infinity category parametrized over objects of an infinity topos. We obtain a coherent functor formalism describing the relationship of the various adjoint functors associated to base-change and symmetric monoidal structures. Our main applications are to the study of generalized Thom spectra. We obtain fiberwise constructions of twisted Umkehr maps for twisted generalized cohomology theories using a geometric fiberwise construction of Atiyah duality. In order to characterize the algebraic structures on generalized Thom spectra and twisted (co)homology, we characterize the generalized Thom spectrum as a categorification of the well-known adjunction between units and group rings.

Submission draft. Various changes, including rewrite in terms of infinity topoi and corrected discussion of functoriality of Atiyah duality

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• Bordism for the 2-group symmetries of the heterotic and CHL strings
• Spectra of units for periodic ring spectra and group completion of graded E-infinity spaces
• Multiplicative parametrized homotopy theory via symmetric spectra in retractive spaces
• Twisted iterated algebraic K-theory and topological T-duality for sphere bundles
• Homotopical and operator algebraic twisted K-theory
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• Chromatic Cyclotomic Extensions
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• A homological approach to chromatic complexity of algebraic K-theory
• Combinatorial parametrised spectra
• Synthetic Spectra via a Monadic and Comonadic Modality
• The Smith Fiber Sequence and Invertible Field Theories
• String topology of finite groups of Lie type
• Thom spectra, higher $THH$ and tensors in $\infty$-categories
• Moduli stack of oriented formal groups and periodic complex bordism
• The Becker-Gottlieb Transfer Is Functorial
• Toward a Galois theory of the integers over the sphere spectrum
• Strict algebraic models for rational parametrised spectra I
• Characters and transfer maps via categorified traces
• Parametrized spectra, a low-tech approach
• Coassembly is a homotopy limit map
• A User's Guide: Relative Thom Spectra via Operadic Kan Extensions
• Morita bundle gerbes
• Twisted Spectra Revisited
• Strict units of commutative ring spectra
• Algebraic K-theory of finitely projective modules on ${\mathbb{E}}_{\infty }$-rings
• $E_n$ ring spectra and Dyer-Lashof operations
• An exposition of the topological half of the Grothendieck-Hirzebruch-Riemann-Roch theorem in the fancy language of spectra