## Geometry & Topology

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

#### Abstract

We introduce a general theory of parametrized objects in the setting of $∞$–categories. Although parametrised spaces and spectra are the most familiar examples, we establish our theory in the generality of families of objects of a presentable $∞$–category parametrized over objects of an $∞$–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 express the generalized Thom spectrum as a categorification of the well-known adjunction between units and group rings.

#### Article information

Source
Geom. Topol., Volume 22, Number 7 (2018), 3761-3825.

Dates
Revised: 25 May 2017
Accepted: 20 July 2017
First available in Project Euclid: 14 December 2018

https://projecteuclid.org/euclid.gt/1544756689

Digital Object Identifier
doi:10.2140/gt.2018.22.3761

Mathematical Reviews number (MathSciNet)
MR3890766

Zentralblatt MATH identifier
06997378

#### Citation

Ando, Matthew; Blumberg, Andrew J; Gepner, David. Parametrized spectra, multiplicative Thom spectra and the twisted Umkehr map. Geom. Topol. 22 (2018), no. 7, 3761--3825. doi:10.2140/gt.2018.22.3761. https://projecteuclid.org/euclid.gt/1544756689

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