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

TL;DR

This paper develops a comprehensive theory of parametrized objects in infinity categories, enabling fiberwise constructions of twisted Umkehr maps and advancing the understanding of generalized Thom spectra and twisted cohomology theories.

Contribution

It introduces a general framework for parametrized objects in infinity categories, with applications to fiberwise constructions of twisted Umkehr maps and Thom spectra.

Findings

Constructed fiberwise twisted Umkehr maps for generalized cohomology theories.

Characterized generalized Thom spectra as categorifications of unit-group adjunctions.

Provided a geometric fiberwise approach to Atiyah duality.

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.