Parametrized spectra, multiplicative Thom spectra, and the twisted Umkehr map
| Autor: | Matthew Ando, David Gepner, Andrew J. Blumberg |
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| Rok vydání: | 2011 |
| Předmět: |
Pure mathematics
Algebraic structure Categorification Duality (mathematics) Homology (mathematics) 01 natural sciences Spectrum (topology) Mathematics::Algebraic Topology 55R70 Mathematics::K-Theory and Homology 55P99 Mathematics::Category Theory 0103 physical sciences FOS: Mathematics twisted Umkehr map Algebraic Topology (math.AT) Mathematics - Algebraic Topology 0101 mathematics Adjoint functors Thom spectrum Mathematics Functor 010102 general mathematics 55N20 55R12 16. Peace & justice Cohomology parametrized spectra 010307 mathematical physics Geometry and Topology |
| Zdroj: | Geom. Topol. 22, no. 7 (2018), 3761-3825 |
| DOI: | 10.48550/arxiv.1112.2203 |
| Popis: | 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. Comment: Submission draft. Various changes, including rewrite in terms of infinity topoi and corrected discussion of functoriality of Atiyah duality |
| Databáze: | OpenAIRE |
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