Lax algebra bundles are defined to generalize the topological Brauer group as a direct summand while capturing higher homotopy data, with the homotopy type of their classifying space described for use in twisted K-theory.
Parametrized spectra, multiplicative Thom spectra, and the twisted Umkehr map
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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.
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math.KT 1years
2020 1verdicts
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On a generalization of the topological Brauer group
Lax algebra bundles are defined to generalize the topological Brauer group as a direct summand while capturing higher homotopy data, with the homotopy type of their classifying space described for use in twisted K-theory.