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 spaces and spectra parametrized over spaces are the most familiar examples, we establish our theory in the generality 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 characterize the generalized Thom spectrum as a categorification of the well-known adjunction between units and group rings.Pre(Orb G ), or sheaves of spaces on a Grothendieck site, such as the site associated to a topological space.Let S denote the ∞-category of spaces. Since S is freely generated under colimits by its final object (the point), for any ∞-category M with small limits, the ∞category of limit-preserving functors S op → M is equivalent (via evaluation at the point) to M itself. If C is an object of M, then we will writefor the resulting functor. Now, to say that C /(−) preserves limits is to say that it satisfies descent, and so we call such a functor a sheaf on S with values in M. For example, the ∞-category Cat ∞ of (not necessarily small) ∞-categories is complete, and so any ∞-category C uniquely determines, and is determined by, a sheaf

A simple universal property of Thom ring spectra

“…Remark That the core of an ‐monoidal ‐category inherits an ‐monoidal structure has been used implicitly before (see for example, [, Definition 8.5, Remark 8.6; , Proposition 2.2.3]), but as far the authors know, no explicit description has appeared in the literature, which is why we feel justified in giving it in such detail.…”

“…Applying Theorem to we directly obtain Lewis's theorem, or rather the generalization from the sphere spectrum to arbitrary given in : Corollary If is an ‐space and is an ‐map, then becomes an ‐algebra.…”

“…The Thom spectrum for a topological group arises classically in the theory of orientations of vector bundles, representing the universal cohomology theory that orients manifolds with structure group . This point of view admits a generalization to ‐ring spectra, as we briefly summarize now; see for a comparison of the various notions of orientations in the and cases, as well as .…”

“…The corresponding Thom isomorphism is an equivalence of ‐algebras, whose underlying ‐module equivalence is a morphism . This Thom isomorphism is due to Mahowald, see [, Corollary 1.8; , Theorem 1.2].…”

“…In , the authors describe a convenient model of Thom spectra as homotopy colimits of diagrams whose shape is given by the base space. Their approach is closely related and essentially equivalent to the earlier models cited above, but is expressed in the language of ‐categories.…”