Diagram spaces, diagram spectra and spectra of units

Lind, J.

doi:10.2140/agt.2013.13.1857

Cited by 46 publications

(50 citation statements)

References 30 publications

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“…An I c -FCP gives rise to an E ∞ -space structured by the linear isometries operad L; specifically, T (R ∞ ) = colim V T (V ) is an L-space with the operad maps induced by the Whitney sum [14, 1.9; 15, 23.6.3]. In fact, as alluded to above, one can set up a Quillen equivalence between the category of I c -FCPs and the category of E ∞ -spaces, although we do not discuss this matter herein (see [10] for a nice treatment of this comparison).…”

Section: The 'Neo-classical' Thom Spectrum Functormentioning

confidence: 99%

An ∞-categorical approach to R -line bundles, R -module Thom spectra, and twisted R -homology

Ando

Blumberg

Gepner

et al. 2013

Journal of Topology

115

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We develop a generalization of the theory of Thom spectra using the language of ∞-categories. This treatment exposes the conceptual underpinnings of the Thom spectrum functor: we use a new model of parameterized spectra, and our definition is motivated by the geometric definition of Thom spectra of May-Sigurdsson. For an A∞-ring spectrum R, we associate a Thom spectrum to a map of ∞-categories from the ∞-groupoid of a space X to the ∞-category of free rank one R-modules, which we show is a model for BGL1R; we show that BGL1R classifies homotopy sheaves of rank one R-modules, which we call R-line bundles. We use our R-module Thom spectrum to define the twisted R-homology and cohomology of R-line bundles over a space classified by a map X → BGL1R, and we recover the generalized theory of orientations in this context. In order to compare this approach to the classical theory, we characterize the Thom spectrum functor axiomatically, from the perspective of Morita theory.

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Section: The 'Neo-classical' Thom Spectrum Functormentioning

confidence: 99%

An ∞-categorical approach to R -line bundles, R -module Thom spectra, and twisted R -homology

Ando

Blumberg

Gepner

et al. 2013

Journal of Topology

115

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“…When F = e is the global family of trivial groups, we recover the Quillen equivalence established by Lind in [16,Thm. 9.9].…”

Section: L-spaces and Orthogonal Spacesmentioning

confidence: 65%

“…By [16,Lemma 8.3], the functor Q⊗ L − from orthogonal spaces to L-spaces is strong symmetric monoidal for the box product of orthogonal spaces (compare [21, Sec. 1.3]) and the operadic product ⊠ L of L-spaces.…”

Section: L-spaces and Orthogonal Spacesmentioning

confidence: 99%

Orbispaces, orthogonal spaces, and the universal compact Lie group

Schwede

2019

Math. Z.

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This paper identifies the homotopy theories of topological stacks and orbispaces with unstable global homotopy theory. At the same time, we provide a new perspective by interpreting it as the homotopy theory of 'spaces with an action of the universal compact Lie group'. The upshot is a novel way to construct and study genuine cohomology theories on stacks, orbifolds, and orbispaces, defined from stable global homotopy types represented by orthogonal spectra.The universal compact Lie group (which is neither compact nor a Lie group) is a wellknown object, namely the topological monoid L of linear isometric self-embeddings of R ∞ . The underlying space of L is contractible, and the homotopy theory of L-spaces with respect to underlying weak equivalences is just another model for the homotopy theory of spaces. However, the monoid L contains copies of all compact Lie groups in a specific way, and we define global equivalences of L-spaces by testing on corresponding fixed points. We establish a global model structure on the category of L-spaces and prove it to be Quillen equivalent to the global model category of orthogonal spaces, and to the category of orbispaces, i.e., presheaves of spaces on the global orbit category.

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“…Of course, one can instead replace the given commutative S-algebra by an associative S-algebra instead, but in this case it is impossible to recover the E ∞ structure on GL 1 R. To describe GL 1 R in this setting requires a different construction; see [32] or [19] for a description.…”

Section: The Space Of Unitsmentioning

confidence: 99%

“…The observation that one could carry out the program of [12] in the setting of spaces is due to Mike Mandell, and was worked out in the thesis of the second author [6]. A detailed presentation of the theory (along with complete proofs) has appeared in [7] (and see also [19]).…”

Section: A ∞ Thom Spectra and Orientationsmentioning

confidence: 99%

Units of ring spectra, orientations, and Thom spectra via rigid infinite loop space theory

Ando

Blumberg

Gepner

et al. 2014

Journal of Topology

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Abstract. We extend the theory of Thom spectra and the associated obstruction theory for orientations in order to support the construction of the E∞ string orientation of tmf , the spectrum of topological modular forms. Specifically, we show that for an E∞ ring spectrum A, the classical construction of gl 1 A, the spectrum of units, is the right adjoint of the functorTo a map of spectra f : b → bgl 1 A, we associate an E∞ A-algebra Thom spectrum M f , which admits an E∞ A-algebra map to R if and only if the composition b → bgl 1 A → bgl 1 R is null; the classical case developed by [28] arises when A is the sphere spectrum. We develop the analogous theory for A∞ ring spectra: if A is an A∞ ring spectrum, then to a map of spaceswe associate an A-module Thom spectrum M f, which admits an R-orientation if and only if B → BGL 1 A → BGL 1 R is null. Our work is based on a new model of the Thom spectrum as a derived smash product.

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Diagram spaces, diagram spectra and spectra of units

Cited by 46 publications

References 30 publications

An ∞-categorical approach to R -line bundles, R -module Thom spectra, and twisted R -homology

An ∞-categorical approach to R -line bundles, R -module Thom spectra, and twisted R -homology

Orbispaces, orthogonal spaces, and the universal compact Lie group

Units of ring spectra, orientations, and Thom spectra via rigid infinite loop space theory

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