Diagram spaces and symmetric spectra

Abstract

We present a general homotopical analysis of structured diagram spaces and discuss the relation to symmetric spectra. The main motivating examples are the I-spaces, which are diagrams indexed by finite sets and injections, and J-spaces, which are diagrams indexed by the Grayson-Quillen construction on the category of finite sets and bijections. We show that the category of I-spaces provides a convenient model for the homotopy category of spaces in which every E-infinity space can be rectified to a strictly commutative monoid. Similarly, the commutative monoids in the category of J-spaces model graded E-infinity spaces. Using the theory of J-spaces we introduce the graded units of a symmetric ring spectrum. The graded units detect periodicity phenomena in stable homotopy and we show how this can be applied to the theory of topological logarithmic structures.

v2: 70 pages. Minor revisions, to appear in Adv. Math

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• On the non-existence of elements of Kervaire invariant one
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• Group completion and units in I-spaces

Cited by in corpus (37)

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• Global homotopy theory
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• Presentably symmetric monoidal infinity-categories are represented by symmetric monoidal model categories
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• Localization sequences for logarithmic topological Hochschild homology
• Group completion and units in I-spaces
• Logarithmic structures on topological K-theory spectra
• Logarithmic topological Hochschild homology of topological K-theory spectra
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• Symmetric spectra model global homotopy theory of finite groups
• Infinite loop spaces and nilpotent K-theory
• Spectra of units for periodic ring spectra and group completion of graded E-infinity spaces
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• Symmetric operads in abstract symmetric spectra
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• Algebraic K-theory of strict ring spectra
• Homotopical and operator algebraic twisted K-theory
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• Generalized Thom spectra and their topological Hochschild homology
• A strictly commutative model for the cochain algebra of a space
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• G-Global Homotopy Theory and Algebraic K-Theory
• Operads in Unstable Global Homotopy Theory
• Genuine vs. naïve symmetric monoidal G-categories
• Parametrized stability and the universal property of global spectra
• Braided injections and double loop spaces
• Another viewpoint on J-spaces
• Admissible replacements for simplicial monoidal model categories
• $E_n$ ring spectra and Dyer-Lashof operations
• On Deformation Theory in Higher Logarithmic Geometry
• Bundles of strongly self-absorbing $C^{\ast }$-algebras with a Clifford grading