Diagram spaces and symmetric spectra

By S. Sagave, Christian Schlichtkrull. 2011

65 citations recorded by the source corpus.

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 Quillen’s localization construction Σ − 1 Σ on the category Σ 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 ∞ space can be rectified to a strictly commutative monoid. Similarly, the commutative monoids in the category of J -spaces model graded E ∞ 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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e5fa7f1cb82b6a15ee068c2bdd0c74d12c79771f

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Citations in the corpus

Model structures on commutative monoids in general model categories (2014, background): “The interested reader is encouraged to consult [14], [33], and [30] for more information about rectification for general model cate gories.”
FI-modules and stability for representations of symmetric groups (2012, background): “utes the innite symmetric product Sym1M’colim FI M, which by the Dold{Thom theorem satises ˇ i(Sym1M) ’He i(M;Z). In fact, Mprovides an example of an I-monoid in the sense of Sagave{Schlichtkrull [SSc] (a commutative monoid in the category of FI-spaces). Sagave{Schlichtkrull prove that any innite lo”
Topological Hochschild homology of Thom spectra and the free loop space (2008, background): “on we observe that if α classiﬁes a well-based I-space over A, then the symmetric spectrum T(α) is levelwise well-based. The statement therefore follows from Proposition 7.10. Remark. It is proved in [34] that there is a model structure on IUwhose weak equivalences are the maps that induce weak homo”