Rings, modules, and algebras in infinite loop space theory

By A. Elmendorf, Michael A. Mandell. 2004

174 citations recorded by the source corpus.

We give a new construction of the algebraic K-theory of small permutative categories that preserves multiplicative structure, and therefore allows us to give a unified treatment of rings, modules, and algebras in both the input and output. This requires us to define multiplicative structure on the category of small permutative categories. The framework we use is the concept of multicategory (elsewhere also called colored operad), a generalization of symmetric monoidal category that precisely captures the multiplicative structure we have present at all stages of the construction. Our method ends up in the Hovey?Shipley?Smith category of symmetric spectra, with an intermediate stop at a category of functors out of a particular wreath product.

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

View Paper: https://www.semanticscholar.org/paper/66c0b259cf329e696fe12f34580ef0f0a5b9fa10

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In Algebra and Number Theory

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Citations in the corpus

RESOLUTION OF COLOURED OPERADS AND RECTIFICATION OF HOMOTOPY ALGEBRAS (2005, background): “This concept goes back to Boardman and Vogt [7], and also occurs in homotopy theory under the name multicategory (cf. e.g. Elmendorf-Mandell [ 12 ]).”
What precisely are E∞ring spaces and E∞ring spectra? (2009, background): “The constructions of [37] and [14] bear no relationship to E∞ ring spaces as they stand and therefore cannot be used to retrieve the earlier applications or to achieve analogous future applications.”
Motivic slices and coloured operads (2010, background): “For more recent accounts, we refer to [3, 5, 9, 16].”
A unified framework for generalized multicategories (2009, unknown): “In most applications of operads (see [ EM06 , BM03] for some recent ones), both symmetry and enrichment are essential.”
Permutative categories, multicategories and algebraicK–theory (2007, background): “Permutative categories, multicategories and algebraic K –theory”
On the algebraic K‐theory of higher categories (2012, methodology): “Very similar results appear in work of Cisinski [19, Th. 2.15] and Blumberg–Mandell [16, Th. 1.3], and an interesting generalization has recently appeared in a preprint of Fiore [26].”
Homotopy completion and topological Quillen homology of structured ring spectra (2011, background): “Part (b) follows immediately from part (a) together with Proposition 6.11, Theorem 6.18, and [27, VII.3.4].”
Uniqueness of the multiplicative cyclotomic trace (2011, background): “In this situation, the algebraic K -theory spectrum inherits the structure of an E ∞ ring spectrum [10,23, 40,53 .”
Homotopy theory of modules over operads in symmetric spectra (2007, background): “One of the theorems of Elmendorf and Mandell [ 6 ] is that for symmetric spectra the category of algebras over any operad O in simplicial sets has a natural model structure inherited from the stable positive model structure on symmetric spectra.”