Rings, modules, and algebras in infinite loop space theory

Abstract

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, 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 Smith's category of symmetric spectra, with an intermediate stop at a new category that may be of interest in its own right, whose objects we call symmetric functors.

59 pages, 1 figure

Cited by in corpus (60)

• Global homotopy theory
• Diagram spaces and symmetric spectra
• On the algebraic K-theory of higher categories
• Universality of multiplicative infinite loop space machines
• Admissibility and rectification of colored symmetric operads
• Permutative categories, multicategories, and algebraic K-theory
• Khovanov homotopy type, Burnside category, and products
• Homotopy theory of modules over operads in symmetric spectra
• Derived Koszul Duality and Involutions in the Algebraic K-Theory of Spaces
• Diagram spaces, diagram spectra, and spectra of units
• An Inverse K-Theory Functor
• On operads, bimodules and analytic functors
• Bar constructions and Quillen homology of modules over operads
• Equivariant iterated loop space theory and permutative G-categories
• Skew monoidal categories and skew multicategories
• Higher topological Hochschild homology of Thom spectra
• Stable bundles over rig categories
• Extending homotopy theories across adjunctions
• Homotopy completion and topological Quillen homology of structured ring spectra
• An algebraic model for rational torus-equivariant spectra
• Localization of algebras over coloured operads
• A model structure for coloured operads in symmetric spectra
• On the category of props
• Infinite loop spaces and nilpotent K-theory
• The Homotopy Theory of Simplicially Enriched Multicategories
• Cofibrancy of operadic constructions in positive symmetric spectra
• Strictification of categories weakly enriched in symmetric monoidal categories
• E_n Genera
• A higher chromatic analogue of the image of J
• Operad bimodules, and composition products on Andre-Quillen filtrations of algebras
• Derived Representation Theory and the Algebraic K-theory of Fields
• Constructing equivariant spectra via categorical Mackey functors
• Symmetric operads in abstract symmetric spectra
• Multiplicative structures on algebraic K-theory
• Yang-Mills theory over surfaces and the Atiyah-Segal theorem
• The spectrum for commutative complex $K$-theory
• A multiplicative comparison of Waldhausen and Segal K-theory
• Khovanov spectra for tangles
• On the definition and K-theory realization of a modular functor
• Chen-Khovanov spectra for tangles
• Spectra associated to symmetric monoidal bicategories
• Preorientations of the derived motivic multiplicative group
• Commutativity properties of Quinn spectra
• On the Goodwillie derivatives of the identity in structured ring spectra
• On autoequivalences of the (\infty, 1)-category of \infty-operads
• Algebraic K-theory, K-regularity, and T-duality of ${\mathcal{O}}_{\infty }$-stable $C^{\ast }$-algebras
• Operads in Unstable Global Homotopy Theory
• K-theoretic descent and a motivic Atiyah-Segal theorem
• Multifunctorial Inverse $K$-Theory
• Ample groupoids, topological full groups, algebraic K-theory spectra and infinite loop spaces
• An involution on the K-theory of bimonoidal categories with anti-involution
• Generalizations of Loday's assembly maps for Lawvere's algebraic theories
• Homotopy functoriality for Khovanov spectra
• Multicategories Model All Connective Spectra
• The algebra of secondary homotopy operations in ring spectra
• The enriched Thomason model structure on 2-categories
• On Cofibrations of Permutative categories
• Multifunctorial $K$-Theory is an Equivalence of Homotopy Theories
• A variant of algebraic K-theory
• Recognition of commutative algebra spectra through an idempotent quasiadjunction