scholarly journals The spectra associated to ℐ-monoids

1978 ◽  
Vol 84 (2) ◽  
pp. 313-322 ◽  
Author(s):  
J. P. May
Keyword(s):  
Loop Space ◽  
Geometric Topology ◽  
Space Theory ◽  
Classifying Spaces ◽  
Infinite Loop ◽  
Infinite Loop Space

In this final sequel to (9), I shall prove a general consistency statement which seems to me to complete the foundations of infinite loop space theory. In particular, this result will specialize to yield the last step of the proof of the following theorem about the stable classifying spaces of geometric topology.

scholarly journals SYMMETRIC MONOIDAL G-CATEGORIES AND THEIR STRICTIFICATION

2019 ◽  
Vol 71 (1) ◽  
pp. 207-246
Author(s):  
Bertrand J Guillou ◽  
J Peter May ◽  
Mona Merling ◽  
Angélica M Osorno
Keyword(s):  
Loop Space ◽  
General Category ◽  
Space Theory ◽  
Classifying Space ◽  
General Internal ◽  
Categorical Framework ◽  
Infinite Loop ◽  
Definition Of ◽  
A Finite Group ◽  
Infinite Loop Space

Abstract We give an operadic definition of a genuine symmetric monoidal $G$-category, and we prove that its classifying space is a genuine $E_\infty $$G$-space. We do this by developing some very general categorical coherence theory. We combine results of Corner and Gurski, Power and Lack to develop a strictification theory for pseudoalgebras over operads and monads. It specializes to strictify genuine symmetric monoidal $G$-categories to genuine permutative $G$-categories. All of our work takes place in a general internal categorical framework that has many quite different specializations. When $G$ is a finite group, the theory here combines with previous work to generalize equivariant infinite loop space theory from strict space level input to considerably more general category level input. It takes genuine symmetric monoidal $G$-categories as input to an equivariant infinite loop space machine that gives genuine $\Omega $-$G$-spectra as output.

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

2014 ◽  
Vol 7 (4) ◽  
pp. 1077-1117 ◽  
Author(s):  
Matthew Ando ◽  
Andrew J. Blumberg ◽  
David Gepner ◽  
Michael J. Hopkins ◽  
Charles Rezk
Keyword(s):  
Loop Space ◽  
Space Theory ◽  
Thom Spectra ◽  
Ring Spectra ◽  
Infinite Loop ◽  
Infinite Loop Space

scholarly journals Rings, modules, and algebras in infinite loop space theory

2006 ◽  
Vol 205 (1) ◽  
pp. 163-228 ◽  
Author(s):  
A.D. Elmendorf ◽  
M.A. Mandell
Keyword(s):  
Loop Space ◽  
Space Theory ◽  
Infinite Loop ◽  
Infinite Loop Space

Infinite loop structures on the algebraicK-theory of spaces

1981 ◽  
Vol 90 (1) ◽  
pp. 85-111 ◽  
Author(s):  
Richard J. Steiner
Keyword(s):  
Topological Space ◽  
Loop Space ◽  
Geometric Realization ◽  
Base Point ◽  
Geometric Method ◽  
Geometric Topology ◽  
Group Completion ◽  
Infinite Loop ◽  
K Theory ◽  
Infinite Loop Space

LetXbe a topological space with base-point. Thealgebraic K-theory AXofXis a space invented by Waldhausen in (15) for use in geometric topology. It can be defined in two ways, which I shall callgeometricandring-theoretic; Steinberger ((12)) has shown them to be equivalent.The geometric method ((15), corollary to lemma 2·1) givesAXas the group-completion of the geometric realization of a permutative category. It follows from the machinery of May ((4), 4) or Segal (11) thatAXis an infinite loop space in a well-defined way ((6)).

On the quotients of mapping class groups of surfaces by the Johnson subgroups

2019 ◽  
pp. 1-23
Author(s):  
TOMÁŠ ZEMAN
Keyword(s):  
Boundary Component ◽  
Loop Space ◽  
Mapping Class Groups ◽  
Mapping Class ◽  
Classifying Spaces ◽  
Class Groups ◽  
Loop Spaces ◽  
Infinite Loop Spaces ◽  
Infinite Loop ◽  
Plus Construction

Abstract We study quotients of mapping class groups ${\Gamma _{g,1}}$ of oriented surfaces with one boundary component by the subgroups ${{\cal I}_{g,1}}(k)$ in the Johnson filtrations, and we show that the stable classifying spaces ${\mathbb {Z}} \times B{({\Gamma _\infty }/{{\cal I}_\infty }(k))^ + }$ after plus-construction are infinite loop spaces, fitting into a tower of infinite loop space maps that interpolates between the infinite loop spaces ${\mathbb {Z}} \times B\Gamma _\infty ^ + $ and ${\mathbb {Z}} \times B{({\Gamma _\infty }/{{\cal I}_\infty }(1))^ + } \simeq {\mathbb {Z}} \times B{\rm{Sp}}{({\mathbb {Z}})^ + }$ . We also show that for each level k of the Johnson filtration, the homology of these quotients with suitable systems of twisted coefficients stabilises as the genus of the surface goes to infinity.

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