Infinite loop space machine

In topology, a branch of mathematics, given a topological monoid X up to homotopy (in a nice way), an infinite loop space machine produces a <a href="/facts/Group_completion/HSLHgFvh">group completion</a> of X together with <a href="/facts/Infinite_loop_space/S5xkdCkx">infinite loop space</a> structure. For example, one can take X to be the <a href="/facts/Classifying_space/gGIAmhtL">classifying space</a> of a <a href="/facts/Symmetric_monoidal_category/AxhumNny">symmetric monoidal category</a> S; that is, 
 
 
 
 X
 =
 B
 S
 
 
 {\displaystyle X=BS}
 
. Then the machine produces the group completion 
 
 
 
 B
 S
 →
 K
 (
 S
 )
 
 
 {\displaystyle BS\to K(S)}
 
. The space 
 
 
 
 K
 (
 S
 )
 
 
 {\displaystyle K(S)}
 
 may be described by the <a href="/facts/K-theory_spectrum/tRuBCj4o">K-theory spectrum</a> of S.
In 1977 <a href="/facts/Robert_Wayne_Thomason/PIE1jolN"> Robert Thomason</a> proved the
equivalence of all infinite loop space machines (he was just 25 years old at the moment.) He published this result next year in a joint paper with John Peter May.

<ul><li>J. P. May and R. Thomason <a href="http://www.math.uchicago.edu/~may/PAPERS/22.pdf">The uniqueness of infinite loop space machines</a></li></ul>

Infinite loop space machine open-in-new

Infinite loop space machine