The concept of -spaces is a model for ∞-groupoids equipped with a multiplication that is unital, associative, and commutative up to higher coherent homotopies: they are models for E-∞ spaces and hence, if grouplike (“very special” -spaces), for infinite loop spaces / connective spectra / abelian ∞-groups.
The notion of -space is a close variant of that of Segal category for the case that the underlying (∞,1)-category happens to be an ∞-groupoid, happens to be connected and is equipped with extra structure.
Let (see Segal's category) be the skeleton of the category of finite pointed sets. We write for the finite pointed set with non-basepoint elements. Then a -space is a functor (or to simplicial sets, or whatever other model one prefers).
We think of as the “underlying space” of a -space , with being a “model for the cartesian power ”. In order for this to be valid, and thus for to present an infinite loop space, a -space must satisfy the further condition that all the Segal maps
are weak equivalences. We include in this the th Segal map , which therefore requires that is contractible. Sometimes the very definition of -space includes this homotopical condition as well.
We have a functor , where is the simplex category, which takes to . Thus, every -space has an underlying simplicial space. This simplicial space is in fact a special Delta-space which exhibits the 1-fold delooping of the corresponding -space.
|(∞,1)-operad||∞-algebra||grouplike version||in Top||generally|
|A-∞ operad||A-∞ algebra||∞-group||A-∞ space, e.g. loop space||loop space object|
|E-k operad||E-k algebra||k-monoidal ∞-group||iterated loop space||iterated loop space object|
|E-∞ operad||E-∞ algebra||abelian ∞-group||E-∞ space, if grouplike: infinite loop space Γ-space||infinite loop space object|
|connective spectrum||connective spectrum object|
The concept goes back to
Discussion of the smash product of spectra on connective spectra via -spaces is due to
Discussion in relation to symmetric spectra includes
B. Badzioch, Algebraic Theories in Homotopy Theory, Annals of Mathematics, 155, 895–913 (2002).