nLab
Gamma-space

Γ\Gamma-spaces

Idea

A Γ\Gamma-space is a model for an ∞-groupoid equipped with a multiplication that is unital, associative, and commutative up to higher coherent homotopies: they are models for groupal E-∞ spaces / infinite loop spaces / abelian ∞-groups.

The notion of Γ\Gamma-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.

Therefore a Γ\Gamma-space can be delooped infinitely many times to produce a connective spectrum.

Γ\Gamma-spaces differ from operadic models for E E_\infty-spaces, such as in terms of algebras over an E-∞ operad, in that their multiplication is specified “geometrically” rather than algebraically.

Definition

Let Γ op\Gamma^{op} denote Segal's category: the skeleton of the category of finite pointed sets. We write n̲\underline{n} for the finite pointed set with nn non-basepoint elements. Then a Γ\Gamma-space is a functor X:Γ opTopX\colon \Gamma^{op}\to Top (or to simplicial sets, or whatever other model one prefers).

We think of X(1̲)X(\underline{1}) as the “underlying space” of a Γ\Gamma-space XX, with X(n̲)X(\underline{n}) being a “model for the cartesian power X nX^n”. In order for this to be valid, and thus for XX to present an infinite loop space, a Γ\Gamma-space must satisfy the further condition that all the Segal maps

X(n̲)X(1̲)××X(1̲) X(\underline{n}) \to X(\underline{1}) \times \dots \times X(\underline{1})

are weak equivalences. We include in this the 00th Segal map X(0̲)*X(\underline{0}) \to *, which therefore requires that X(0̲)X(\underline{0}) is contractible. Sometimes the very definition of Γ\Gamma-space includes this homotopical condition as well.

Properties

  • Note that we have a functor ΔΓ\Delta\to\Gamma, where Δ\Delta is the simplex category, which takes [n][n] to n̲\underline{n}. Thus, every Γ\Gamma-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 Γ\Gamma-space.

  • The topos Set Γ op\Set^{\Gamma^{op}} of Γ\Gamma-sets is the classifying topos for pointed objects (MO question).

  • A model structure on Γ\Gamma-spaces can be found in Bousfield and Friedlander below.

(∞,1)-operad∞-algebragrouplike versionin Topgenerally
A-∞ operadA-∞ algebra∞-groupA-∞ space, e.g. loop spaceloop space object
E-k operadE-k algebrak-monoidal ∞-groupiterated loop spaceiterated loop space object
E-∞ operadE-∞ algebraabelian ∞-groupE-∞ space, if grouplike: infinite loop space \simeq Γ-spaceinfinite loop space object
\simeq connective spectrum\simeq connective spectrum object
stabilizationspectrumspectrum object

References

The notion goes back to

  • G. Segal, “Categories and Cohomology Theories”, Topology 13 (1974).

The model category structure on Γ\Gamma-spaces (a generalized Reedy model structure) was established in

See also

  • C. Balteanu, Z. Fiedorowicz, R. Schwanzl and R. Vogt, Iterated Monoidal Categories, Advances in Mathematics (2003).

  • B. Badzioch, Algebraic Theories in Homotopy Theory, Annals of Mathematics, 155, 895–913 (2002).

Discussion of Γ\Gamma-spaces in the broader context of higher algebra in (infinity,1)-operad theory is around remark 2.4.2.2 of

Revised on June 19, 2013 21:00:32 by Urs Schreiber (82.169.65.155)