$\Gamma$-spaces

Idea

A $\Gamma$-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 E-∞ spaces / infinite loop spaces.

The notion of $\Gamma$-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 $\Gamma$-space can be delooped infinitely many times to produce a connective spectrum.

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

Definition

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

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

$X\left(\underline{n}\right)\to X\left(\underline{1}\right)×\dots ×X\left(\underline{1}\right)$X(\underline{n}) \to X(\underline{1}) \times \dots \times X(\underline{1})

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

Properties

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

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

• A model structure on $\Gamma$-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 $\Gamma$-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 $\Gamma$-spaces in the broader context of higher algebra in (infinity,1)-operad theory is around remark 2.4.2.2 of

Revised on February 18, 2013 19:58:41 by Toby Bartels (64.89.53.62)