nLab Gamma-space

Context

Stable Homotopy theory

stable homotopy theory

$\Gamma$-spaces

Idea

The concept of $\Gamma$-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” $\Gamma$-spaces), for infinite loop spaces / connective spectra / abelian ∞-groups.

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.

$\Gamma$-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 $\Gamma^{op}$ (see Segal's category) be 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 $X\colon \Gamma^{op}\to Top$ (or to simplicial sets, or whatever other model one prefers).

We think of $X(\underline{1})$ as the “underlying space” of a $\Gamma$-space $X$, with $X(\underline{n})$ 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 maps

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

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

Properties

Relation to simplicial sets

We have a functor $\Delta\to\Gamma$, where $\Delta$ is the simplex category, which takes $[n]$ 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.

As classifying topos for pointed objects

The topos $\Set^{\Gamma^{op}}$ of $\Gamma$-sets is the classifying topos for pointed objects (MO question). For more on this see also at classifying topos for the theory of objects.

Model category structure

A model category structure on $\Gamma$-spaces is due to (Bousfield-Friedlander 77)

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 concept 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

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