nLab Gamma-space

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.

Delooping

One of the main advantages of $\Gamma$-spaces (and, more generally, $\Gamma$-objects) is that the delooping construction is very easy to express in this language.

The delooping construction is a functor

$$B\colon Fun(\Gamma^{op},M) \to Fun(\Gamma^{op},M),$$

where $M$ is the relative category for which we are considering $\Gamma$-objects. The most common choices are $M=sSet$, the model category of simplicial sets?, and $M=Top$, the model category of topological spaces.

We define

$$(B A)(S) := hocolim (T\mapsto A(S\times T)),$$

where $T\in\Delta^\op$ and the argument of the homotopy colimit functor is a simplicial object in $M$. Here $T\in\Delta$ is converted first to an object of $\Gamma$ via the functor $\Delta\to\Gamma$ described below.

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.

Model category structure

A model category structure on $\Gamma$-spaces is due to (Bousfield-Friedlander 77). See at model structure for connective spectra.

Related notions

• Gamma-set

	(∞,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 $\simeq$ ∞-space	infinite loop space object
				$\simeq$ connective spectrum	$\simeq$ connective spectrum object
stabilization				spectrum	spectrum object

• looping and delooping, stabilization hypothesis

References

The concept goes back to

• Graeme Segal, Categories and cohomology theories, Topology 13 (1974).

Another early reference considers $\Gamma$-objects in simplicial groups. It is also the first reference that uses the terms “special Γ-spaces” and “very special Γ-spaces”, which it attributes to Segal.

• Donald W. Anderson, Chain functors and homology theories, Symposium on Algebraic Topology, Lecture Notes in Mathematics (1971), 1–12. doi.

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

• A. K. Bousfield, E. M. Friedlander, Homotopy Theory of Γ-spaces, Spectra, and Bisimplicial Sets, Geometric Applications of Homotopy Theory (1977) (pdf)

Discussion of the smash product of spectra on connective spectra via $\Gamma$-spaces

• Manos Lydakis, Smash products and $\Gamma$-spaces, Math. Proc. Cam. Phil. Soc. 126 (1999) 311-328 [doi:10.1017/S0305004198003260, pdf, pdf]

and of the corresponding monoid objects, hence ring spectra:

• Stefan Schwede, Stable homotopical algebra and $\Gamma$-spaces, Math. Proc. Camb. Phil. Soc. (1999), 126, 329 (pdf)

• Tyler Lawson, Commutative Γ-rings do not model all commutative ring spectra, Homology Homotopy Appl. Volume 11, Number 2 (2009), 189-194. (Euclid)

Discussion in relation to symmetric spectra includes

• Stefan Schwede, chapter I, section 7.4 of Symmetric spectra (2012)

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

• Jacob Lurie, Higher Algebra

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).