nLab
Gamma-space

Idea

The concept of Γ\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” Γ\Gamma-spaces), for infinite loop spaces / connective spectra / 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.

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} (see Segal's category) be 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.

Delooping

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

The delooping construction is a functor

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

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

We define

(BA)(S):=hocolim(TA(S×T)),(B A)(S) := hocolim (T\mapsto A(S\times T)),

where TΔ opT\in\Delta^\op and the argument of the homotopy colimit functor is a simplicial object in MM. Here TΔT\in\Delta is converted first to an object of Γ\Gamma via the functor ΔΓ\Delta\to\Gamma described below.

Properties

Relation to simplicial sets

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.

Model category structure

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

(∞,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 ∞-spaceinfinite loop space object
\simeq connective spectrum\simeq connective spectrum object
stabilizationspectrumspectrum object

References

The concept goes back to

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

Another early reference considers Γ\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 Γ\Gamma-spaces (a generalized Reedy model structure) was established in

Discussion of the smash product of spectra on connective spectra via Γ\Gamma-spaces is due to

  • Lydakis, Smash products and Γ\Gamma-spaces, Math. Proc. Cam. Phil. Soc. 126 (1999), 311-328 (pdf)

and of the corresponding monoid objects, hence ring spectra, in

  • Stefan Schwede, Stable homotopical algebra and Γ\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

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

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).

Last revised on June 18, 2021 at 17:53:45. See the history of this page for a list of all contributions to it.