category object in an (∞,1)-category, groupoid object
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.
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
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.
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
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
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.
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.
A model category structure on $\Gamma$-spaces is due to (Bousfield-Friedlander 77). See at model structure for connective spectra.
The concept goes back to
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.
The model category structure on $\Gamma$-spaces (a generalized Reedy model structure) was established in
Discussion of the smash product of spectra on connective spectra via $\Gamma$-spaces
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
Discussion of $\Gamma$-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 April 18, 2024 at 07:55:44. See the history of this page for a list of all contributions to it.