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\begin{document}

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\section*{Gamma-space}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{stable_homotopy_theory}{}\paragraph*{{Stable Homotopy theory}}\label{stable_homotopy_theory}

[[!include stable homotopy theory - contents]]

\hypertarget{internal_categories}{}\paragraph*{{Internal $(\infty,1)$-Categories}}\label{internal_categories}

[[!include internal infinity-categories contents]]

\hypertarget{spaces}{}\section*{{$\Gamma$-spaces}}\label{spaces}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_simplicial_sets}{Relation to simplicial sets}\dotfill \pageref*{relation_to_simplicial_sets} \linebreak
\noindent\hyperlink{model_category_structure}{Model category structure}\dotfill \pageref*{model_category_structure} \linebreak
\noindent\hyperlink{related_notions}{Related notions}\dotfill \pageref*{related_notions} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The concept of \emph{$\Gamma$-spaces} is a model for [[∞-groupoids]] equipped with a multiplication that is unital, associative, and commutative up to higher [[coherence|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 [[n-connected object of an (∞,1)-category|connected]] and is equipped with extra structure.

$\Gamma$-spaces differ from [[operad|operadic]] models for $E_\infty$-spaces, such as in terms of [[algebra over an operad|algebras]] over an [[E-∞ operad]], in that their multiplication is specified ``[[geometric definition of higher categories|geometrically]]'' rather than [[algebraic definition of higher categories|algebraically]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Let $\Gamma^{op}$ (see [[Segal's category]]) be the [[skeleton]] of the category of [[finite set|finite]] [[pointed sets]]. We write $\underline{n}$ for the finite pointed set with $n$ non-basepoint elements. Then a \textbf{$\Gamma$-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 map]]s

\begin{displaymath}
X(\underline{n}) \to X(\underline{1}) \times \dots \times X(\underline{1})
\end{displaymath}
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 \emph{$\Gamma$-space} includes this homotopical condition as well.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{relation_to_simplicial_sets}{}\subsubsection*{{Relation to simplicial sets}}\label{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.

\hypertarget{model_category_structure}{}\subsubsection*{{Model category structure}}\label{model_category_structure}

A [[model category]] structure on $\Gamma$-spaces is due to (\hyperlink{BousfieldFriedlander77}{Bousfield-Friedlander 77}). See at \emph{[[model structure for connective spectra]]}.

\hypertarget{related_notions}{}\subsection*{{Related notions}}\label{related_notions}

\begin{itemize}%
\item [[Gamma-set]]

\end{itemize}
[[!include k-monoidal table]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

The concept goes back to

\begin{itemize}%
\item [[Graeme Segal]], \emph{Categories and cohomology theories}, Topology 13 (1974).

\end{itemize}
The [[model category]] structure on $\Gamma$-spaces (a [[generalized Reedy model structure]]) was established in

\begin{itemize}%
\item [[A. K. Bousfield]], [[E. M. Friedlander]], \emph{Homotopy Theory of $\Gamma$-spaces, Spectra, and Bisimplicial Sets}, Geometric Applications of Homotopy Theory (1977) (\href{http://dodo.pdmi.ras.ru/~topology/books/bousfield-friedlander.pdf}{pdf})

\end{itemize}
Discussion of the [[smash product of spectra]] on connective spectra via $\Gamma$-spaces is due to

\begin{itemize}%
\item Lydakis, \emph{Smash products and $\Gamma$-spaces}, Math. Proc. Cam. Phil. Soc. 126 (1999), 311-328 (\href{http://hopf.math.purdue.edu/Lydakis/smash_gamma.pdf}{pdf})

\end{itemize}
and of the corresponding [[monoid objects]], hence [[ring spectra]], in

\begin{itemize}%
\item [[Stefan Schwede]], \emph{Stable homotopical algebra and $\Gamma$-spaces}, Math. Proc. Camb. Phil. Soc. (1999), 126, 329 (\href{http://www.math.uni-bonn.de/people/schwede/Gamma.pdf}{pdf})


\item [[Tyler Lawson]], \emph{Commutative $\Gamma$-rings do not model all commutative ring spectra}, Homology Homotopy Appl. Volume 11, Number 2 (2009), 189-194. (\href{http://projecteuclid.org/euclid.hha/1296138517}{Euclid})



\end{itemize}
Discussion in relation to [[symmetric spectra]] includes

\begin{itemize}%
\item [[Stefan Schwede]], chapter I, section 7.4 of \emph{[[Symmetric spectra]]} (2012)

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

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Higher Algebra]]}

\end{itemize}
See also

\begin{itemize}%
\item C. Balteanu, Z. Fiedorowicz, [[R. Schwanzl]] and [[R. Vogt]], \emph{Iterated Monoidal Categories}, Advances in Mathematics (2003).


\item B. Badzioch, \emph{Algebraic Theories in Homotopy Theory}, Annals of Mathematics, 155, 895--913 (2002).



\end{itemize}
[[!redirects Gamma-spaces]] [[!redirects ∞-space]] [[!redirects ∞-spaces]]

[[!redirects Gamma space]] [[!redirects Gamma spaces]]

[[!redirects Segal's category Gamma]] [[!redirects Segal's category ?]]

[[!redirects model structure for Gamma-spaces]] [[!redirects model structure for ∞-spaces]]



\end{document}