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\begin{document}

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\section*{symmetric monoidal (infinity,1)-category}

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

\hypertarget{category_theory}{}\paragraph*{{$(\infty,1)$-Category theory}}\label{category_theory}

[[!include quasi-category theory contents]]

\hypertarget{monoidal_categories}{}\paragraph*{{Monoidal categories}}\label{monoidal_categories}

[[!include monoidal categories - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition_in_terms_of_quasicategories}{Definition in terms of quasi-categories}\dotfill \pageref*{definition_in_terms_of_quasicategories} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{classes_of_examples}{Classes of examples}\dotfill \pageref*{classes_of_examples} \linebreak
\noindent\hyperlink{specific_examples}{Specific examples}\dotfill \pageref*{specific_examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{model_category_structure}{Model category structure}\dotfill \pageref*{model_category_structure} \linebreak
\noindent\hyperlink{commutative_monoids}{Commutative $\infty$-monoids}\dotfill \pageref*{commutative_monoids} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \emph{symmetric monoidal $(\infty,1)$-category} is

\begin{itemize}%
\item an [[(∞,1)-category]]


\item which is ``$\infty$-[[k-tuply monoidal n-category|tuply monoidal]]'', or ``stably monoidal''.



\end{itemize}
This means that it is

\begin{itemize}%
\item a [[monoidal (∞,1)-category]];


\item for which the [[tensor product]] is commutative up to infinite coherent homotopy.



\end{itemize}
This can be understood as a special case of an [[(∞,1)-operad]] (\ldots{}to be expanded on\ldots{})

Equivalently, a symmetric monoidal $(\infty,1)$-category is a [[commutative algebra in an (infinity,1)-category]] in the [[(infinity,1)-category of (infinity,1)-categories]].

Just as many ordinary $(\infty,1)$-categories (particularly, all of those that are [[locally presentable (infinity,1)-category|locally presentable]]) can be presented by [[model categories]], many symmetric monoidal $(\infty,1)$-categories can be presented by symmetric [[monoidal model categories]]. See for instance \hyperlink{NikolausSagave15}{NikolausSagave15}.

\hypertarget{definition_in_terms_of_quasicategories}{}\subsection*{{Definition in terms of quasi-categories}}\label{definition_in_terms_of_quasicategories}

Recall that in terms of [[quasi-category|quasi-categories]] a general [[monoidal (infinity,1)-category]] is conceived as a coCartesian fibration $C^\otimes \to N(\Delta)^{op}$ of [[simplicial set]]s over the ([[opposite category|opposite]] of) the [[nerve]] $N(\Delta)^{op}$ of the [[simplex category]] satisfying a certain property.

The fiber of this fibration over the 1-[[simplex]] $[1]$ is the [[monoidal (infinity,1)-category]] $C$ itself, its value over a map $[n] \to [1]$ encodes the tensor product of $n$ factors of $C$ with itself.

The following definition encodes the \emph{commutativity} of all these operations by replacing $\Delta$ with the category $FinSet_*$ of pointed finite sets.

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{symmetric monoidal $(\infty,1)$-category} is a [[coCartesian fibration]] of [[simplicial set]]s

\begin{displaymath}
p : C^\otimes \to N(FinSet_*)
\end{displaymath}
such that

\begin{itemize}%
\item for each $n \geq 0$ the associated functors $C^\otimes_{[n]} \to C^\otimes_{[1]}$ determine an equivalence of $(\infty,1)$-categories $C^\otimes_{[n]} \stackrel{\simeq}{\to} C_{[1]}^n$.

\end{itemize}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
In other words, a symmetric monoidal $(\infty,1)$-category is an $\mathcal{O}$-[[monoidal (∞,1)-category]] for

\begin{displaymath}
\mathcal{O} = Com
\end{displaymath}
the [[commutative operad|commutative]] [[(∞,1)-operad]].

\end{remark}
See (\hyperlink{LurieAlgebra}{Lurie, def. 2.0.0.7}).

\begin{prop}
\label{}\hypertarget{}{}
The [[homotopy category of an (infinity,1)-category|homotopy category]] of a symmetric monoidal $(\infty,1)$-category is an ordinary [[symmetric monoidal category]].

\end{prop}
\begin{remark}
\label{}\hypertarget{}{}
There is a functor $\varphi : \Delta^{op} \to FinSet_*$ such that the [[monoidal (infinity,1)-category]] \emph{underlying} a symmetric monoidal $(\infty,1)$-category $p : C^\otimes \to N(FinSet_*)$ is the [[(infinity,1)-pullback]] of $p$ along $\varphi$.

\end{remark}
\hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples}

\hypertarget{classes_of_examples}{}\subsubsection*{{Classes of examples}}\label{classes_of_examples}

\begin{itemize}%
\item [[cartesian monoidal (∞,1)-category]]

\end{itemize}
\hypertarget{specific_examples}{}\subsubsection*{{Specific examples}}\label{specific_examples}

\begin{itemize}%
\item [[stable (infinity,1)-category of spectra]]


\item [[symmetric monoidal (infinity,1)-category of presentable (infinity,1)-categories]]



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

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

A [[presentable (infinity,1)-category|presentation]] of the [[(∞,1)-category]] of all symmetric monoidal $(\infty,1)$-categories is provided by the [[model structure for dendroidal coCartesian fibrations]].

\hypertarget{commutative_monoids}{}\subsubsection*{{Commutative $\infty$-monoids}}\label{commutative_monoids}

See [[commutative monoid in a symmetric monoidal (∞,1)-category]].

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[monoidal category]], [[monoidal (∞,1)-category]]


\item [[symmetric monoidal category]], \textbf{symmetric monoidal $(\infty,1)$-category}, [[symmetric monoidal (∞,n)-category]]


\item [[tensor (∞,1)-category]]


\item [[closed monoidal category]] , [[closed monoidal (∞,1)-category]]


\item [[K-theory of a symmetric monoidal (∞,1)-category]]


\item [[prime spectrum of a symmetric monoidal stable (∞,1)-category]]



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

The defintion of symmetric monoidal quasi-category is definition 1.2 in

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

\end{itemize}
and definition 2.0.0.7 in

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

\end{itemize}
A concise treatment is also available in

\begin{itemize}%
\item [[Moritz Groth]], \emph{A short course on infinity-categories}, \href{http://www.math.ru.nl/~mgroth/preprints/groth_scinfinity.pdf}{pdf}.

\end{itemize}
Relation to [[monoidal model categories]] (in particular, that every locally presentable symmetric monoidal $(\infty,1)$-category arises from a symmetric monoidal model category) is discussed in

\begin{itemize}%
\item [[Thomas Nikolaus]], [[Steffen Sagave]], \emph{Presentably symmetric monoidal infinity-categories are represented by symmetric monoidal model categories}. Algebr. Geom. Topol. 17 (2017), no. 5, 3189--3212. (\href{http://arxiv.org/abs/1506.01475}{arXiv:1506.01475})

\end{itemize}
[[!redirects symmetric monoidal (infinity,1)-categories]] [[!redirects symmetric monoidal (∞,1)-category]] [[!redirects symmetric monoidal (∞,1)-categories]]



\end{document}