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\begin{document}

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

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

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

[[!include monoidal categories - contents]]

\hypertarget{topos_theory}{}\paragraph*{{$(\infty,1)$-topos theory}}\label{topos_theory}

[[!include (infinity,1)-topos - contents]]

\hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra}

[[!include higher algebra - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{idea_of_the_simplicial_definition}{Idea of the simplicial definition}\dotfill \pageref*{idea_of_the_simplicial_definition} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{plain_monoidal_category}{Plain monoidal $(\infty,1)$-category}\dotfill \pageref*{plain_monoidal_category} \linebreak
\noindent\hyperlink{monoidal_category}{$\mathcal{O}$-monoidal $(\infty,1)$-category}\dotfill \pageref*{monoidal_category} \linebreak
\noindent\hyperlink{higher_monoidal_structure}{Higher monoidal structure}\dotfill \pageref*{higher_monoidal_structure} \linebreak
\noindent\hyperlink{operadicalgebraic_definition_of_monoidal_structure}{Operadic/algebraic definition of monoidal structure}\dotfill \pageref*{operadicalgebraic_definition_of_monoidal_structure} \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}

The notion of \emph{monoidal $(\infty,1)$-category} is the analogue of the notion of [[monoidal category]] in the context of [[(infinity,1)-category|(∞,1)-categories]].

There are various ways to state the monoidal structure. One is in terms of fibrations over the [[simplex category]]. This is the approach taken in

\begin{itemize}%
\item [[Jacob Lurie]], \emph{Noncommutative algebra} (\href{http://arxiv.org/abs/math/0702299}{arXiv})

\end{itemize}
Another is in terms of [[(∞,1)-operad]]s (see there). This approach has been taken in (\hyperlink{Francis}{Francis}). Both are described below.

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 monoidal $(\infty,1)$-categories can be presented by [[monoidal model categories]]. See for instance \hyperlink{NikolausSagave15}{NikolausSagave15}.

\hypertarget{idea_of_the_simplicial_definition}{}\subsubsection*{{Idea of the simplicial definition}}\label{idea_of_the_simplicial_definition}

As discussed at the end of the entry on [[monoidal category]], an ordinary [[monoidal category]] may be thought of as a lax functor

\begin{displaymath}
* \to \mathbf{B} Cat
\end{displaymath}
from the terminal category to the one-object 3-category whose single [[hom-object]] is the [[2-category]] [[Cat]] of all categories and for which composition is the [[cartesian monoidal category|cartesian monoidal structure]] on [[Cat]].

More concretely, as also described there, such a lax functor is a kind of [[descent]] object in a [[weighted limit]] $lim^{\Delta} const_{\mathbf{B}Cat}$, namely a diagram

\begin{displaymath}
\itexarray{
    F \Delta^4 &\stackrel{=}{\to}& \mathbf{B} Cat
    \\
    \uparrow && \uparrow^{Id} 
    \\
    F \Delta^3 &\stackrel{\alpha}{\to}& \mathbf{B} Cat
    \\
    \uparrow && \uparrow^{Id} 
    \\
    F \Delta^2 &\stackrel{\otimes}{\to}& \mathbf{B} Cat
    \\
    \uparrow && \uparrow^{Id} 
    \\
    F \Delta^1 &\stackrel{C}{\to}& \mathbf{B}Cat
    \\
    \uparrow && \uparrow^{Id} 
    \\
    F\Delta^0 &\stackrel{\bullet}{\to}& \mathbf{B}Cat
  }
\end{displaymath}
where $F \Delta^n$ is the free 3-category of the $n$-[[simplex]] (an [[oriental]]), where the horizontal morphisms are the chosen data -- the category $C$, product $\otimes$, associator $\alpha$ and, in degree 4, the respect for the pentagon identity -- and the condition is that this commutes for all vertical morphisms $F(\Delta^n \to \Delta^m)$.

So this is a 4-functor

\begin{displaymath}
F(\Delta^4) \to \mathbf{B}Cat
\end{displaymath}
subject to a certain constraint.

Using the general mechanism of [[generalized universal bundle]]s, this classifies a [[Cat]]-bundle

\begin{displaymath}
\itexarray{
    C^\otimes \to F(\Delta^4)
  }
  \,.
\end{displaymath}
With a bit more time than I have on the train one can figure out that conversely suitable such fibrations are equivalent to monoidal categories. Alternatively, one can read pages 5 and 6 of \emph{LurieNonCom} cited below.

In any case, this motivates the following definition.

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

\hypertarget{plain_monoidal_category}{}\subsubsection*{{Plain monoidal $(\infty,1)$-category}}\label{plain_monoidal_category}

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{monoidal ($\infty,1$)-category} $(C, \otimes)$ is

\begin{itemize}%
\item a [[simplicial set]] $C^\otimes$;


\item and a [[coCartesian fibration]] of simplicial sets $p_\otimes : C^\otimes \to N(\Delta)^{op}$


\item such that for each $n \in \mathbb{N}$ the induced [[(infinity,1)-functor]] $C^\otimes_{[n]} \to C^\otimes_{\{i,i+1\}}$ determines an equivalence of [[(infinity,1)-category|(infinity,1)-categories]]



\end{itemize}
\begin{displaymath}
C^\otimes_{[n]}
  \to
  C^\otimes_{\{0,1\}}
  \times
  \cdots
  \times
  C^\otimes_{\{n-1,n\}}
  \simeq
  (C^\otimes_{[1]})^n
  \,.
\end{displaymath}
Here $\Delta$ is the [[simplex category]] and $N(\Delta)$ its [[nerve]].

\end{defn}
\hypertarget{monoidal_category}{}\subsubsection*{{$\mathcal{O}$-monoidal $(\infty,1)$-category}}\label{monoidal_category}

The following defines [[symmetric monoidal (∞,1)-categories]] and their variants, where the [[commutative operad]] is replaced by any other [[(∞,1)-operad]].

\begin{defn}
\label{}\hypertarget{}{}
Let $\mathcal{O}^\otimes$ be an [[(∞,1)-operad]]. A \textbf{[[coCartesian fibration of (∞,1)-operads]]} is

\begin{enumerate}%
\item a [[coCartesian fibration]] $p : \mathcal{C}^\otimes \to \mathcal{O}^\otimes$ of the underlying [[quasi-categories]];


\item such that the composite

\begin{displaymath}
\mathcal{C}^\otimes \stackrel{p}{\to} \mathcal{O}^\otimes \to Fin_* = Comm
\end{displaymath}
exhibits $\mathcal{C}^\otimes$ as an [[(∞,1)-operad]].



\end{enumerate}
In this case we say that the underlying [[(∞,1)-category]]

\begin{displaymath}
\mathcal{C}  = \mathcal{C}^\otimes \times_{\mathcal{O}^\otimes} \mathcal{O}
\end{displaymath}
is equipped by $p$ with the [[structure]] of an \textbf{$\mathcal{O}$-monoidal $(\infty,1)$-category}.

\end{defn}
This is (\hyperlink{Lurie}{Lurie, def. 2.1.2.13}).

\begin{defn}
\label{}\hypertarget{}{}
For $\mathcal{O}$ = [[commutative operad|Comm]], an $\mathcal{O}$-monoidal $(\infty,1)$-category is a [[symmetric monoidal (∞,1)-category]].

\end{defn}
\hypertarget{higher_monoidal_structure}{}\subsection*{{Higher monoidal structure}}\label{higher_monoidal_structure}

While for an ordinary [[monoid]] there is just one notion of commutativity (either it is or it is not commutative), already a [[monoidal category]] distinguishes between being just [[braided monoidal category|braided monoidal]] or fully [[symmetric monoidal category|symmetric monoidal]].

This pattern continues, as expressed by the [[k-tuply monoidal n-category|periodic table of k-tuply monoidal categories]].

A [[higher category theory|higher category]] may be a [[k-tuply monoidal n-category]] or more generally [[k-tuply monoidal (n,r)-category]] for different values of $k$. The lowest value of $k= 1$ (since for $k = 0$ there is no monoidal structure at all) corresponds to monoidal product which is $\infty$-associative, i.e. associative up to higher coherent homotopies, but need not have any degree of \emph{commutativity}.

One says that an $n$-category is \emph{symmetric monoiodal} if it is ``as monoidal as possible'', i.e. $\infty$-tuply monoidal. In particular, in [[higher algebra|Noncommutative algebra]] and [[higher algebra|Commutative algebra]] we have

\begin{itemize}%
\item the 1-fold monoidal (∞,1)-categories described here;


\item [[symmetric monoidal (infinity,1)-category|∞-tuply monoidal (∞,1)-categories]].



\end{itemize}
\hypertarget{operadicalgebraic_definition_of_monoidal_structure}{}\subsubsection*{{Operadic/algebraic definition of monoidal structure}}\label{operadicalgebraic_definition_of_monoidal_structure}

For each $1 \leq n \leq \infty$ let $E_n$ denote the [[little n-disk operad|little n-disk]] [[operad]] whose [[topological space]] of $E_n^k$ of $k$-ary operations is the space of embeddings of $k$ $n$-dimensional disks (balls) in one $n$-dimensional disk without intersection, and whose composition operation is the obvious one obtained from gluing the big outer disks into given inner disks.

In \hyperlink{Francis}{John Francis' PhD thesis} the theory of [[(∞,1)-categories]] equipped with an action of the $E_n$-[[operad]] is established, so that

\begin{itemize}%
\item $(\infty,1)$-categories with an $E_1$-action are precisely [[monoidal (∞,1)-categories]] -- 1-fold monoidal $(\infty,1)$-categories;


\item $(\infty,1)$-categories with an $E_\infty$-action are precisely [[symmetric monoidal (∞,1)-categories]] -- $\infty$-tuply monoidal $(\infty,1)$-categories;


\item $(\infty,1)$-categories with an $E_n$-action for $1 \lt n \lt \infty$ are the corresponding $n$-tuply monoidal $(\infty,1)$-categories in between.



\end{itemize}
\begin{quote}%
\textbf{Remark} The second statement is example 2.3.8 in \hyperlink{Francis}{EnAction}. The first seems to be clear but is maybe not in the literature. Jacob Lurie is currently rewriting [[higher algebra|Higher Algebra]] such as to build in a discussion of $E_n$-operadic structures in the definition of $k$-tuply monoidal $(\infty,1)$-categories.


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

\begin{itemize}%
\item [[monoidal category]], [[monoidal (2,1)-category]], \textbf{monoidal $(\infty,1)$-category}

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


\item [[Picard ∞-group]]



\end{itemize}

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

\begin{itemize}%
\item [[braided ∞-group]]

\end{itemize}

\item [[symmetric monoidal category]], [[symmetric monoidal (2,1)-category]], [[symmetric monoidal (∞,1)-category]]

\begin{itemize}%
\item [[abelian ∞-group]]

\end{itemize}

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


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



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

The simplicial definition for plain monoidal $(\infty,1)$-categories is definition 1.1.2 in

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Noncommutative Algebra]]} (\href{http://arxiv.org/abs/math/0702299}{arXiv})

\end{itemize}
John Francis' work on [[little cubes operad]]-actions on $(\infty,1)$-categories is in

\begin{itemize}%
\item [[John Francis]], PhD thesis (\href{http://dspace.mit.edu/handle/1721.1/43792}{web})

\end{itemize}
The general notion of an $\mathcal{O}$-monoidal $(\infty,1)$-category is around definition 2.1.2.13 of

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

\end{itemize}
An introductory survey is in

\begin{itemize}%
\item [[Moritz Groth]], \emph{A short course on $\infty$-categories} (\href{http://www.math.uni-bonn.de/~mgroth/InfinityCategories.pdf}{pdf})

\end{itemize}
A relation to [[monoidal model categories]] (in particular, that every locally presentable symmetric monoidal $(\infty,1)$-category arises from a symmetric monoidal model category) is described 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 monoidal (infinity,1)-categories]] [[!redirects monoidal (∞,1)-category]] [[!redirects monoidal (∞,1)-categories]]



\end{document}