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\begin{document}

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\section*{tensor product of infinity-modules}

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

\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{definition}{Definition}\dotfill \pageref*{definition} \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 generalization of the notion of [[tensor product of modules]] to [[∞-modules]].

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

We start by defining a collection of [[colored operad|colored]] [[symmetric operad]] $Tens^\otimes$ parameterized by the [[simplex category]] $\Delta$ such that for each $k$-[[simplex]] $[k] \in Delta$ the [[algebras over an operad]] over $Tens^\otimes_{[k]}$ are $(n+1)$-tuples of [[associative algebras]] $(A_i)$ together with a consecutive sequence of [[bimodules]] over these (the right algebra of every bimodule being the left algebra of the next one).

The definition is a straightforward generalization of the of the [[operad for modules]] and the [[operad for bimodules]].

\begin{defn}
\label{}\hypertarget{}{}
Write $Tens^\otimes$ for the category (to be thought of as a family of [[categories of operators]] of [[symmetric operads]]) whose

\begin{itemize}%
\item [[objects]] are triples consisting of

\begin{itemize}%
\item an object $\langle n\rangle \in Assoc^\otimes$ of the [[category of operators]] of the [[associative operad]];


\item an object $[k] \in \Delta$ of the [[simplex category]];


\item two [[functions]] $c_-, c_+ \colon \langle n\rangle^\circ \to [k]$ such that for all $i in \langle n\rangle^\circ$ either $c_+(i) = c_-(i)$ or $c_+(i) = c_-(i) + 1$;



\end{itemize}

\item [[morphisms]] consist if

\begin{itemize}%
\item a morphism $\alpha \colon \langle n\rangle \to \langle n'\rangle$ in $Assoc^\otimes$


\item a morphism $\lambda \colon [k'] \to [k]$ in $\Delta$



\end{itemize}
such that (\ldots{})



\end{itemize}
\end{defn}
(\hyperlink{Lurie}{Lurie, def. 4.3.4.1})

We disuss how an object of this category is to be thought of as labeled with ``algebra labels $\mathfrak{a}_i$'' for vertices of a simplex, an ``bimodule lables $\mathfrak{n}_{i, j}$'' for edges of the simplex.

\begin{remark}
\label{}\hypertarget{}{}
By construction there are [[forgetful functors]]

\begin{displaymath}
\Delta^{op}
  \leftarrow
  Tens^\otimes
  \rightarrow
  \mathcal{Ass}^\otimes
  \,.
\end{displaymath}
\end{remark}
(\hyperlink{Lurie}{Lurie, 4.3.4.1})

\begin{defn}
\label{NotationForTensS}\hypertarget{NotationForTensS}{}
For $S \to \Delta^{op}$ an [[(∞,1)-functor]] (given as a map of simplicial sets from a [[quasi-category]] $S$ to the [[nerve]] of the [[simplex category]]), write

\begin{displaymath}
Tens^\otimes_{S} 
   \coloneqq
  Tens^\otimes \underset{\Delta^{op}}{\times} S
\end{displaymath}
for the [[fiber product]] in [[sSet]].

\end{defn}
(\hyperlink{Lurie}{Lurie, notation 4.3.4.5, 4.3.4.15})

\begin{prop}
\label{SegalConditionOfTens}\hypertarget{SegalConditionOfTens}{}
We have

\begin{itemize}%
\item $Tens^\otimes_{[0]} \simeq Assoc^\otimes$, the [[associative operad]];


\item $Tens^\otimes_{[1]} \simeq BM^\otimes$ the [[operad for bimodules]].


\item $Tens^\otimes_{[k]} 
  \simeq 
 Tens^\otimes_{\{0,1\}} 
  \underset{Tens^\otimes_{\{1\}}}{\coprod} 
 Tens^\otimes_{\{1,2\}} 
   \underset{Tens^\otimes_{\{2\}}}{\coprod} 
   \cdots 
   \underset{Tens^\otimes_{\{k-1\}}}{\coprod} 
 Tens^\otimes_{\{k-1,k\}}$

as an [[(∞,1)-colimit]] in the [[(∞,1)-category]] of [[(∞,1)-operads]] (a dual \emph{[[Segal condition]]})



\end{itemize}
\end{prop}
(\hyperlink{Lurie}{Lurie, example 4.3.4.6, 4.3.4.7, prop. 4.3.4.11})

\begin{remark}
\label{}\hypertarget{}{}
Prop. \ref{SegalConditionOfTens} implies that for $\mathcal{C}^\otimes$ an [[(∞,1)-operad]], the [[(∞,1)-algebras over an (∞,1)-operad]] over the fiber $Tens^\otimes_{[k]}$ in $\mathcal{C}$ form the [[(∞,1)-category]]

\begin{displaymath}
Alg_{Tens^\otimes_{[k]}}(\mathcal{C})
  \simeq
  \underbrace{
  BMod(\mathcal{C}) 
   \underset{Alg(\mathcal{C})}{\times}
  BMod(\mathcal{C})
   \underset{Alg(\mathcal{C})}{\times}
  \cdots
   \underset{Alg(\mathcal{C})}{\times}
  BMod(\mathcal{C})
  }_{k\;factors}
  \,.
\end{displaymath}
\end{remark}
(\hyperlink{Lurie}{Lurie, 4.3.5})

\begin{defn}
\label{NotationForAlS}\hypertarget{NotationForAlS}{}
For $\mathcal{C}^\otimes \to Tens^\otimes_S$ a [[fibration]] in the [[model structure for quasi-categories]] which exhibits $\mathcal{C}^\otimes$ as an $S$-[[family of (∞,1)-operads]], write

\begin{displaymath}
Alg_S(\mathcal{C}) \hookrightarrow Fun_{Tens^\otimes_S}(Step_S, \mathcal{C}^\otimes)
\end{displaymath}
for the full [[sub-(∞,1)-category]] on those [[(∞,1)-functors]] which send inert morphisms to inert morphisms.

\end{defn}
(\hyperlink{Lurie}{Lurie, notation 4.3.4.15})

\begin{prop}
\label{}\hypertarget{}{}
For an [[(∞,1)-functor]] $S \to \Delta^{op}$ and a [[fibration]] in the [[model structure for quasicategories]] $q \colon \mathcal{C}^\otimes \to Tens_S^\otimes$ exhibiting $\mathcal{C}^\otimes$ as an $S$-[[family of (∞,1)-operads]], then there is an [[equivalence of (∞,1)-categories]]

\begin{displaymath}
Alg_{/Tens_S}(\mathcal{C})
  \to 
  Alg_S(\mathcal{C})
  \,.
\end{displaymath}
\end{prop}
(\hyperlink{Lurie}{Lurie, prop. 4.3.4.17}).

\begin{defn}
\label{NotationForTensGt}\hypertarget{NotationForTensGt}{}
Let $\Delta^1 \to \Delta^{op}$ be the map that picks the morphism $\{0,2\} \hookrightarrow \Delta^2$ in the [[simplex category]]. With def. \ref{NotationForTensS} write

\begin{displaymath}
Tens^\otimes_{\gt}
  \coloneqq
  Tens_{\Delta^1}^\otimes
  \coloneqq
  Tens^\otimes \underset{\Delta^{op}}{\times} \Delta^1
  \,.
\end{displaymath}
\end{defn}
(\hyperlink{Lurie}{Lurie, notation 4.3.5.1})

\begin{remark}
\label{BilinearInfinityMap}\hypertarget{BilinearInfinityMap}{}
The $Tens^\otimes_{\gt}$ of def. \ref{NotationForTensGt} is a [[correspondence]] of [[(∞,1)-operads]] which exhibits bilinear maps as follows:

An [[∞-algebra over an (∞,1)-operad]] $\gamma_1 \colon Tens^\otimes_{\gt} \times_{\Delta^1} \{1\} \to \mathcal{C}^\otimes$ is equivalently a bimodule

\begin{displaymath}
X \in {}_{A'} Mod(\mathcal{C})_{C'}
  \,,
\end{displaymath}
while an $\infty$-algebra $\gamma_0 \colon Tens^\otimes_{\gt} \times_{\Delta^1} \{0\} \to \mathcal{C}^\otimes$ is equivalently a pair of bimodules

\begin{displaymath}
N_1 \in {}_A Mod(\mathcal{C})_B
  \;\;, \;\;
  N_2 \in {}_B Mod(\mathcal{C})_C
 \,.
\end{displaymath}
An extension of $(\gamma_0, \gamma_1)$ through the correspondence hence to a map of [[generalized (∞,1)-operads]] $Tens^\otimes_{\gt} \to \mathcal{C}^\otimes$ is equivalently a pair of [[A-∞ algebra]] maps $A \to A'$ and $B \to B'$ together with a bilinear map $N_1 \otimes N_2 \to X$.

\end{remark}
(Lurie, beginning of 4.3.4).

\begin{defn}
\label{RelativeTensorProduct}\hypertarget{RelativeTensorProduct}{}
(\textbf{relative tensor product of $\infty$-bimodules})

For $q \colon \mathcal{C}^\otimes \to \mathcal{O}^\otimes$ a [[fibration of (∞,1)-operads]], consider a morphism of [[generalized (∞,1)-operads]]

\begin{displaymath}
F \colon Tens_{\gt}^\otimes \to \mathcal{C}^{\otimes}
  \,.
\end{displaymath}
This exhibits three [[A-∞ algebras]] $A_i \coloneqq F|_{\{i\}}$, a pair of bimodule objects

\begin{displaymath}
(N_1, N_2) = F|_{[2]}
\end{displaymath}
over $A_0$-$A_1$ and over $A_1$-$A_2$, respectively, and a bimodule object $N = F|_{[1]}$ over $A_0$-$A_2$. We say that $N$ exhibits the \textbf{relative [[tensor product of ∞-modules]]} of $N_1$ with $N_2$ over $A_1$

\begin{displaymath}
N \simeq N_1 \otimes_{A_1} N_2
\end{displaymath}
if $F$ is an operadic $q$-[[(∞,1)-colimit]]-[[diagram]].

\end{defn}
(\hyperlink{Lurie}{Lurie, def. 4.3.5.3}).

\begin{remark}
\label{TensorProductFunctor}\hypertarget{TensorProductFunctor}{}
Let $\mathcal{C}^\otimes \to Assoc^\otimes$ exhibit a [[monoidal (∞,1)-category]] such that $\mathcal{C}$ has [[geometric realization]] of [[simplicial objects in an (∞,1)-category|simplicial objects]] and the tensor product preserves these separately in each argument.

Then the tensor product of $\infty$-modules def. \ref{RelativeTensorProduct} extends to an [[(∞,1)-functor]]

\begin{displaymath}
BMod(\mathcal{C})
  \underset{Alg(\mathcal{C})}{\times}
  BMod(\mathcal{C})
  \to 
  BMod(\mathcal{C})
  \,.
\end{displaymath}
\end{remark}
(\hyperlink{Lurie}{Lurie, example 4.3.5.11})

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

\begin{itemize}%
\item [[(∞,1)-bimodules]], [[(∞,2)-category of (∞,1)-bimodules]]

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

Section 4.3.5 of

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

\end{itemize}
[[!redirects tensor product of ∞-modules]]

[[!redirects tensor product of (∞,1)-modules]] [[!redirects tensor product of (infinity,1)-modules]]



\end{document}