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\begin{document}

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\section*{(infinity,1)-bimodule}

\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{category_of_bimodules_and_intertwiners}{$(\infty,1)$-Category of $(\infty,1)$-Bimodules and intertwiners}\dotfill \pageref*{category_of_bimodules_and_intertwiners} \linebreak
\noindent\hyperlink{TensorProductOfBimodules}{Tensor products of $(\infty,1)$-Bimodules}\dotfill \pageref*{TensorProductOfBimodules} \linebreak
\noindent\hyperlink{the_category_of_algebras_and_bimodules}{The $(\infty,2)$-Category of $(\infty,1)$-algebras and -bimodules}\dotfill \pageref*{the_category_of_algebras_and_bimodules} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

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

\hypertarget{category_of_bimodules_and_intertwiners}{}\subsubsection*{{$(\infty,1)$-Category of $(\infty,1)$-Bimodules and intertwiners}}\label{category_of_bimodules_and_intertwiners}

Write $BMod^\otimes$ for the [[(∞,1)-category of operators]] of the [[(∞,1)-operad]] [[operad for bimodules]]. Write

\begin{displaymath}
\iota_{\pm} \colon Assoc \to BMod
\end{displaymath}
for the two canonical inclusions of the [[associative operad]] (as discussed at \emph{\hyperlink{RelationToTheAssociativeOperad}{operad for bimodules - relation to the associative operad}}).

\begin{defn}
\label{NotationForWeaklyBiEnrichedInfinityCategory}\hypertarget{NotationForWeaklyBiEnrichedInfinityCategory}{}
For $p \colon \mathcal{C}^\otimes \to BMod^\otimes$ a [[fibration of (∞,1)-operads]], write

\begin{displaymath}
\mathcal{C}^\otimes_{\pm}
  \coloneqq
  \mathcal{C}^\otimes \underset{BMod^\otimes}{\times}^\pm Assoc^\otimes
\end{displaymath}
for the two [[fiber products]] of $p$ with the inclusions $\iota_\pm$. The canonical [[projection]] maps

\begin{displaymath}
\mathcal{C}^\otimes_{\pm}
  \to 
  Assoc^\otimes
\end{displaymath}
exhibit these as two [[planar (∞,1)-operads]].

Finally write

\begin{displaymath}
\mathcal{C} 
    \coloneqq 
  \mathcal{C}^\otimes \underset{BMod^\otimes}{\times} \{\mathfrak{n}\}
\end{displaymath}
for the [[(∞,1)-category]] over the object labeled $\mathfrak{n}$.

\end{defn}
(\hyperlink{Lurie}{Lurie, notation 4.3.1.11}).

\begin{remark}
\label{}\hypertarget{}{}
This exhibits $\mathcal{C}$ as equipped with [[weak tensoring]] over $\mathcal{C}_-$ and reverse weak tensoring over $\mathcal{C}_+$.

\end{remark}
The most familiar special case of these definitions to keep in mind is the following.

\begin{remark}
\label{MonoidalCategoryAsBitensoredOverItself}\hypertarget{MonoidalCategoryAsBitensoredOverItself}{}
For $\mathcal{C}^\otimes \to Assoc^\otimes$ a [[coCartesian fibration of (∞,1)-operads]], hence exhibiting $\mathcal{C}^\otimes$ as a [[monoidal (∞,1)-category]], [[pullback]] along the canonical map $\phi \colon BMod^\otimes \to Assoc^\otimes$ gives a fibration

\begin{displaymath}
\phi^* \mathcal{C}^\otimes \to BMod^\otimes
\end{displaymath}
as in def. \ref{NotationForWeaklyBiEnrichedInfinityCategory} above. In the terminology there this exhibts $\mathcal{C}$ as weakly enriched (weakly [[tensoring|tensored]]) over itself from the left and from the right.

This is the special case for which bimodules are traditionally considered.

\end{remark}
(\hyperlink{Lurie}{Lurie, example 4.3.1.15})

\begin{defn}
\label{}\hypertarget{}{}
For $\mathcal{C}^\otimes \to BMod^\otimes$ a [[fibration of (∞,1)-operads]] we say that the corresponding [[(∞,1)-category]] of [[(∞,1)-algebras over an (∞,1)-operad]]

\begin{displaymath}
BMod(\mathcal{C}) \coloneqq Alg_{/BMod}(\mathcal{C})
\end{displaymath}
is the \textbf{$(\infty,1)$-category of $(\infty,1)$-bimodules} in $\mathcal{C}$.

Composition with the two inclusions $\iota_{1,2}\colon Assoc BMod$ of the [[associative operad]] yields a [[fibration]] in the [[model structure for quasi-categories]] $BMod(\mathcal{C}) \to Alg(\mathcal{C}_-)\times Alg(\mathcal{C}_+)$. Then for $A_- \in Alg_{\mathcal{C}_-}$ and $A_+ \in Alg_{\mathcal{C}_+}$ two algebras the [[fiber product]]

\begin{displaymath}
{}_A BMod_{B}(\mathcal{C})
  \coloneqq
  \{A\} 
   \underset{Alg(\mathcal{C}_-)}{\times}
  BMod(\mathcal{C})
   \underset{Alg(\mathcal{C}_-)}{\times}
  \{B\}
\end{displaymath}
we call the \textbf{$(\infty,1)$-category of $A$-$B$-bimodules}.

\end{defn}
(\hyperlink{Lurie}{Lurie, def. 4.3.1.12})

\begin{example}
\label{}\hypertarget{}{}
For the special case of remark \ref{MonoidalCategoryAsBitensoredOverItself} where the bitensored structure on $\mathcal{C}$ is induced from a monoidal structure $\mathcal{C}^\otimes \to Asoc^\otimes$, we have by the [[universal property]] of the [[pullback]] that

\begin{displaymath}
BMod(\mathcal{C})
  \simeq
  {Alg_{BMod}}_{/Assoc}(\mathcal{C})
  \simeq
  \left\{
    \itexarray{
       && \mathcal{C}
       \\
       &{}^{\mathllap{(A,B,N)}}\nearrow& \downarrow 
       \\
       BMod^\otimes &\to& Assoc^\otimes
    }
  \right\}
\end{displaymath}
\end{example}
\begin{remark}
\label{}\hypertarget{}{}
Let $\mathcal{C}$ be a [[1-category]], for simplicity. Then a [[morphism]]

\begin{displaymath}
(A_1,B_1,N_1) \to $(A_2,B_2,N_2)$
\end{displaymath}
in $BMod(\mathcal{C})$ is a pair $\phi_1 \colon A_1 \to A_1$, $\rho \colon B_1 \to B_2$ of algebra homomorphisms and a morphism $\kappa \colon N_1 \to N_2$ which is ``linear in both $A$ and $B$'' or ``is an [[intertwiner]]'' with respect to $\phi$ and $\rho$ in that for all $a \in A$, $b \in B$ and $n \in N$ we have

\begin{displaymath}
\kappa(a \cdot n \cdot b) = \phi(a) \cdot \kappa(n)    
  \,.
\end{displaymath}
It is natural to depict this by the square diagram

\begin{displaymath}
\itexarray{
    A_1 &\stackrel{N_1}{\to}& B_1
    \\
    {}^{\mathllap{\phi}}\downarrow
    &
    \Downarrow^{\kappa}
    &
    \downarrow^{\mathrlap{\rho}}
    \\
    A_2 &\underset{N_2}{\to}& B_2
  }
  \,.
\end{displaymath}
This notation is naturally suggestive of the existence of the further [[horizontal composition]] by [[tensor product of (∞,1)-modules]], which we come to \hyperlink{TensorProductOfBimodules}{below}.

On the other hand, a morphism $N_1 \to N_2$ in ${}_A BMod(\mathcal{C})_B$ is given by the special case of the above for $\phi = id$ and $\rho = id$.

\end{remark}
\hypertarget{TensorProductOfBimodules}{}\subsubsection*{{Tensor products of $(\infty,1)$-Bimodules}}\label{TensorProductOfBimodules}

\begin{defn}
\label{NotationForTensS}\hypertarget{NotationForTensS}{}
Write $Tens^\otimes$ for the [[generalized (∞,1)-operad]] discussed at \emph{[[tensor product of ∞-modules]]}.

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]].

Moreover, 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})

\hypertarget{the_category_of_algebras_and_bimodules}{}\subsubsection*{{The $(\infty,2)$-Category of $(\infty,1)$-algebras and -bimodules}}\label{the_category_of_algebras_and_bimodules}

We discuss the generalization of the notion of bimodules to [[homotopy theory]], hence the generalization from [[category theory]] to [[(∞,1)-category theory]]. (\hyperlink{Lurie}{Lurie, section 4.3}).

Let $\mathcal{C}$ be [[monoidal (∞,1)-category]] such that

\begin{enumerate}%
\item it admits [[geometric realization]] of [[simplicial objects in an (∞,1)-category]] (hence a [[left adjoint|left]] [[adjoint (∞,1)-functor]] ${\vert-\vert} \colon \mathcal{C}^{\Delta^{op}} \to \mathcal{C}$ to the constant simplicial object functor), true notably when $\mathcal{C}$ is a [[presentable (∞,1)-category]];


\item the [[tensor product]] $\otimes \colon \mathcal{C}\times \mathcal{C} \to \mathcal{C}$ preserves this geometric realization separately in each argument.



\end{enumerate}
Then there is an [[(∞,2)-category]] $Mod(\mathcal{C})$ which given as an [[(∞,1)-category object]] internal to [[(∞,1)Cat]] has

\begin{itemize}%
\item $(\infty,1)$-category of objects

\begin{displaymath}
Mod(\mathcal{C})_{[0]} \simeq Alg(\mathcal{C})
\end{displaymath}
the [[A-∞ algebras]] and [[∞-algebra]] [[homomorphisms]] in $\mathcal{C}$;


\item $(\infty,1)$-category of morphisms

\begin{displaymath}
Mod(\mathcal{C})_{[1]} \simeq BMod(\mathcal{C})
\end{displaymath}
the $\infty$-bimodules and bimodule homomorphisms ([[intertwiners]]) in $\mathcal{C}$



\end{itemize}
This is (\hyperlink{Lurie}{Lurie, def. 4.3.6.10, remark 4.3.6.11}).

Morover, the [[horizontal composition]] of bimodules in this [[(∞,2)-category]] is indeed the relative [[tensor product of ∞-modules]]

\begin{displaymath}
\circ_{A,B,C}
  =
  (-) \otimes_B (-)
  \;\colon\;
  {}_A Mod_{B} \times {}_{B}Mod_C \to {}_A Mod_C
  \,.
\end{displaymath}
This is (\hyperlink{Lurie}{Lurie, lemma 4.3.6.9 (3)}).

Here are some steps in the construction:

The \textbf{idea} of the following constructions is this: we start with a [[generalized (∞,1)-operad]] $Tens^\otimes \to FinSet_* \times \Delta^{op}$ which is such that the [[(∞,1)-algebras over an (∞,1)-operad]] over its fiber over $[k] \in \Delta^{op}$ is equivalently the collection of $(k+1)$-tuples of [[A-∞ algebras]] in $\mathcal{C}$ together with a string of $k$ $\infty$-bimodules between them. Then we turn that into a [[simplicial object in an (∞,1)-category|simplicial object]] in [[(∞,1)Cat]]

\begin{displaymath}
Mod(\mathcal{C}) \in ((\infty,1)Cat)^{\Delta^{op}}
  \,.
\end{displaymath}
This turns out to be an [[internal (∞,1)-category]] object in [[(∞,1)Cat]], hence an [[(∞,2)-category]] whose object of objects is the category $Alg(\mathcal{C})$ of [[A-∞ algebras]] and [[homomorphisms]] in $\mathcal{C}$ between them, and whose object of morphisms is the category $BMod(\mathcal{C})$ of $\infty$-bimodules and [[intertwiners]].

\begin{defn}
\label{}\hypertarget{}{}
Define $Mod(\mathcal{C}) \to \Delta^{op}$ as the map of [[simplicial sets]] with the [[universal property]] that for every other map of simplicial set $K \to \Delta^{op}$ there is a canonical bijection

\begin{displaymath}
Hom_{sSet/\Delta^{op}}(K, Mod(\mathcal{C}))
  \simeq
  Alg_{Tens_K / Assoc}( \mathcal{C} )
  \,,
\end{displaymath}
where

\begin{itemize}%
\item on the left we have the hom-simplicial set in the [[slice category]]


\item on the right we have the [[(∞,1)-category]] of [[(∞,1)-algebras over an (∞,1)-operad]] given by lifts $\mathcal{A}$ in

\begin{displaymath}
\itexarray{
     && \mathcal{C}^\otimes
     \\
     &{}^{\mathcal{A}}\nearrow& \downarrow
     \\
    Tens_K &\to& Assoc
  }
  \,.
\end{displaymath}


\end{itemize}
\end{defn}
This is (\hyperlink{Lurie}{Lurie, cor. 4.3.6.2}) specified to the case of (\hyperlink{Lurie}{Lurie, lemma 4.3.6.9}). Also (\hyperlink{Lurie}{Lurie, def. 4.3.4.19})

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

The general theory in terms of [[higher algebra]] of [[(∞,1)-operads]] is discussed in section 4.3 of

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

\end{itemize}
Specifically the homotopy theory of [[A-infinity bimodules]] is discussed in

\begin{itemize}%
\item Volodymyr Lyubashenko, Oleksandr Manzyuk, \emph{A-infinity-bimodules and Serre A-infinity-functors} (\href{http://arxiv.org/abs/math/0701165}{arXiv:math/0701165})

\end{itemize}
and section 5.4.1 of

\begin{itemize}%
\item [[Boris Tsygan]], \emph{Noncommutative calculus and operads} in Guillermo Cortinas (ed.) \emph{Topics in Noncommutative geometry}, Clay Mathematics Proceedings volume 16

\end{itemize}
The generalization to [[(infinity,n)-modules]] is in

\begin{itemize}%
\item [[Rune Haugseng]], \emph{The higher Morita category of $E_n$-algebras} (\href{http://arxiv.org/abs/1412.8459}{arXiv:1412.8459})

\end{itemize}
[[!redirects (infinity,1)-bimodules]]

[[!redirects ∞-bimodule]] [[!redirects ∞-bimodules]] [[!redirects (∞,1)-bimodule]] [[!redirects (∞,1)-bimodules]]

[[!redirects (∞,1)-category of (∞,1)-bimodules]] [[!redirects (infinity,1)-category of (∞,1)-bimodules]]

[[!redirects (∞,1)-categories of (∞,1)-bimodules]] [[!redirects (infinity,1)-categories of (∞,1)-bimodules]]

[[!redirects infinity-bimodule]] [[!redirects infinity-bimodules]]



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