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\begin{document}

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\section*{external tensor product}

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

\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}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_fiberwise_tensor_product}{Relation to fiberwise tensor product}\dotfill \pageref*{relation_to_fiberwise_tensor_product} \linebreak
\noindent\hyperlink{generation_of__from_external_tensor_products}{Generation of $Mod(X_1 \times X_2)$ from external tensor products}\dotfill \pageref*{generation_of__from_external_tensor_products} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \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 concept of \emph{external tensor product} is a variant of that of [[tensor product]] in a [[monoidal category]] when the latter is generalized to [[indexed monoidal categories]] ([[dependent linear type theory]]).

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

Consider an [[indexed monoidal category]] given by a [[Cartesian fibration]]

\begin{displaymath}
\itexarray{
    Mod(-)
    \\
    \downarrow
    \\
    \mathbf{H}
  }
\end{displaymath}
over a [[cartesian monoidal category]] $\mathbf{H}$.

\begin{defn}
\label{}\hypertarget{}{}
Given $X_1, X_2 \in \mathbf{H}$ the \emph{external tensor product} over these is the [[functor]]

\begin{displaymath}
\boxtimes \;\colon\;
  Mod(X_1)\times Mod(X_2)
  \longrightarrow
  Mod(X_1 \times X_2)
\end{displaymath}
given on $A_1 \in Mod(X_1)$ with $A_2 \in Mod(X_2)$ by

\begin{displaymath}
A_1 \boxtimes A_2 \coloneqq (p_1^\ast A_1) \otimes_{X_1 \times X_2} (p_2^\ast A_2)
  \in Mod(X_1 \times X_2)
  \,,
\end{displaymath}
where $p_1, p_2$ denote the [[projection]] maps out of the [[Cartesian product]] $X_1 \times X_2 \in \mathbf{H}$.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
The external tensor product constitutes a [[tensor product]] on the total category $Mod$ of the given [[Grothendieck fibration]] $Mod(-)\to \mathbf{H}$; and with respect to this it is a [[monoidal fibration]].

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

\hypertarget{relation_to_fiberwise_tensor_product}{}\subsubsection*{{Relation to fiberwise tensor product}}\label{relation_to_fiberwise_tensor_product}

\begin{prop}
\label{}\hypertarget{}{}
The fiberwise (``internal'') tensor product over $X\in \mathbf{H}$ is recovered form the external one via a [[natural equivalence]]

\begin{displaymath}
A_1 \otimes_X A_2 \simeq \Delta_X^\ast (A_1 \boxtimes A_2)
\end{displaymath}
for $A_1, A_2 \in Mod(X)$, where $\Delta \colon X \longrightarrow X \times X$ is the [[diagonal]] in $\mathbf{H}$ on $X$.

\end{prop}
\hypertarget{generation_of__from_external_tensor_products}{}\subsubsection*{{Generation of $Mod(X_1 \times X_2)$ from external tensor products}}\label{generation_of__from_external_tensor_products}

Under suitable conditions on compact generation of $Mod(-)$ then one may deduce that $Mod(X_1 \times X_2)$ is generated under external product from $Mod(X_1)$ and $Mod(X_2)$.

(\hyperlink{BondalvdBerg03}{Bondal-vdBerg 03}, \hyperlink{BFN08}{BFN 08, proof of prop. 3.24})

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item [[external tensor product of vector bundles]]

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

\begin{itemize}%
\item \href{direct+product+group#Representations}{direct product group representations}

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

For general abstract literature dealing with the external tensor products see the references at \emph{[[indexed monoidal category]]} and at \emph{[[dependent linear type theory]]}.

For discussion in the context of categories of [[quasicoherent sheaves]] in ([[derived algebraic geometry|derived]]) see for instance

\begin{itemize}%
\item [[Alexei Bondal]] and M. Van den Bergh, \emph{Generators and representability of functors in commutative and noncommutative geometry}, Mosc. Math. J. 3 (2003), no. 1, 1--36, 258.


\item [[David Ben-Zvi]], [[John Francis]], [[David Nadler]], \emph{Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry}, J. Amer. Math. Soc. 23 (2010), no. 4, 909-966 (\href{http://arxiv.org/abs/0805.0157}{arXiv:0805.0157})



\end{itemize}
[[!redirects external tensor products]]

[[!redirects external product]] [[!redirects external products]]



\end{document}