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\begin{document}

%-------------------------------------------------------------------


\section*{pushout-product axiom}

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

\hypertarget{model_category_theory}{}\paragraph*{{Model category theory}}\label{model_category_theory}

[[!include model 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}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{remarks}{Remarks}\dotfill \pageref*{remarks} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


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

The \emph{pushout-product axiom} is a compatibility condition between

\begin{enumerate}%
\item a [[closed monoidal category|closed]] [[monoidal category|symmetric monoidal]] structure


\item a [[model category]] structure



\end{enumerate}
on a category.

[[closed monoidal category|Closed]] [[monoidal category|symmetric monoidal categories]] satisfying the pushout-product axiom, together with a unit condition, are called [[monoidal model category|monoidal model categories]] and hence are in particular [[closed monoidal homotopical category|closed monoidal homotopical categories]].

This is relevant in [[enriched homotopy theory]], which pairs [[enriched category theory]] with [[homotopy theory]].

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

Let $C$ be a [[closed monoidal category|closed]] [[monoidal category|symmetric monoidal category]] equipped with a [[model category]] structure.

Then $C$ satisfies the \emph{pushout-product axiom} if for any pair of cofibrations $f : X \to Y$ and $f' : X' \to Y'$ their [[pushout-product]], hence the induced morphism out of the [[coproduct]]

\begin{displaymath}
(X \otimes Y') \coprod_{X \otimes X'} (Y \otimes X')
  \to
  Y \otimes Y'
  \,,
\end{displaymath}
is itself a cofibration, which, furthermore, is acyclic if $f$ or $f'$ is.

This means that the [[tensor product]]

\begin{displaymath}
\otimes : C \times C \to C
\end{displaymath}
is a left [[Quillen bifunctor]].

\hypertarget{remarks}{}\subsection*{{Remarks}}\label{remarks}

\begin{itemize}%
\item This implies in particular that tensoring with cofibrant objects preserves cofibrations and acyclic cofibrations.


\item However the tensor product of two (acyclic) cofibrations is in general not an (acyclic) cofibration.


\item The pushout-product axiom makes sense more generally in the context of a [[two-variable adjunction]] between model categories. This is important in [[enriched homotopy theory]].



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

\begin{itemize}%
\item [[monoidal model category]], [[enriched model category]]


\item [[Joyal-Tierney calculus]]



\end{itemize}
[[!redirects pushout product axiom]]



\end{document}