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\begin{document}

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\section*{smash product}

\begin{quote}%
This article is about smash products in [[topology]]/[[homotopy theory]]. For the notion of \emph{Hopf smash product} see at [[crossed product algebra]].


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

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{stable_homotopy_theory}{}\paragraph*{{Stable Homotopy theory}}\label{stable_homotopy_theory}

[[!include stable homotopy theory - 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{ForPointedSets}{For pointed sets}\dotfill \pageref*{ForPointedSets} \linebreak
\noindent\hyperlink{ForGeneralPointedObjects}{For general pointed objects}\dotfill \pageref*{ForGeneralPointedObjects} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{of_pointed_topological_spaces}{Of pointed topological spaces}\dotfill \pageref*{of_pointed_topological_spaces} \linebreak
\noindent\hyperlink{of_spectra}{Of spectra}\dotfill \pageref*{of_spectra} \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 \emph{smash product} is the canonical [[tensor product]] of [[pointed objects]] in an ambient [[monoidal category]]. It is essentially given by taking the tensor product of the underlying objects and then identifying all pieces that contain the base point of either with a new basepoint.

An archetypical special case is the smash product of [[pointed topological spaces]] and hence of pointed [[homotopy types]]. Under [[stabilization]] this induces the important [[smash product of spectra]] in [[stable homotopy theory]].

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

\hypertarget{ForPointedSets}{}\subsubsection*{{For pointed sets}}\label{ForPointedSets}

\begin{defn}
\label{}\hypertarget{}{}
The \textbf{smash product} $A \wedge B$ of two [[pointed sets]] $A$ and $B$ is the [[quotient set]] of the [[cartesian product]] $A \times B$ where all points with the basepoint as a coordinate (the one from $A$ or the one from $B$) are identified.

The subset that is `smashed' here can be identified with the [[wedge sum]] $A \vee B$, so the definition of the smash product can be summarised as follows:

\begin{displaymath}
A \wedge B = \frac{A \times B}{A \vee B}
\end{displaymath}
\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
The smash product is the [[tensor product]] in the [[closed monoidal category]] of [[pointed sets]].\newline That is, it is characterized by the existence of [[natural isomorphisms]]

\begin{displaymath}
Fun_*(A \wedge B, C) \cong Fun_*(A, Fun_*(B, C))
\end{displaymath}
where $Fun_*(A,B)$ is the set of basepoint-preserving [[functions]] from $A$ to $B$, itself made into a pointed set by taking as basepoint the [[constant function]] from all of $A$ to the basepoint in $B$.

\end{prop}
This is a special case of the general discussion \hyperlink{ForGeneralPointedObjects}{below}.

\hypertarget{ForGeneralPointedObjects}{}\subsubsection*{{For general pointed objects}}\label{ForGeneralPointedObjects}

Let $(\mathcal{C}, \otimes, 1_{\mathcal{C}})$ be a [[closed monoidal category|closed]] [[symmetric monoidal category]] with ([[finite limit|finite]]) [[limits]] and [[colimits]]. Write $\ast \in \mathcal{C}$ for the [[terminal object]] of $\mathcal{C}$. Write $\mathcal{C}^{\ast/}$ for the category of [[pointed objects]] in $\mathcal{C}$.

\begin{defn}
\label{GeneralSmashProduct}\hypertarget{GeneralSmashProduct}{}
For $X,Y \in \mathcal{C}^{\ast/}$ two [[pointed objects]] in $\mathcal{C}$, their \emph{smash product} is given by the following [[pushout]] of [[pushouts]] and [[tensor products]] all formed in $\mathcal{C}$

\begin{displaymath}
X \wedge Y
  \coloneqq
  \ast \underset{(X \otimes \ast)\coprod (Y \otimes \ast)}{\coprod} (X \otimes Y)
\end{displaymath}
regarded as a pointed object via the induced [[co-projection]] from $\ast$.

\end{defn}
In this generality this appears as (\hyperlink{ElmendorfMandell07}{Elmendorf-Mandell 07, construction 4.19}).

\begin{prop}
\label{}\hypertarget{}{}
The smash product of def. \ref{GeneralSmashProduct} makes $\mathcal{C}^{\ast/}$ be a [[closed monoidal category|closed]] [[symmetric monoidal category]] with ([[finite limit|finite]]) [[limits]] and [[colimits]].

\end{prop}
A proof appears as (\hyperlink{ElmendorfMandell07}{Elmendorf-Mandell 07, lemma 4.20}). For more of these details see at \emph{\href{pointed+object#ClosedMonoidalStructure}{Pointed object -- Closed and monoidal structure}}. For [[base change]] functoriality of these structures see at \emph{\href{Wirthm&#252;ller+context#PointedObjects}{Wirthm\"u{}ller context -- Examples -- On pointed objects}}.

\begin{remark}
\label{}\hypertarget{}{}
The formula for the smash product in def. \ref{GeneralSmashProduct} can be considered in any [[category]] $\mathcal{C}$ with [[finite limits]] and [[colimits]], but unless $\mathcal{C}$ is closed symmetric monoidal, it will not have all these properties.

If finite [[products]] in $C$ preserve finite colimits, then the smash product is [[associativity|associative]], and if $C$ is also [[cartesian closed category|cartesian closed]], then it makes the category of pointed objects in $C$ [[closed monoidal category|closed monoidal]]. However, if finite products in $C$ do not preserve finite colimits, the smash product can fail to be associative.

\end{remark}
\begin{example}
\label{}\hypertarget{}{}
Examples of closed symmetric monoidal categories $(\mathcal{C}, \otimes 1_{\mathcal{C}})$ include in particular [[toposes]] with their [[cartesian monoidal category|cartesian monoidal structure]]. For the topos $\mathcal{C} =$ [[Set]] the general discussion here reduces to that \hyperlink{ForPointedSets}{above}.

\end{example}
There is a general abstract way to obtain this smash product monoidal structure:

\begin{prop}
\label{}\hypertarget{}{}
The category of [[pointed objects]] is the [[Eilenberg-Moore category]] of [[algebras over a monad]] for the ``[[maybe monad]]'', $X \mapsto X \coprod \ast$. This being a suitably [[monoidal monad]] it canonically induces a monoidal structure on its EM-category, and that is the smash product.

\end{prop}
For more on this see at \emph{\href{maybe+monad#EMCategoryAndRelationToPointedObjects}{maybe monad -- EM-Category and Relation to pointed objects}}.

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

\hypertarget{of_pointed_topological_spaces}{}\subsubsection*{{Of pointed topological spaces}}\label{of_pointed_topological_spaces}

The most common case when $C$ is a category of [[topological spaces]]. In that case, the [[natural transformation|natural]] [[continuous function|map]] $A\wedge (B\wedge C)\to (A\wedge B)\wedge C$ is a [[homeomorphism]] provided $C$ is a [[locally compact Hausdorff space]]. Thus if both $A$ and $C$ are locally compact Hausdorff, then we have the [[associativity]] $A\wedge(B\wedge C)\cong (A\wedge B)\wedge C$.

Associativity fails in general for the category [[Top]] of all topological spaces; however, it is satisfied for pointed objects in any [[convenient category of topological spaces]], since such a category is cartesian closed. In particular, the smash product is associative for pointed [[compactly generated spaces]].

\hypertarget{of_spectra}{}\subsubsection*{{Of spectra}}\label{of_spectra}

See at \emph{[[symmetric smash product of spectra]]}.

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

\begin{itemize}%
\item [[pointed object]]


\item [[wedge sum]]


\item [[pointed mapping space]]



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

On the general definition of smash products via [[closed monoidal category]] structure on [[pointed objects]]

\begin{itemize}%
\item [[Anthony Elmendorf]], [[Michael Mandell]], \emph{Permutative categories, multicategories, and algebraic K-theory}, Algebraic \& Geometric Topology 9 (2009) 2391-2441 (\href{http://arxiv.org/abs/0710.0082}{arXiv:0710.0082})

\end{itemize}
On commutativity of smashing with [[homotopy limits]]:

\begin{itemize}%
\item [[Wolfgang Lueck]], Holger Reich, Marco Varisco, \emph{Commuting homotopy limits and smash products}, K-Theory, 30 (2): 137--165, 2003 (\href{http://arxiv.org/abs/math/0302116}{arXiv:math/0302116})

\end{itemize}
[[!redirects smash product]] [[!redirects smash products]] [[!redirects smash produc]]



\end{document}