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\begin{document}

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\section*{symmetric smash product of spectra}

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

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

[[!include stable homotopy theory - contents]]

\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{details}{Details}\dotfill \pageref*{details} \linebreak
\noindent\hyperlink{for_modules}{For $S$-modules}\dotfill \pageref*{for_modules} \linebreak
\noindent\hyperlink{for_excisive_functors}{For excisive functors}\dotfill \pageref*{for_excisive_functors} \linebreak
\noindent\hyperlink{TopologicalEndsAndCoends}{Topological ends and coends}\dotfill \pageref*{TopologicalEndsAndCoends} \linebreak
\noindent\hyperlink{monoidal_topological_categories}{Monoidal topological categories}\dotfill \pageref*{monoidal_topological_categories} \linebreak
\noindent\hyperlink{AlgebrasAndModules}{Algebras and modules}\dotfill \pageref*{AlgebrasAndModules} \linebreak
\noindent\hyperlink{day_convolution}{Day convolution}\dotfill \pageref*{day_convolution} \linebreak
\noindent\hyperlink{FunctorsWithSmashProduct}{Functors with smash product}\dotfill \pageref*{FunctorsWithSmashProduct} \linebreak
\noindent\hyperlink{OnPreExcisiveFunctors}{Pre-Excisive functors}\dotfill \pageref*{OnPreExcisiveFunctors} \linebreak
\noindent\hyperlink{ForSymmetricAndOrthogonalSpectra}{For symmetric and orthogonal spectra}\dotfill \pageref*{ForSymmetricAndOrthogonalSpectra} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{original_sources}{Original sources}\dotfill \pageref*{original_sources} \linebreak
\noindent\hyperlink{reviews_and_introductions}{Reviews and introductions}\dotfill \pageref*{reviews_and_introductions} \linebreak


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

A \emph{symmetric smash product of spectra} is a realization of the [[smash product of spectra]] such as to make a [[symmetric monoidal category|symmetric]] [[monoidal model category]] [[presentable (infinity,1)-category|presentation]] of the [[symmetric monoidal (infinity,1)-category of spectra]].

In [[higher algebra]] and [[stable homotopy theory]] one is interested in [[monoid in a monoidal (∞,1)-category|monoid objects]] in the [[stable (∞,1)-category of spectra]] -- called $A_\infty$-[[A-∞-ring|rings]] -- and [[commutative monoid in an (infinity,1)-category|commutative monoid objects]] -- called $E_\infty$-[[E-∞-ring|rings]]. These monoid objects satisfy associativity, uniticity and, in the $E_\infty$-case, commutativity up to [[coherence|coherent]] [[higher homotopies]].

For concretely working with these objects, it is often useful to have concrete [[category theory|1-categorical]] algebraic models for these intricate [[higher category theory|higher categorical]]/homotopical entities. The \emph{symmetric monoidal smash product of spectra} is a structure that allows to model [[A-infinity rings]] as ordinary [[monoids]] and [[E-infinity rings]] as ordinary [[commutative monoids]] in a suitable ordinary [[category]] -- one speaks of \emph{[[highly structured ring spectra]]}.

Historically, this had been desired but out of reach for a long time, due to the initial focus on the model by plain [[sequential spectra]]. By \href{smash+product+of+spectra#WhySequentialSpectraHaveNoSymmetricSmashProduct}{this remark} at \emph{[[smash product of spectra]]}, plain sequential spectra do not reflect the graded-commutativity implicit in the [[braiding]] of the [[smash product]] of [[n-spheres]] and thus do not admit a symmetric smash product of spectra.

When the relevant [[highly structured ring spectra]] were finally found that do admit symmetric smash products, the relief was substantial and led to terminology such as ``[[brave new algebra]]''. More recently maybe the term [[higher algebra]] is becoming more popular.

Then, model structures were found which also admit symmetric monoidal smash products, but which are not of the form ``highly structured spectra'': [[model structure for excisive functors]].

As a first step one wants a [[model category of spectra]] $\mathcal{S}$ that [[presentable (infinity,1)-category|presents]] the full [[(infinity,1)-category of spectra]]. This allows to model the notion of [[equivalence]] of spectra and of [[homotopies]] between maps of spectra. Several such model categories have been known for a long time; all are Quillen equivalent and their common [[homotopy category]] is called ``the'' [[stable homotopy category]] $Ho \mathcal{S}$.

Now, for some of the model categories $\mathcal{S}$ of spectra, the smash product on $Ho \mathcal{S}$ can be lifted to a functor

\begin{displaymath}
\wedge\colon \mathcal{S} \times \mathcal{S} \to \mathcal{S}
  \,,
\end{displaymath}
but for the most part these functors were neither associative nor unital nor commutative at the level of the 1-category $\mathcal{S}$. In fact (\hyperlink{Lewis91}{Lewis 91}) proved a theorem that there could be \emph{no} symmetric monoidal category $\mathcal{S}$ modeling the stable homotopy category and satisfying a couple of other natural requirements.

However, in the 1990s it was realized that by dropping one or another of Lewis' other requirements, symmetric monoidal categories of spectra could be produced. The first such category was the category of \textbf{[[S-module]]s} described by \hyperlink{ElmendorfKrizMandellMay97}{Elmendorf-Kriz-Mandell-May 97}, but others soon followed, including \textbf{[[symmetric spectra]]} and \textbf{[[orthogonal spectra]]}. All of these form symmetric [[monoidal model categories]] which are symmetric-monoidally [[Quillen equivalence|Quillen equivalent]].

Moreover, in all of these cases, the monoidal structure on the model category $\mathcal{S}$ absorbs all the higher coherent homotopies that used to be supplied by the action of an $A_\infty$ or $E_\infty$ operad. Thus, honest (commutative) monoids in $\mathcal{S}$ model the same ``(commutative) ring objects up to all coherent higher homotopies'' that are modeled by the classical $A_\infty$ and $E_\infty$ ring spectra, and for this reason they are often still referred to as $A_\infty$ or $E_\infty$ ring spectra, respectively.

\hypertarget{details}{}\subsection*{{Details}}\label{details}

\hypertarget{for_modules}{}\subsubsection*{{For $S$-modules}}\label{for_modules}

The construction of [[S-modules]] by EKMM begins with the notion of [[coordinate-free spectrum|coordinate free Lewis-May spectra]]. Using the [[linear isometries operad]], one can construct a [[monad]] $\mathbb{L}$ on the category $\mathcal{S}$ of such spectra, and the category of $\mathbb{L}$-algebras is a well-behaved model for the stable homotopy category, and moreover admits a smash product which is associative up to isomorphism, but unital only up to weak equivalence. However, the subcategory of the $\mathbb{L}$-algebras for which the unit transformations are isomorphisms is again a well-behaved model for $Ho \mathbb{S}$, which is moreover symmetric monoidal.

Since the unit transformation is of the form $S\wedge E \to E$, where $S$ is the [[sphere spectrum]], and this map looks like the action of a ring on a module, the objects of this subcategory are called \textbf{$S$-modules} and the category is called $Mod_S$. The intuition is that just as an abelian group is a [[module]] over the archetypical ring $\mathbb{Z}$ of [[integer|integers]], a spectrum should be regarded as a module over the archetypal ring spectrum, namely the sphere spectrum.

Similarly, just as an ordinary [[ring]] is a [[monoid]] in the category $Mod_\mathbb{Z}$ of $\mathbb{Z}$-[[module]]s, i.e. a $\mathbb{Z}$-algebra, an $A_\infty$ or $E_\infty$ ring spectrum is a (possibly commutative) monoid in the category of $S$-modules, and thus referred to as an \textbf{$S$-algebra}. More generally, for any $A_\infty$-[[A-infinity-ring|ring spectrum]] $R$, there is a notion of $R$-[[module]] spectra forming a category $Mod_R$, which in turn carries an associative and commutative smash product $\wedge_R$ and a [[model category]] structure on $Mod_R$ such that $\wedge_R$ becomes unital in the [[homotopy category]]. All this is such that an $A_\infty$-[[A-infinity-algebra|algebra]] over $R$ is a [[monoid object]] in $(Mod_R, \wedge_R)$. Similarly $E_\infty$-[[E-infinity-algebra|algebras]] are commutative monoid objects in $(Mod_R, \wedge_R)$.

\hypertarget{for_excisive_functors}{}\subsubsection*{{For excisive functors}}\label{for_excisive_functors}

\hypertarget{TopologicalEndsAndCoends}{}\paragraph*{{Topological ends and coends}}\label{TopologicalEndsAndCoends}

For working with pointed [[topologically enriched functors]], a certain shape of [[limits]]/[[colimits]] is particularly relevant: these are called (pointed topological enriched) \emph{[[ends]]} and \emph{[[coends]]}. We here introduce these and then derive some of their basic properties, such as notably the expression for topological [[left Kan extension]] in terms of [[coends]] (prop. \ref{TopologicalLeftKanExtensionBCoend} below). Further below it is via left Kan extension along the ordinary smash product of pointed topological spaces (``[[Day convolution]]'') that the [[symmetric monoidal smash product of spectra]] is induced.

\begin{defn}
\label{OppositeAndProductOfPointedTopologicallyEnrichedCategory}\hypertarget{OppositeAndProductOfPointedTopologicallyEnrichedCategory}{}
Let $\mathcal{C}, \mathcal{D}$ be pointed [[topologically enriched categories]] (\href{Introduction+to+Stable+homotopy+theory+--+P#TopEnrichedCategory}{def.}), i.e. [[enriched categories]] over $(Top_{cg}^{\ast/}, \wedge, S^0)$ from example \ref{PointedTopologicalSpacesWithSmashIsSymmetricMonoidalCategory}.

\begin{enumerate}%
\item The \textbf{pointed topologically enriched [[opposite category]]} $\mathcal{C}^{op}$ is the [[topologically enriched category]] with the same [[objects]] as $\mathcal{C}$, with [[hom-spaces]]

\begin{displaymath}
\mathcal{C}^{op}(X,Y)
  \coloneqq
  \mathcal{C}(Y,X)
\end{displaymath}
and with [[composition]] given by [[braiding]] followed by the composition in $\mathcal{C}$:

\begin{displaymath}
\mathcal{C}^{op}(X,Y)
  \wedge
  \mathcal{C}^{op}(Y,Z)
  =
  \mathcal{C}(Y,X)\wedge \mathcal{C}(Z,Y)
  \underoverset{\simeq}{\tau}{\longrightarrow}
  \mathcal{C}(Z,Y) \wedge \mathcal{C}(Y,X)
    \overset{\circ_{Z,Y,X}}{\longrightarrow}
  \mathcal{C}(Z,X)
  =
  \mathcal{C}^{op}(X,Z) 
  \,.
\end{displaymath}

\item the \textbf{pointed topological [[product category]]} $\mathcal{C} \times \mathcal{D}$ is the [[topologically enriched category]] whose [[objects]] are [[pairs]] of objects $(c,d)$ with $c \in \mathcal{C}$ and $d\in \mathcal{D}$, whose [[hom-spaces]] are the [[smash product]] of the separate hom-spaces

\begin{displaymath}
(\mathcal{C}\times \mathcal{D})((c_1,d_1),\;(c_2,d_2) )
    \coloneqq
  \mathcal{C}(c_1,c_2)\wedge \mathcal{D}(d_1,d_2)
\end{displaymath}
and whose [[composition]] operation is the [[braiding]] followed by the [[smash product]] of the separate composition operations:

\begin{displaymath}
\itexarray{
    (\mathcal{C}\times \mathcal{D})((c_1,d_1), \; (c_2,d_2))
      \wedge
    (\mathcal{C}\times \mathcal{D})((c_2,d_2), \; (c_3,d_3))
    \\
    {}^{\mathllap{=}}\downarrow
    \\
    \left(\mathcal{C}(c_1,c_2) \wedge \mathcal{D}(d_1,d_2)\right)
      \wedge
    \left(\mathcal{C}(c_2,c_3) \wedge \mathcal{D}(d_2,d_3)\right)
    \\
    \downarrow^{\mathrlap{\tau}}_{\mathrlap{\simeq}}
    \\
    \left(\mathcal{C}(c_1,c_2)\wedge \mathcal{C}(c_2,c_3)\right)
      \wedge
    \left( \mathcal{D}(d_1,d_2)\wedge \mathcal{D}(d_2,d_3)\right)
      &\overset{
    (\circ_{c_1,c_2,c_3})\wedge (\circ_{d_1,d_2,d_3})
    }{\longrightarrow}
      &
    \mathcal{C}(c_1,c_3)\wedge \mathcal{D}(d_1,d_3)
    \\
    && \downarrow^{\mathrlap{=}}
    \\
    && (\mathcal{C}\times \mathcal{D})((c_1,d_1),\; (c_3,d_3))
  }
 \,.
\end{displaymath}


\end{enumerate}
\end{defn}
\begin{example}
\label{PointedTopologicalFunctorOnProductCategory}\hypertarget{PointedTopologicalFunctorOnProductCategory}{}
A pointed [[topologically enriched functor]] (\href{Introduction+to+Stable+homotopy+theory+--+P#TopologicallyEnrichedFunctor}{def.}) into $Top^{\ast/}_{cg}$ (\href{Introduction+to+Stable+homotopy+theory+--+P#TopologicallyEnrichedFunctorsToTopk}{exmpl.}) out of a pointed topological [[product category]] as in def. \ref{OppositeAndProductOfPointedTopologicallyEnrichedCategory}

\begin{displaymath}
F 
    \;\colon\;
  \mathcal{C} \times \mathcal{D}
    \longrightarrow
  Top^{\ast/}_{cg}
\end{displaymath}
(a ``pointed topological [[bifunctor]]'') has component maps of the form

\begin{displaymath}
F_{(c_1,d_1),(c_2,d_2)}
    \;\colon\;
  \mathcal{C}(c_1,c_2)
  \wedge
  \mathcal{D}(d_1,d_2)
    \longrightarrow
  Maps(F_0((c_1,d_1)),F_0((c_2,d_2)))_\ast
  \,.
\end{displaymath}
By functoriallity and under passing to [[adjuncts]] (\href{Introduction+to+Stable+homotopy+theory+--+P#SmashHomAdjunctionOnPointedCompactlyGeneratedTopologicalSpaces}{cor.}) this is equivalent to two commuting \emph{[[actions]]}

\begin{displaymath}
\rho_{c_1,c_2}(d)
   \;\colon\;
  \mathcal{C}(c_1,c_2) \wedge F_0((c_1,d)) 
    \longrightarrow
  F_0((c_2,d))
\end{displaymath}
and

\begin{displaymath}
\rho_{d_1,d_2}(c)
    \;\colon\;
  \mathcal{D}(d_1,d_2) \wedge F_0((c,d_1))
    \longrightarrow
  F_0((c,d_2))
  \,.
\end{displaymath}
In the special case of a functor out of the [[product category]] of some $\mathcal{C}$ with its [[opposite category]] (def. \ref{OppositeAndProductOfPointedTopologicallyEnrichedCategory})

\begin{displaymath}
F 
    \;\colon\;
  \mathcal{C}^{op} \times \mathcal{C}
    \longrightarrow
  Top^{\ast/}_{cg}
\end{displaymath}
then this takes the form

\begin{displaymath}
\rho_{c_2,c_1}(d)
   \;\colon\;
  \mathcal{C}(c_1,c_2) \wedge F_0((c_2,d)) 
    \longrightarrow
  F_0((c_1,d))
\end{displaymath}
and

\begin{displaymath}
\rho_{d_1,d_2}(c)
    \;\colon\;
  \mathcal{C}(d_1,d_2) \wedge F_0((c,d_1))
    \longrightarrow
  F_0((c,d_2))
  \,.
\end{displaymath}
\end{example}
\begin{defn}
\label{EndAndCoendInTopcgSmash}\hypertarget{EndAndCoendInTopcgSmash}{}
Let $\mathcal{C}$ be a [[small category|small]] pointed [[topologically enriched category]] (\href{Introduction+to+Stable+homotopy+theory+--+P#TopEnrichedCategory}{def.}), i.e. an [[enriched category]] over $(Top_{cg}^{\ast/}, \wedge, S^0)$ from example \ref{PointedTopologicalSpacesWithSmashIsSymmetricMonoidalCategory}. Let

\begin{displaymath}
F 
    \;\colon\;
  \mathcal{C}^{op} \times \mathcal{C}
    \longrightarrow
  Top^{\ast/}_{cg}
\end{displaymath}
be a pointed [[topologically enriched functor]] (\href{Introduction+to+Stable+homotopy+theory+--+P#TopologicallyEnrichedFunctor}{def.}) out of the pointed topological [[product category]] of $\mathcal{C}$ with its [[opposite category]], according to def. \ref{OppositeAndProductOfPointedTopologicallyEnrichedCategory}.

\begin{enumerate}%
\item The \textbf{[[coend]]} of $F$, denoted $\overset{c \in \mathcal{C}}{\int} F(c,c)$, is the [[coequalizer]] in $Top_{cg}^{\ast}$ (\href{Introduction+to+Stable+homotopy+theory+--+P#DescriptionOfLimitsAndColimitsInTop}{prop.}, \href{Introduction+to+Stable+homotopy+theory+--+P#CoequalizerInTop}{exmpl.}, \href{Introduction+to+Stable+homotopy+theory+--+P#LimitsAndColimitsOfPointedObjects}{prop.}, \href{Introduction+to+Stable+homotopy+theory+--+P#kTopIsCoreflectiveSubcategory}{cor.}) of the two actions encoded in $F$ via example \ref{PointedTopologicalFunctorOnProductCategory}:

\begin{displaymath}
\underset{c,d \in \mathcal{C}}{\coprod} \mathcal{C}(c,d) \wedge F(d,c)
   \underoverset
     {\underset{\underset{c,d}{\sqcup} \rho_{(d,c)}(c) }{\longrightarrow}}
     {\overset{\underset{c,d}{\sqcup} \rho_{(c,d)}(d) }{\longrightarrow}}
     {\phantom{AAAAAAAA}}
  \underset{c \in \mathcal{C}}{\coprod} F(c,c)
   \overset{coeq}{\longrightarrow}
  \overset{c\in \mathcal{C}}{\int} F(c,c)
  \,.
\end{displaymath}

\item The \textbf{[[end]]} of $F$, denoted $\underset{c\in \mathcal{C}}{\int} F(c,c)$, is the \textbf{[[equalizer]]} in $Top_{cg}^{\ast/}$ (\href{Introduction+to+Stable+homotopy+theory+--+P#DescriptionOfLimitsAndColimitsInTop}{prop.}, \href{Introduction+to+Stable+homotopy+theory+--+P#EqualizerInTop}{exmpl.}, \href{Introduction+to+Stable+homotopy+theory+--+P#LimitsAndColimitsOfPointedObjects}{prop.}, \href{Introduction+to+Stable+homotopy+theory+--+P#kTopIsCoreflectiveSubcategory}{cor.}) of the [[adjuncts]] of the two actions encoded in $F$ via example \ref{PointedTopologicalFunctorOnProductCategory}:

\begin{displaymath}
\underset{c\in \mathcal{C}}{\int} F(c,c)
    \overset{\;\;equ\;\;}{\longrightarrow}
  \underset{c \in \mathcal{C}}{\prod} F(c,c)
   \underoverset   
     {\underset{\underset{c,d}{\sqcup} \tilde \rho_{(c,d)}(c)  }{\longrightarrow}}
     {\overset{\underset{c,d}{\sqcup} \tilde\rho_{d,c}(d)}{\longrightarrow}}
     {\phantom{AAAAAAAA}}
   \underset{c\in \mathcal{C}}{\prod}
   Maps\left( \mathcal{C}\left(c,d\right), \;  F\left(c,d\right) \right)_\ast
  \,.
\end{displaymath}


\end{enumerate}
\end{defn}
\begin{example}
\label{NaturalTransformationsViaEnds}\hypertarget{NaturalTransformationsViaEnds}{}
Let $\mathcal{C}$ be a [[small category|small]] pointed [[topologically enriched category]] (\href{Introduction+to+Stable+homotopy+theory+--+P#TopEnrichedCategory}{def.}). For $F,G
  \;\colon\;
  \mathcal{C}
    \longrightarrow
  Top^{\ast/}_{cg}$ two pointed [[topologically enriched functors]], then the [[end]] (def. \ref{EndAndCoendInTopcgSmash}) of $Maps(F(-),G(-))_\ast$ is a topological space whose underlying [[pointed set]] is the pointed set of [[natural transformations]] $F\to G$ (\href{Introduction+to+Stable+homotopy+theory+--+P#TopologicallyEnrichedFunctor}{def.})

\begin{displaymath}
U
  \left(
    \underset{c \in \mathcal{C}}{\int} Maps(F(c),G(c))_\ast
  \right)
  \;\simeq\;
  Hom_{[\mathcal{C},Top^{\ast/}_{cg}]}(F,G)
  \,.
\end{displaymath}
\end{example}
\begin{proof}
The underlying pointed set functor $U\colon Top^{\ast/}_{cg}\to Set^{\ast/}$ [[preserved limit|preserves]] all [[limits]] (\href{Introduction+to+Stable+homotopy+theory+--+P#DescriptionOfLimitsAndColimitsInTop}{prop.}, \href{Introduction+to+Stable+homotopy+theory+--+P#LimitsAndColimitsOfPointedObjects}{prop.}, \href{Introduction+to+Stable+homotopy+theory+--+P#kTopIsCoreflectiveSubcategory}{prop.}). Therefore there is an [[equalizer]] diagram in $Set^{\ast/}$ of the form

\begin{displaymath}
U
  \left(
    \underset{c\in \mathcal{C}}{\int}
    Maps(F(c),G(c))_\ast
  \right)
    \overset{equ}{\longrightarrow}
  \underset{c\in \mathcal{C}}{\prod} Hom_{Top^{\ast/}_{cg}}(F(c),G(c))
      \underoverset   
        {\underset{\underset{c,d}{\sqcup} U(\tilde \rho_{(c,d)}(d))  }{\longrightarrow}}
        {\overset{\underset{c,d}{\sqcup} U(\tilde\rho_{d,c}(c))}{\longrightarrow}}
        {\phantom{AAAAAAAA}}
   \underset{c,d\in \mathcal{C}}{\prod}
    Hom_{Top^{\ast/}_{cg}}(
      \mathcal{C}(c,d),
      Maps(F(c),G(d))_\ast
    )
  \,.
\end{displaymath}
Here the object in the middle is just the set of collections of component morphisms $\left\{ F(c)\overset{\eta_c}{\to} G(c)\right\}_{c\in \mathcal{C}}$. The two parallel maps in the equalizer diagram take such a collection to the functions which send any $c \overset{f}{\to} d$ to the result of precomposing

\begin{displaymath}
\itexarray{
    F(c)
    \\
    {}^{\mathllap{f(f)}}\downarrow
    \\
    F(d) &\underset{\eta_d}{\longrightarrow}& G(d)
  }
\end{displaymath}
and of postcomposing

\begin{displaymath}
\itexarray{
    F(c) &\overset{\eta_c}{\longrightarrow}& G(c)
    \\
    && \downarrow^{\mathrlap{G(f)}}
    \\
    && G(d)
  }
\end{displaymath}
each component in such a collection, respectively. These two functions being equal, hence the collection $\{\eta_c\}_{c\in \mathcal{C}}$ being in the equalizer, means precisley that for all $c,d$ and all $f\colon c \to d$ the square

\begin{displaymath}
\itexarray{
    F(c) &\overset{\eta_c}{\longrightarrow}& G(c)
    \\
    {}^{\mathllap{F(f)}}\downarrow && \downarrow^{\mathrlap{G(f)}}
    \\
    F(d) &\underset{\eta_d}{\longrightarrow}& G(g)
  }
\end{displaymath}
is a [[commuting square]]. This is precisley the condition that the collection $\{\eta_c\}_{c\in \mathcal{C}}$ be a [[natural transformation]].

\end{proof}
Conversely, example \ref{NaturalTransformationsViaEnds} says that [[ends]] over [[bifunctors]] of the form $Maps(F(-),G(-)))_\ast$ constitute [[hom-spaces]] between pointed [[topologically enriched functors]]:

\begin{defn}
\label{PointedTopologicalFunctorCategory}\hypertarget{PointedTopologicalFunctorCategory}{}
Let $\mathcal{C}$ be a [[small category|small]] pointed [[topologically enriched categories]] (\href{Introduction+to+Stable+homotopy+theory+--+P#TopEnrichedCategory}{def.}). Define the structure of a pointed [[topologically enriched category]] on the category $[\mathcal{C}, Top_{cg}^{\ast/}]$ of pointed [[topologically enriched functors]] to $Top^{\ast/}_{cg}$ (\href{Introduction+to+Stable+homotopy+theory+--+P#TopologicallyEnrichedFunctorsToTopk}{exmpl.}) by taking the [[hom-spaces]] to be given by the [[ends]] (def. \ref{EndAndCoendInTopcgSmash}) of example \ref{NaturalTransformationsViaEnds}:

\begin{displaymath}
[\mathcal{C},Top^{\ast/}_{cg}](F,G)
    \;\coloneqq\;
  \int_{c\in \mathcal{C}} Maps(F(c),G(c))_\ast
\end{displaymath}
and by taking the composition maps to be the morphisms induced by the maps

\begin{displaymath}
\left(
    \underset{c\in \mathcal{C}}{\int} Maps(F(c),G(c))_\ast
  \right)
  \wedge
  \left(
    \underset{c \in \mathcal{C}}{\int} Maps(G(c),H(c))_\ast
  \right)
    \overset{}{\longrightarrow}
  \underset{c\in \mathcal{C}}{\prod}
    Maps(F(c),G(c))_\ast \wedge Maps(G(c),H(c))_\ast
  \overset{(\circ_{F(c),G(c),H(c)})_{c\in \mathcal{C}}}{\longrightarrow}
  \underset{c \in \mathcal{C}}{\prod}
    Maps(F(c),H(c))_\ast
\end{displaymath}
by observing that these equalize the two actions in the definition of the [[end]].

The resulting pointed [[topologically enriched category]] $[\mathcal{C},Top^{\ast/}_{cg}]$ is also called the \textbf{$Top^{\ast/}_{cg}$-[[enriched functor category]]} over $\mathcal{C}$ with coefficients in $Top^{\ast/}_{cg}$.

\end{defn}
First of all this yields a concise statement of the pointed topologically [[enriched Yoneda lemma]] (\href{Introduction+to+Stable+homotopy+theory+--+P#TopologicallyEnrichedYonedaLemma}{prop.})

\begin{prop}
\label{YonedaReductionTopological}\hypertarget{YonedaReductionTopological}{}
\textbf{(topologically [[enriched Yoneda lemma]])}

Let $\mathcal{C}$ be a [[small category|small]] pointed [[topologically enriched categories]] (\href{Introduction+to+Stable+homotopy+theory+--+P#TopEnrichedCategory}{def.}). For $F \colon \mathcal{C}\to Top^{\ast/}_{cg}$ a pointed [[topologically enriched functor]] (\href{Introduction+to+Stable+homotopy+theory+--+P#TopologicallyEnrichedFunctor}{def.}) and for $c\in \mathcal{C}$ an object, there is a [[natural isomorphism]]

\begin{displaymath}
[\mathcal{C}, Top^{\ast/}_{cg}](\mathcal{C}(c,-),\; F)
  \;\simeq\;
  F(c)
\end{displaymath}
between the [[hom-space]] of the pointed topological functor category, according to def. \ref{PointedTopologicalFunctorCategory}, from the [[representable functor|functor represented]] by $c$ to $F$, and the value of $F$ on $c$.

In terms of the [[ends]] (def. \ref{EndAndCoendInTopcgSmash}) defining these [[hom-spaces]], this means that

\begin{displaymath}
\underset{d\in \mathcal{C}}{\int} Maps(\mathcal{C}(c,d), F(d))_\ast
    \;\simeq\;
  F(c)
  \,.
\end{displaymath}
In this form the statement is also known as \textbf{[[Yoneda reduction]]}.

\end{prop}
The \textbf{proof} of prop. \ref{YonedaReductionTopological} is essentially dual to the proof of the next prop. \ref{TopologicalCoYonedaLemma}.

Now that [[natural transformations]] are phrased in terms of [[ends]] (example \ref{NaturalTransformationsViaEnds}), as is the Yoneda lemma (prop. \ref{YonedaReductionTopological}), it is natural to consider the [[formal duality|dual]] statement involving [[coends]]:

\begin{prop}
\label{TopologicalCoYonedaLemma}\hypertarget{TopologicalCoYonedaLemma}{}
\textbf{([[co-Yoneda lemma]])}

Let $\mathcal{C}$ be a [[small category|small]] pointed [[topologically enriched categories]] (\href{Introduction+to+Stable+homotopy+theory+--+P#TopEnrichedCategory}{def.}). For $F \colon \mathcal{C}\to Top^{\ast/}_{cg}$ a pointed [[topologically enriched functor]] (\href{Introduction+to+Stable+homotopy+theory+--+P#TopologicallyEnrichedFunctor}{def.}) and for $c\in \mathcal{C}$ an object, there is a [[natural isomorphism]]

\begin{displaymath}
F(-)
    \simeq
  \overset{c \in \mathcal{C}}{\int}
  \mathcal{C}(c,-) \wedge F(c)
  \,.
\end{displaymath}
Moreover, the morphism that hence exhibits $F(c)$ as the [[coequalizer]] of the two morphisms in def. \ref{EndAndCoendInTopcgSmash} is componentwise the canonical action

\begin{displaymath}
\mathcal{C}(d,c) \wedge F(c)
   \longrightarrow
  F(d)
\end{displaymath}
which is [[adjunct]] to the component map $\mathcal{C}(d,c) \to Maps(F(c),F(d))_{\ast}$ of the [[topologically enriched functor]] $F$.

\end{prop}
(e.g. \hyperlink{MMSS00}{MMSS 00, lemma 1.6})

\begin{proof}
The coequalizer of pointed topological spaces that we need to consider has underlying it a coequalizer of underlying pointed sets (\href{Introduction+to+Stable+homotopy+theory+--+P#DescriptionOfLimitsAndColimitsInTop}{prop.}, \href{Introduction+to+Stable+homotopy+theory+--+P#LimitsAndColimitsOfPointedObjects}{prop.}, \href{Introduction+to+Stable+homotopy+theory+--+P#kTopIsCoreflectiveSubcategory}{prop.}). That in turn is the colimit over the diagram of underlying sets with the basepointe adjoined to the diagram (\href{Introduction+to+Stable+homotopy+theory+--+P#LimitsAndColimitsOfPointedObjects}{prop.}). For a coequalizer diagram adding that extra point to the diagram clearly does not change the colimit, and so we need to consider the plain coequalizer of sets.

That is just the set of [[equivalence classes]] of [[pairs]]

\begin{displaymath}
( c \overset{}{\to} c_0,\; x \in F(c) )
  \,,
\end{displaymath}
where two such pairs

\begin{displaymath}
( c \overset{f}{\to} c_0,\; x \in F(c) )
  \,,\;\;\;\;
  ( d \overset{g}{\to} c_0,\; y \in F(d) )
\end{displaymath}
are regarded as equivalent if there exists

\begin{displaymath}
c \overset{\phi}{\to} d
\end{displaymath}
such that

\begin{displaymath}
f = g \circ \phi
  \,,
  \;\;\;\;\;and\;\;\;\;\;
  y = \phi(x)
  \,.
\end{displaymath}
(Because then the two pairs are the two images of the pair $(g,x)$ under the two morphisms being coequalized.)

But now considering the case that $d = c_0$ and $g = id_{c_0}$, so that $f = \phi$ shows that any pair

\begin{displaymath}
( c \overset{\phi}{\to} c_0, \; x \in F(c))
\end{displaymath}
is identified, in the coequalizer, with the pair

\begin{displaymath}
(id_{c_0},\; \phi(x) \in F(c_0))
  \,,
\end{displaymath}
hence with $\phi(x)\in F(c_0)$.

This shows the claim at the level of the underlying sets. To conclude it is now sufficient (\href{Introduction+to+Stable+homotopy+theory+--+P#DescriptionOfLimitsAndColimitsInTop}{prop.}) to show that the topology on $F(c_0) \in Top^{\ast/}_{cg}$ is the [[final topology]] (\href{Introduction+to+Stable+homotopy+theory+--+P#InitialAndFinalTopologies}{def.}) of the system of component morphisms

\begin{displaymath}
\mathcal{C}(d,c) \wedge F(c)
    \longrightarrow
  \overset{c}{\int} \mathcal{C}(c,c_0) \wedge F(c)
\end{displaymath}
which we just found. But that system includes

\begin{displaymath}
\mathcal{C}(c,c) \wedge F(c) \longrightarrow F(c)
\end{displaymath}
which is a [[retraction]]

\begin{displaymath}
id \;\colon\; F(c) \longrightarrow \mathcal{C}(c,c) \wedge F(c)
  \longrightarrow F(c)
\end{displaymath}
and so if all the preimages of a given subset of the coequalizer under these component maps is open, it must have already been open in $F(c)$.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
The statement of the [[co-Yoneda lemma]] in prop. \ref{TopologicalCoYonedaLemma} is a kind of [[categorification]] of the following statement in [[analysis]] (whence the notation with the integral signs):

For $X$ a [[topological space]], $f \colon X \to\mathbb{R}$ a [[continuous function]] and $\delta(-,x_0)$ denoting the [[Dirac distribution]], then

\begin{displaymath}
\int_{x \in X} \delta(x,x_0) f(x)
  = 
  f(x_0)
  \,.
\end{displaymath}
\end{remark}
It is this analogy that gives the name to the following statement:

\begin{prop}
\label{CoendsCommuteWithEachOther}\hypertarget{CoendsCommuteWithEachOther}{}
\textbf{([[Fubini theorem]] for (co)-ends)}

For $F$ a pointed topologically enriched [[bifunctor]] on a small pointed topological [[product category]] $\mathcal{C}_1 \times \mathcal{C}_2$ (def. \ref{OppositeAndProductOfPointedTopologicallyEnrichedCategory}), i.e.

\begin{displaymath}
F 
  \;\colon\;
  \left(
    \mathcal{C}_1\times\mathcal{C}_2
  \right)^{op}
  \times
  (\mathcal{C}_1 \times\mathcal{C}_2)
  \longrightarrow
  Top^{\ast/}_{cg}
\end{displaymath}
then its [[end]] and [[coend]] (def. \ref{EndAndCoendInTopcgSmash}) is equivalently formed consecutively over each variable, in either order:

\begin{displaymath}
\overset{(c_1,c_2)}{\int} F((c_1,c_2), (c_1,c_2))
  \simeq
  \overset{c_1}{\int}
  \overset{c_2}{\int}
  F((c_1,c_2), (c_1,c_2))
  \simeq
  \overset{c_2}{\int}
  \overset{c_1}{\int}
  F((c_1,c_2), (c_1,c_2))
\end{displaymath}
and

\begin{displaymath}
\underset{(c_1,c_2)}{\int} F((c_1,c_2), (c_1,c_2))
  \simeq
  \underset{c_1}{\int}
  \underset{c_2}{\int}
  F((c_1,c_2), (c_1,c_2))
  \simeq
  \underset{c_2}{\int}
  \underset{c_1}{\int}
  F((c_1,c_2), (c_1,c_2))
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
Because [[limits]] commute with limits, and [[colimits]] commute with colimits.

\end{proof}
\begin{remark}
\label{MappingSpacePreservesEnds}\hypertarget{MappingSpacePreservesEnds}{}
Because the pointed compactly generated [[mapping space]] functor (\href{Introduction+to+Stable+homotopy+theory+--+P#PointedMappingSpace}{exmpl.})

\begin{displaymath}
Maps(-,-)_\ast
  \;\colon\;
  \left(Top^{\ast/}_{cg}\right)^{op}
  \times
  Top^{\ast/}_{cg}
  \longrightarrow
  Top^{\ast/}_{cg}
\end{displaymath}
takes [[colimits]] in the first argument and [[limits]] in the second argument to limits (\href{Introduction+to+Stable+homotopy+theory+--+P#MappingSpacesSendsColimitsInFirstArgumentToLimits}{cor.}), it also takes [[coends]] in the first argument and [[ends]] in the second argument, to ends (def. \ref{EndAndCoendInTopcgSmash}):

\begin{displaymath}
Maps( X, \; \int_{c} F(c,c))_\ast
  \simeq
  \int_c Maps(X, F(c,c)_\ast)
\end{displaymath}
and

\begin{displaymath}
Maps( \int^{c} F(c,c),\; Y  )_\ast
  \simeq
  \underset{c}{\int} Maps( F(c,c),\; Y )_\ast
  \,.
\end{displaymath}
\end{remark}
\begin{prop}
\label{TopologicalLeftKanExtensionBCoend}\hypertarget{TopologicalLeftKanExtensionBCoend}{}
\textbf{(left Kan extension via coends)}

Let $\mathcal{C}, \mathcal{D}$ be [[small category|small]] pointed [[topologically enriched categories]] (\href{Introduction+to+Stable+homotopy+theory+--+P#TopEnrichedCategory}{def.}) and let

\begin{displaymath}
p \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}
\end{displaymath}
be a pointed [[topologically enriched functor]] (\href{Introduction+to+Stable+homotopy+theory+--+P#TopologicallyEnrichedFunctor}{def.}). Then precomposition with $p$ constitutes a functor

\begin{displaymath}
p^\ast 
  \;\colon\;
  [\mathcal{D}, Top^{\ast/}_{cg}]
  \longrightarrow
  [\mathcal{C}, Top^{\ast/}_{cg}]
\end{displaymath}
$G\mapsto G\circ p$. This functor has a [[left adjoint]] $Lan_p$, called \textbf{left [[Kan extension]]} along $p$

\begin{displaymath}
[\mathcal{D}, Top^{\ast/}_{cg}]
    \underoverset
      {\underset{p^\ast}{\longrightarrow}}
      {\overset{Lan_p }{\longleftarrow}}
      {\bot}
  [\mathcal{C}, Top^{\ast/}_{cg}]
\end{displaymath}
which is given objectwise by a [[coend]] (def. \ref{EndAndCoendInTopcgSmash}):

\begin{displaymath}
(Lan_p F)
  \;\colon\;
  d
  \;\mapsto \;
  \overset{c\in \mathcal{C}}{\int}
   \mathcal{D}(p(c),d) \wedge F(c)
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
Use the expression of natural transformations in terms of ends (example \ref{NaturalTransformationsViaEnds} and def. \ref{PointedTopologicalFunctorCategory}), then use the respect of $Maps(-,-)_\ast$ for ends/coends (remark \ref{MappingSpacePreservesEnds}), use the smash/mapping space adjunction (\href{Introduction+to+Stable+homotopy+theory+--+P#SmashHomAdjunctionOnPointedCompactlyGeneratedTopologicalSpaces}{cor.}), use the [[Fubini theorem]] (prop. \ref{CoendsCommuteWithEachOther}) and finally use [[Yoneda reduction]] (prop. \ref{YonedaReductionTopological}) to obtain a sequence of [[natural isomorphisms]] as follows:

\begin{displaymath}
\begin{aligned}
    [\mathcal{D},Top^{\ast/}_{cg}]( Lan_p F, \, G )
    & =
   \underset{d \in \mathcal{D}}{\int}
    Maps( (Lan_p F)(d), \, G(d) )_\ast
   \\
   & =
   \underset{d\in \mathcal{D}}{\int}
   Maps\left(
     \overset{c \in \mathcal{C}}{\int} \mathcal{D}(p(c),d) \wedge F(c)
     ,\;
     G(d)
   \right)_\ast
    \\
  &\simeq
  \underset{d \in \mathcal{D}}{\int}
  \underset{c \in \mathcal{C}}{\int}
  Maps(
    \mathcal{D}(p(c),d)\wedge F(c) \,,\; G(d)
  )_\ast
  \\
  & \simeq
  \underset{c\in \mathcal{C}}{\int}
  \underset{d\in \mathcal{D}}{\int}
  Maps(F(c), 
    Maps(
     \mathcal{D}(p(c),d) , \, G(d)
    )_\ast
  )_\ast
  \\
  & \simeq
  \underset{c\in \mathcal{C}}{\int}
  Maps(F(c), 
   \underset{d\in \mathcal{D}}{\int}
   Maps(
    \mathcal{D}(p(c),d) , \, G(d)
   )_\ast
  )_\ast
  \\
  & \simeq
  \underset{c\in \mathcal{C}}{\int}
  Maps(F(c), G(p(c))
  )_\ast  
  \\
  & =
  [\mathcal{C}, Top^{\ast/}_{cg}](F,p^\ast G)
  \end{aligned}
  \,.
\end{displaymath}
\end{proof}
\hypertarget{monoidal_topological_categories}{}\paragraph*{{Monoidal topological categories}}\label{monoidal_topological_categories}

We recall the basic definitions of [[monoidal categories]] and of [[monoid in a monoidal category|monoids]] and [[module object|modules]] [[internalization|internal]] to monoidal categories. All examples are at the end of this section, starting with example \ref{TopAsASymmetricMonoidalCategory} below.

\begin{defn}
\label{MonoidalCategory}\hypertarget{MonoidalCategory}{}
A \textbf{(pointed) [[topologically enriched category|topologically enriched]] [[monoidal category]]} is a (pointed) [[topologically enriched category]] $\mathcal{C}$ (\href{Introduction+to+Stable+homotopy+theory+--+P#TopEnrichedCategory}{def.}) equipped with

\begin{enumerate}%
\item a (pointed) [[topologically enriched functor]] (\href{Introduction+to+Stable+homotopy+theory+--+P#TopologicallyEnrichedFunctor}{def.})

\begin{displaymath}
\otimes 
     \;\colon\; 
   \mathcal{C} \times \mathcal{C}  
    \longrightarrow
   \mathcal{C}
\end{displaymath}
out of the (pointed) topologival [[product category]] of $\mathcal{C}$ with itself (def. \ref{OppositeAndProductOfPointedTopologicallyEnrichedCategory}), called the \textbf{[[tensor product]]},


\item an object

\begin{displaymath}
1 \in \mathcal{C}
\end{displaymath}
called the \textbf{[[unit object]]} or \textbf{[[tensor unit]]},


\item a [[natural isomorphism]] (\href{Introduction+to+Stable+homotopy+theory+--+P#TopologicallyEnrichedFunctor}{def.})

\begin{displaymath}
a 
    \;\colon\; 
  ((-)\otimes (-)) \otimes (-)
    \overset{\simeq}{\longrightarrow}
  (-) \otimes ((-)\otimes(-))
\end{displaymath}
called the \textbf{[[associator]]},


\item a [[natural isomorphism]]

\begin{displaymath}
\ell
    \;\colon\; 
  (1 \otimes (-)) 
    \overset{\simeq}{\longrightarrow}
  (-)
\end{displaymath}
called the \textbf{[[left unitor]]}, and a natural isomorphism

\begin{displaymath}
r \;\colon\; (-) \otimes 1 \overset{\simeq}{\longrightarrow} (-)
\end{displaymath}
called the \textbf{[[right unitor]]},



\end{enumerate}
such that the following two kinds of [[commuting diagram|diagrams commute]], for all objects involved:

\begin{enumerate}%
\item \textbf{triangle identity}:

\begin{displaymath}
\itexarray{
     & (x \otimes 1) \otimes y &\stackrel{a_{x,1,y}}{\longrightarrow} & x \otimes (1 \otimes y)
      \\     
      & {}_{\rho_x \otimes 1_y}\searrow       
      && \swarrow_{1_x \otimes \lambda_y}
      & 
     \\
     &&
     x \otimes y
     &&
  }
\end{displaymath}

\item the \textbf{[[pentagon identity]]}:



\end{enumerate}
[[!include monoidal category {\tt \symbol{62}} pentagon]]

\end{defn}
\begin{lemma}
\label{kel1}\hypertarget{kel1}{}
\textbf{(\href{monoidal+category#kel1}{Kelly 64})}

Let $(\mathcal{C}, \otimes, 1)$ be a [[monoidal category]], def. \ref{MonoidalCategory}. Then the left and right [[unitors]] $\ell$ and $r$ satisfy the following conditions:

\begin{enumerate}%
\item $\ell_1 = r_1 \;\colon\; 1 \otimes 1 \overset{\simeq}{\longrightarrow} 1$;


\item for all objects $x,y \in \mathcal{C}$ the following [[commuting diagram|diagram commutes]]:

\begin{displaymath}
\itexarray{
    (1 \otimes x) \otimes y  & & 
    \\
    {}^\mathllap{\alpha_{1, x, y}} \downarrow 
    & \searrow^\mathrlap{\ell_x y} & 
    \\
    1 \otimes (x \otimes y) 
    & \underset{\ell_{x \otimes y}}{\longrightarrow} & x \otimes y
  }
  \,.
\end{displaymath}
Analogously for the right unitor.



\end{enumerate}
\end{lemma}
\begin{defn}
\label{BraidedMonoidalCategory}\hypertarget{BraidedMonoidalCategory}{}
A \textbf{(pointed) [[topologically enriched category|topological]] [[braided monoidal category]]}, is a (pointed) [[topologically enriched category|topological]] [[monoidal category]] $\mathcal{C}$ (def. \ref{MonoidalCategory}) equipped with a [[natural isomorphism]]

\begin{displaymath}
\tau_{x,y} \colon x \otimes y \to y \otimes x
\end{displaymath}
called the \textbf{[[braiding]]}, such that the following two kinds of [[commuting diagram|diagrams commute]] for all [[objects]] involved:

\begin{displaymath}
\itexarray{
   (x \otimes y) \otimes z 
   &\stackrel{a_{x,y,z}}{\to}&
   x \otimes (y \otimes z)
   &\stackrel{\tau_{x,y \otimes z}}{\to}&
   (y \otimes z) \otimes x
   \\
   \downarrow^{\tau_{x,y}\otimes Id}
   &&&&
   \downarrow^{a_{y,z,x}}
   \\
   (y \otimes x) \otimes z
   &\stackrel{a_{y,x,z}}{\to}&
   y \otimes (x \otimes z)
   &\stackrel{Id \otimes \tau_{x,z}}{\to}&
   y \otimes (z \otimes x)
  }
\end{displaymath}
and

\begin{displaymath}
\itexarray{
   x \otimes (y \otimes z) 
   &\stackrel{a^{-1}_{x,y,z}}{\to}&
   (x \otimes y) \otimes z
   &\stackrel{\tau_{x \otimes y, z}}{\to}&
   z \otimes (x \otimes y)
   \\
   \downarrow^{Id \otimes \tau_{y,z}}
   &&&&
   \downarrow^{a^{-1}_{z,x,y}}
   \\
   x \otimes (z \otimes y)
   &\stackrel{a^{-1}_{x,z,y}}{\to}&
   (x \otimes z) \otimes y
   &\stackrel{\tau_{x,z} \otimes Id}{\to}&
   (z \otimes x) \otimes y
  }
  \,,
\end{displaymath}
where $a_{x,y,z} \colon (x \otimes y) \otimes z \to x \otimes (y \otimes z)$ denotes the components of the [[associator]] of $\mathcal{C}^\otimes$.

\end{defn}
\begin{defn}
\label{SymmetricMonoidalCategory}\hypertarget{SymmetricMonoidalCategory}{}
A \textbf{(pointed) [[topologically enriched category|topological]] [[symmetric monoidal category]]} is a (pointed) topological [[braided monoidal category]] (def. \ref{BraidedMonoidalCategory}) for which the [[braiding]]

\begin{displaymath}
\tau_{x,y} \colon x \otimes y \to y \otimes x
\end{displaymath}
satisfies the condition:

\begin{displaymath}
\tau_{y,x} \circ \tau_{x,y} = 1_{x \otimes y}
\end{displaymath}
for all objects $x, y$

\end{defn}
\begin{defn}
\label{ClosedMonoidalCategory}\hypertarget{ClosedMonoidalCategory}{}
Given a (pointed) [[topologically enriched category|topological]] [[symmetric monoidal category]] $\mathcal{C}$ with [[tensor product]] $\otimes$ (def. \ref{SymmetricMonoidalCategory}) it is called a \textbf{[[closed monoidal category]]} if for each $Y \in \mathcal{C}$ the functor $Y \otimes(-)\simeq (-)\otimes X$ has a [[right adjoint]], denoted $[Y,-]$

\begin{displaymath}
\mathcal{C}
    \underoverset
      {\underset{[Y,-]}{\longrightarrow}}
      {\overset{(-) \otimes Y}{\longleftarrow}}
      {\bot}
  \mathcal{C}
  \,,
\end{displaymath}
hence if there are [[natural isomorphisms]]

\begin{displaymath}
Hom_{\mathcal{C}}(X \otimes Y, Z)
   \;\simeq\;
  Hom_{\mathcal{C}}{C}(X, [Y,Z])
\end{displaymath}
for all objects $X,Z \in \mathcal{C}$.

Since for the case that $X = 1$ is the [[tensor unit]] of $\mathcal{C}$ this means that

\begin{displaymath}
Hom_{\mathcal{C}}(1,[Y,Z]) \simeq Hom_{\mathcal{C}}(Y,Z)
  \,,
\end{displaymath}
the object $[Y,Z] \in \mathcal{C}$ is an enhancement of the ordinary [[hom-set]] $Hom_{\mathcal{C}}(Y,Z)$ to an object in $\mathcal{C}$. Accordingly, it is also called the \textbf{[[internal hom]]} between $Y$ and $Z$.

\end{defn}
\begin{example}
\label{TopAsASymmetricMonoidalCategory}\hypertarget{TopAsASymmetricMonoidalCategory}{}
The category [[Set]] of [[sets]] and [[functions]] between them, regarded as enriched in [[discrete topological spaces]], becomes a [[symmetric monoidal category]] according to def. \ref{SymmetricMonoidalCategory} with [[tensor product]] the [[Cartesian product]] $\times$ of sets. The [[associator]], [[unitor]] and [[braiding]] isomorphism are the evident (almost unnoticable but nevertheless nontrivial) canonical identifications.

Similarly the $Top_{cg}$ of [[compactly generated topological spaces]] (\href{Introduction+to+Stable+homotopy+theory+--+P#kTop}{def.}) becomes a [[symmetric monoidal category]] with [[tensor product]] the corresponding [[Cartesian products]], hence the operation of forming k-ified (\href{Introduction+to+Stable+homotopy+theory+--+P#kTopIsCoreflectiveSubcategory}{cor.}) [[product topological spaces]] (\href{Introduction+to+Stable+homotopy+theory+--+P#ProductTopologicalSpace}{exmpl.}). The underlying functions of the [[associator]], [[unitor]] and [[braiding]] isomorphisms are just those of the underlying sets, as above.

Symmetric monoidal categories, such as these, for which the tensor product is the [[Cartesian product]] are called \emph{[[Cartesian monoidal categories]]}.

\end{example}
\begin{example}
\label{PointedTopologicalSpacesWithSmashIsSymmetricMonoidalCategory}\hypertarget{PointedTopologicalSpacesWithSmashIsSymmetricMonoidalCategory}{}
The category $Top_{cg}^{\ast/}$ of [[pointed topological space|pointed]] [[compactly generated topological spaces]] with [[tensor product]] the [[smash product]] $\wedge$ (\href{Introduction+to+Stable+homotopy+theory+--+P#SmashProductOfPointedObjects}{def.})

\begin{displaymath}
X \wedge Y \coloneqq \frac{X\times Y}{X\vee Y}
\end{displaymath}
is a [[symmetric monoidal category]] (def. \ref{SymmetricMonoidalCategory}) with [[unit object]] the pointed [[0-sphere]] $S^0$.

The components of the [[associator]], the [[unitors]] and the [[braiding]] are those of [[Top]] as in example \ref{TopAsASymmetricMonoidalCategory}, descended to the [[quotient topological spaces]] which appear in the definition of the [[smash product]]). This works for pointed [[compactly generated spaces]] (but not for general pointed topological spaces) by \href{Introduction+to+Stable+homotopy+theory+--+P#SmashProductInTopcgIsAssociative}{this prop.}.

\end{example}
\begin{example}
\label{ExampleAbelianGroupsOfMonoidalCategory}\hypertarget{ExampleAbelianGroupsOfMonoidalCategory}{}
The category [[Ab]] of [[abelian groups]], regarded as enriched in [[discrete topological spaces]], becomes a [[symmetric monoidal category]] with tensor product the actual [[tensor product of abelian groups]] $\otimes_{\mathbb{Z}}$ and with [[tensor unit]] the additive group $\mathbb{Z}$ of [[integers]]. Again the [[associator]], [[unitor]] and [[braiding]] isomorphism are the evident ones coming from the underlying sets, as in example \ref{TopAsASymmetricMonoidalCategory}.

This is the archetypical case that motivates the notation ``$\otimes$'' for the pairing operation in a [[monoidal category]]:

\begin{enumerate}%
\item A [[monoid in a monoidal category|monoid in]] $(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z})$ (def. \ref{MonoidsInMonoidalCategory}) is equivalently a [[ring]].


\item A [[commutative monoid in a symmetric monoidal category|commutative monoid in]] in $(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z})$ (def. \ref{MonoidsInMonoidalCategory}) is equivalently a [[commutative ring]] $R$.


\item An $R$-[[module object]] in $(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z})$ (def. \ref{ModulesInMonoidalCategory}) is equivalently an $R$-[[module]];


\item The tensor product of $R$-module objects (def. \ref{TensorProductOfModulesOverCommutativeMonoidObject}) is the standard [[tensor product of modules]].


\item The [[category of modules|category of module objects]] $R Mod(Ab)$ (def. \ref{TensorProductOfModulesOverCommutativeMonoidObject}) is the standard [[category of modules]] $R Mod$.



\end{enumerate}
\end{example}
\hypertarget{AlgebrasAndModules}{}\paragraph*{{Algebras and modules}}\label{AlgebrasAndModules}

\begin{defn}
\label{MonoidsInMonoidalCategory}\hypertarget{MonoidsInMonoidalCategory}{}
Given a (pointed) [[topologically enriched category|topological]] [[monoidal category]] $(\mathcal{C}, \otimes, 1)$, then a \textbf{[[monoid in a monoidal category|monoid internal to]]} $(\mathcal{C}, \otimes, 1)$ is

\begin{enumerate}%
\item an [[object]] $A \in \mathcal{C}$;


\item a morphism $e \;\colon\; 1 \longrightarrow A$ (called the \emph{[[unit]]})


\item a morphism $\mu \;\colon\; A \otimes A \longrightarrow A$ (called the \emph{product});



\end{enumerate}
such that

\begin{enumerate}%
\item ([[associativity]]) the following [[commuting diagram|diagram commutes]]

\begin{displaymath}
\itexarray{
    (A\otimes A) \otimes A 
      &\underoverset{\simeq}{a_{A,A,A}}{\longrightarrow}&
    A \otimes (A \otimes A)
      &\overset{A \otimes \mu}{\longrightarrow}&
    A \otimes A
    \\
    {}^{\mathllap{\mu \otimes A}}\downarrow  
      && &&
    \downarrow^{\mathrlap{\mu}}
    \\
    A \otimes A
      &\longrightarrow&
      &\overset{\mu}{\longrightarrow}&
    A
  }
  \,,
\end{displaymath}
where $a$ is the associator isomorphism of $\mathcal{C}$;


\item ([[unitality]]) the following [[commuting diagram|diagram commutes]]:

\begin{displaymath}
\itexarray{
    1 \otimes A 
      &\overset{e \otimes id}{\longrightarrow}&
    A \otimes A
      &\overset{id \otimes e}{\longleftarrow}& 
    A \otimes 1
    \\
    & {}_{\mathllap{\ell}}\searrow 
    & \downarrow^{\mathrlap{\mu}} &
    & \swarrow_{\mathrlap{r}}
    \\
    && A
  }
  \,,
\end{displaymath}
where $\ell$ and $r$ are the left and right unitor isomorphisms of $\mathcal{C}$.



\end{enumerate}
Moreover, if $(\mathcal{C}, \otimes , 1)$ has the structure of a [[symmetric monoidal category]] (def. \ref{SymmetricMonoidalCategory}) $(\mathcal{C}, \otimes, 1, B)$ with symmetric [[braiding]] $\tau$, then a monoid $(A,\mu, e)$ as above is called a \textbf{[[commutative monoid in a symmetric monoidal category|commutative monoid in]]} $(\mathcal{C}, \otimes, 1, B)$ if in addition

\begin{itemize}%
\item (commutativity) the following [[commuting diagram|diagram commutes]]

\begin{displaymath}
\itexarray{
    A \otimes A 
      && \underoverset{\simeq}{\tau_{A,A}}{\longrightarrow} &&
    A \otimes A
    \\
    & {}_{\mathllap{\mu}}\searrow && \swarrow_{\mathrlap{\mu}}
    \\
    && A
  }
  \,.
\end{displaymath}


\end{itemize}
A [[homomorphism]] of monoids $(A_1, \mu_1, e_1)\longrightarrow (A_2, \mu_2, f_2)$ is a morphism

\begin{displaymath}
f \;\colon\; A_1 \longrightarrow A_2
\end{displaymath}
in $\mathcal{C}$, such that the following two [[commuting diagram|diagrams commute]]

\begin{displaymath}
\itexarray{
    A_1 \otimes A_1 
      &\overset{f \otimes f}{\longrightarrow}&
    A_2 \otimes A_2
    \\
    {}^{\mathllap{\mu_1}}\downarrow && \downarrow^{\mathrlap{\mu_2}}
    \\
    A_1 &\underset{f}{\longrightarrow}& A_2
  }
\end{displaymath}
and

\begin{displaymath}
\itexarray{
    1_{\mathcal{c}} &\overset{e_1}{\longrightarrow}& A_1
    \\
    & {}_{\mathllap{e_2}}\searrow & \downarrow^{\mathrlap{f}}
    \\
    && A_2
  }
  \,.
\end{displaymath}
Write $Mon(\mathcal{C}, \otimes,1)$ for the [[category of monoids]] in $\mathcal{C}$, and $CMon(\mathcal{C}, \otimes, 1)$ for its subcategory of commutative monoids.

\end{defn}
\begin{example}
\label{MonoidGivenByTensorUnit}\hypertarget{MonoidGivenByTensorUnit}{}
Given a (pointed) [[topologically enriched category|topological]] [[monoidal category]] $(\mathcal{C}, \otimes, 1)$, then the [[tensor unit]] $1$ is a [[monoid in a monoidal category|monoid in]] $\mathcal{C}$ (def. \ref{MonoidsInMonoidalCategory}) with product given by either the left or right [[unitor]]

\begin{displaymath}
\ell_1 = r_1 \;\colon\; 1 \otimes 1 \overset{\simeq}{\longrightarrow} 1
  \,.
\end{displaymath}
By lemma \ref{kel1}, these two morphisms coincide and define an [[associativity|associative]] product with unit the identity $id \colon 1 \to 1$.

If $(\mathcal{C}, \otimes , 1)$ is a [[symmetric monoidal category]] (def. \ref{SymmetricMonoidalCategory}), then this monoid is a [[commutative monoid in a symmetric monoidal category|commutative monoid]].

\end{example}
\begin{defn}
\label{ModulesInMonoidalCategory}\hypertarget{ModulesInMonoidalCategory}{}
Given a (pointed) [[topologically enriched category|topological]] [[monoidal category]] $(\mathcal{C}, \otimes, 1)$ (def. \ref{MonoidalCategory}), and given $(A,\mu,e)$ a [[monoid in a monoidal category|monoid in]] $(\mathcal{C}, \otimes, 1)$ (def. \ref{MonoidsInMonoidalCategory}), then a \textbf{left [[module object]]} in $(\mathcal{C}, \otimes, 1)$ over $(A,\mu,e)$ is

\begin{enumerate}%
\item an [[object]] $N \in \mathcal{C}$;


\item a [[morphism]] $\rho \;\colon\; A \otimes N \longrightarrow N$ (called the \emph{[[action]]});



\end{enumerate}
such that

\begin{enumerate}%
\item ([[unitality]]) the following [[commuting diagram|diagram commutes]]:

\begin{displaymath}
\itexarray{
    1 \otimes N 
      &\overset{e \otimes id}{\longrightarrow}&
    A \otimes N
    \\
    & {}_{\mathllap{\ell}}\searrow 
    & \downarrow^{\mathrlap{\rho}} 
    \\
    && A
  }
  \,,
\end{displaymath}
where $\ell$ is the left unitor isomorphism of $\mathcal{C}$.


\item (action property) the following [[commuting diagram|diagram commutes]]

\begin{displaymath}
\itexarray{
    (A\otimes A) \otimes N
      &\underoverset{\simeq}{a_{A,A,N}}{\longrightarrow}&
    A \otimes (A \otimes N)
      &\overset{A \otimes \rho}{\longrightarrow}&
    A \otimes N
    \\
    {}^{\mathllap{\mu \otimes N}}\downarrow  
      && &&
    \downarrow^{\mathrlap{\rho}}
    \\
    A \otimes N
      &\longrightarrow&
      &\overset{\rho}{\longrightarrow}&
    N
  }
  \,,
\end{displaymath}


\end{enumerate}
A [[homomorphism]] of left $A$-module objects

\begin{displaymath}
(N_1, \rho_1) \longrightarrow (N_2, \rho_2)
\end{displaymath}
is a morphism

\begin{displaymath}
f\;\colon\; N_1 \longrightarrow N_2
\end{displaymath}
in $\mathcal{C}$, such that the following [[commuting diagram|diagram commutes]]:

\begin{displaymath}
\itexarray{
    A\otimes N_1 &\overset{A \otimes f}{\longrightarrow}& A\otimes N_2
    \\
    {}^{\mathllap{\rho_1}}\downarrow 
      && 
    \downarrow^{\mathrlap{\rho_2}}
    \\
    N_1 &\underset{f}{\longrightarrow}& N_2
  }
  \,.
\end{displaymath}
For the resulting \textbf{[[category of modules]]} of left $A$-modules in $\mathcal{C}$ with $A$-module homomorphisms between them, we write

\begin{displaymath}
A Mod(\mathcal{C})
  \,.
\end{displaymath}
This is naturally a (pointed) [[topologically enriched category]] itself.

\end{defn}
\begin{example}
\label{EveryObjectIsModuleOverTensorUnit}\hypertarget{EveryObjectIsModuleOverTensorUnit}{}
Given a [[monoidal category]] $(\mathcal{C},\otimes, 1)$ (def. \ref{MonoidalCategory}) with the [[tensor unit]] $1$ regarded as a [[monoid in a monoidal category]] via example \ref{MonoidGivenByTensorUnit}, then the left [[unitor]]

\begin{displaymath}
\ell_C 
    \;\colon\;
  1\otimes C \longrightarrow C
\end{displaymath}
makes every object $C \in \mathcal{C}$ into a left module, according to def. \ref{ModulesInMonoidalCategory}, over $C$. The action property holds due to lemma \ref{kel1}. This gives an [[equivalence of categories]]

\begin{displaymath}
\mathbb{C} \simeq 1 Mod(\mathcal{C})
\end{displaymath}
of $\mathcal{C}$ with the [[category of modules]] over its tensor unit.

\end{example}
\begin{prop}
\label{MonoidModuleOverItself}\hypertarget{MonoidModuleOverItself}{}
In the situation of def. \ref{ModulesInMonoidalCategory}, the monoid $(A,\mu, e)$ canonically becomes a left module over itself by setting $\rho \coloneqq \mu$. More generally, for $C \in \mathcal{C}$ any object, then $A \otimes C$ naturally becomes a left $A$-module by setting:

\begin{displaymath}
\rho
  \;\colon\;
  A \otimes (A \otimes C)
   \underoverset{\simeq}{a^{-1}_{A,A,C}}{\longrightarrow}
  (A \otimes A) \otimes C
    \overset{\mu \otimes id}{\longrightarrow}
  A \otimes C
  \,.
\end{displaymath}
The $A$-modules of this form are called \textbf{[[free modules]]}.

The [[free functor]] $F$ constructing free $A$-modules is [[left adjoint]] to the [[forgetful functor]] $U$ which sends a module $(N,\rho)$ to the underlying object $U(N,\rho) \coloneqq N$.

\begin{displaymath}
A Mod(\mathcal{C})
    \underoverset
     {\underset{U}{\longrightarrow}}
     {\overset{F}{\longleftarrow}}
     {\bot}
  \mathcal{C}
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
A homomorphism out of a free $A$-module is a morphism in $\mathcal{C}$ of the form

\begin{displaymath}
f \;\colon\; A\otimes C \longrightarrow N
\end{displaymath}
fitting into the diagram (where we are notationally suppressing the [[associator]])

\begin{displaymath}
\itexarray{
    A \otimes A \otimes C
      &\overset{A \otimes f}{\longrightarrow}&
    A \otimes N
    \\
    {}^{\mathllap{\mu \otimes id}}\downarrow 
      && 
    \downarrow^{\mathrlap{\rho}}
    \\
    A \otimes C
      &\underset{f}{\longrightarrow}&
    N
  }
  \,.
\end{displaymath}
Consider the composite

\begin{displaymath}
\tilde f
    \;\colon\;
  C
    \underoverset{\simeq}{\ell_C}{\longrightarrow}
  1 \otimes C
    \overset{e\otimes id}{\longrightarrow}
  A \otimes C
    \overset{f}{\longrightarrow}
  N
  \,,
\end{displaymath}
i.e. the restriction of $f$ to the unit ``in'' $A$. By definition, this fits into a [[commuting square]] of the form (where we are now notationally suppressing the [[associator]] and the [[unitor]])

\begin{displaymath}
\itexarray{
   A \otimes C
     &\overset{id \otimes \tilde f}{\longrightarrow}&
   A \otimes N
   \\
   {}^{\mathllap{id \otimes e \otimes id}}\downarrow 
     && 
   \downarrow^{\mathrlap{=}}
   \\
   A \otimes A \otimes C
    &\underset{id \otimes f}{\longrightarrow}&
   A \otimes N
  }
  \,.
\end{displaymath}
Pasting this square onto the top of the previous one yields

\begin{displaymath}
\itexarray{
   A \otimes C
     &\overset{id \otimes \tilde f}{\longrightarrow}&
   A \otimes N
   \\
   {}^{\mathllap{id \otimes e \otimes id}}\downarrow 
     && 
   \downarrow^{\mathrlap{=}}
    \\
    A \otimes A \otimes C
      &\overset{A \otimes f}{\longrightarrow}&
    A \otimes N
    \\
    {}^{\mathllap{\mu \otimes id}}\downarrow 
      && 
    \downarrow^{\mathrlap{\rho}}
    \\
    A \otimes C
      &\underset{f}{\longrightarrow}&
    N
  }
  \,,
\end{displaymath}
where now the left vertical composite is the identity, by the unit law in $A$. This shows that $f$ is uniquely determined by $\tilde f$ via the relation

\begin{displaymath}
f = \rho \circ (id_A \otimes \tilde f)
  \,.
\end{displaymath}
This natural bijection between $f$ and $\tilde f$ establishes the adjunction.

\end{proof}
\begin{defn}
\label{TensorProductOfModulesOverCommutativeMonoidObject}\hypertarget{TensorProductOfModulesOverCommutativeMonoidObject}{}
Given a (pointed) [[topologically enriched category|topological]] [[symmetric monoidal category]] $(\mathcal{C}, \otimes, 1)$ (def. \ref{SymmetricMonoidalCategory}), given $(A,\mu,e)$ a [[commutative monoid in a symmetric monoidal category|commutative monoid in]] $(\mathcal{C}, \otimes, 1)$ (def. \ref{MonoidsInMonoidalCategory}), and given $(N_1, \rho_1)$ and $(N_2, \rho_2)$ two left $A$-[[module objects]] (def.\ref{MonoidsInMonoidalCategory}), then the \textbf{[[tensor product of modules]]} $N_1 \otimes_A N_2$ is, if it exists, the [[coequalizer]]

\begin{displaymath}
N_1 \otimes A \otimes N_2
  \underoverset
    {\underset{\rho_{1}\circ (\tau_{N_1,A} \otimes N_2)}{\longrightarrow}}
    {\overset{N_1 \otimes \rho_2}{\longrightarrow}}
    {\phantom{AAAA}}
  N_1 \otimes N_1
    \overset{coequ}{\longrightarrow}
  N_1 \otimes_A N_2
\end{displaymath}
\end{defn}
\begin{prop}
\label{MonoidalCategoryOfModules}\hypertarget{MonoidalCategoryOfModules}{}
Given a (pointed) [[topologically enriched category|topological]] [[symmetric monoidal category]] $(\mathcal{C}, \otimes, 1)$ (def. \ref{SymmetricMonoidalCategory}), and given $(A,\mu,e)$ a [[commutative monoid in a symmetric monoidal category|commutative monoid in]] $(\mathcal{C}, \otimes, 1)$ (def. \ref{MonoidsInMonoidalCategory}). If all [[coequalizers]] exist in $\mathcal{C}$, then the [[tensor product of modules]] $\otimes_A$ from def. \ref{TensorProductOfModulesOverCommutativeMonoidObject} makes the [[category of modules]] $A Mod(\mathcal{C})$ into a [[symmetric monoidal category]], $(A Mod, \otimes_A, A)$ with [[tensor unit]] the object $A$ itself, regarded as an $A$-module via prop. \ref{MonoidModuleOverItself}.

\end{prop}
\begin{defn}
\label{AAlgebra}\hypertarget{AAlgebra}{}
Given a [[monoidal category|monoidal]] [[category of modules]] $(A Mod , \otimes_A , A)$ as in prop. \ref{MonoidalCategoryOfModules}, then a [[monoid in a monoidal category|monoid]] $(E, \mu, e)$ in $(A Mod , \otimes_A , A)$ (def. \ref{MonoidsInMonoidalCategory}) is called an \textbf{$A$-[[associative algebra|algebra]]}.

\end{defn}
\begin{prop}
\label{AlgebrasOverAAreMonoidsUnderA}\hypertarget{AlgebrasOverAAreMonoidsUnderA}{}
Given a [[monoidal category|monoidal]] [[category of modules]] $(A Mod , \otimes_A , A)$ in a [[monoidal category]] $(\mathcal{C},\otimes, 1)$ as in prop. \ref{MonoidalCategoryOfModules}, and an $A$-algebra $(E,\mu,e)$ (def. \ref{AAlgebra}), then there is an [[equivalence of categories]]

\begin{displaymath}
A Alg_{comm}(\mathcal{C}) 
    \coloneqq 
  CMon(A Mod)
   \simeq
  CMon(\mathcal{C})^{A/}
\end{displaymath}
between the [[category of commutative monoids]] in $A Mod$ and the [[coslice category]] of commutative monoids in $\mathcal{C}$ under $A$, hence between commutative $A$-algebras in $\mathcal{C}$ and commutative monoids $E$ in $\mathcal{C}$ that are equipped with a homomorphism of monoids $A \longrightarrow E$.

\end{prop}
(e.g. \hyperlink{EKMM97}{EKMM 97, VII lemma 1.3})

\begin{proof}
In one direction, consider a $A$-algebra $E$ with unit $e_E \;\colon\; A \longrightarrow E$ and product $\mu_{E/A} \colon E \otimes_A E \longrightarrow E$. There is the underlying product $\mu_E$

\begin{displaymath}
\itexarray{
    E \otimes A \otimes E
    & 
    \underoverset
      {\underset{}{\longrightarrow}}
      {\overset{}{\longrightarrow}}
      {\phantom{AAA}}
    &
    E \otimes E
     &\overset{coeq}{\longrightarrow}&
    E \otimes_A E
    \\
    && & {}_{\mathllap{\mu_E}}\searrow & \downarrow^{\mathrlap{\mu_{E/A}}}
    \\
    && && E
  }
  \,.
\end{displaymath}
By considering a diagram of such coequalizer diagrams with middle vertical morphism $e_E\circ e_A$, one find that this is a unit for $\mu_E$ and that $(E, \mu_E, e_E \circ e_A)$ is a commutative monoid in $(\mathcal{C}, \otimes, 1)$.

Then consider the two conditions on the unit $e_E \colon A \longrightarrow E$. First of all this is an $A$-module homomorphism, which means that

\begin{displaymath}
(\star)
  \;\;\;\;\;
  \;\;\;\;\;
  \itexarray{
    A \otimes A &\overset{id \otimes e_E}{\longrightarrow}& A \otimes E
    \\
    {}^{\mathllap{\mu_A}}\downarrow && \downarrow^{\mathrlap{\rho}}
    \\
    A &\underset{e_E}{\longrightarrow}& E
  }
\end{displaymath}
[[commuting diagram|commutes]]. Moreover it satisfies the unit property

\begin{displaymath}
\itexarray{
    A \otimes_A E 
      &\overset{e_A \otimes id}{\longrightarrow}&
    E \otimes_A E
    \\
    & {}_{\mathllap{\simeq}}\searrow & \downarrow^{\mathrlap{\mu_{E/A}}}
    \\
    && E
  }
  \,.
\end{displaymath}
By forgetting the tensor product over $A$, the latter gives

\begin{displaymath}
\itexarray{
    A \otimes E 
      &\overset{e \otimes id}{\longrightarrow}&
    E \otimes E
    \\
    \downarrow && \downarrow^{\mathrlap{}}
    \\
    A \otimes_A E 
      &\overset{e_E \otimes id}{\longrightarrow}&
    E \otimes_A E
    \\
    {}^{\mathllap{\simeq}}\downarrow 
      && 
    \downarrow^{\mathrlap{\mu_{E/A}}}
    \\
    E &=& E
  }
  \;\;\;\;\;\;\;\;
   \simeq
  \;\;\;\;\;\;\;\;
  \itexarray{
    A \otimes E 
      &\overset{e_E \otimes id}{\longrightarrow}&
    E \otimes E
    \\
    {}^{\mathllap{\rho}}\downarrow && \downarrow^{\mathrlap{\mu_{E}}}
    \\
    E &\underset{id}{\longrightarrow}& E  
  }
  \,,
\end{displaymath}
where the top vertical morphisms on the left the canonical coequalizers, which identifies the vertical composites on the right as shown. Hence this may be [[pasting|pasted]] to the square $(\star)$ above, to yield a [[commuting square]]

\begin{displaymath}
\itexarray{
    A \otimes A
     &\overset{id\otimes e_E}{\longrightarrow}&
    A \otimes E 
      &\overset{e_E \otimes id}{\longrightarrow}&
    E \otimes E
    \\
    {}^{\mathllap{\mu_A}}\downarrow
      &&
    {}^{\mathllap{\rho}}\downarrow 
      && 
    \downarrow^{\mathrlap{\mu_{E}}}
    \\
    A &\underset{e_E}{\longrightarrow}& E &\underset{id}{\longrightarrow}& E  
  }  
  \;\;\;\;\;\;\;\;\;\;
   =
  \;\;\;\;\;\;\;\;\;\;
  \itexarray{
    A \otimes A 
     &\overset{e_E \otimes e_E}{\longrightarrow}&
    E \otimes E
    \\
    {}^{\mathllap{\mu_A}}\downarrow
      &&
    \downarrow^{\mathrlap{\mu_E}}
    \\
    A &\underset{e_E}{\longrightarrow}& E
  }
  \,.
\end{displaymath}
This shows that the unit $e_A$ is a homomorphism of monoids $(A,\mu_A, e_A) \longrightarrow (E, \mu_E, e_E\circ e_A)$.

Now for the converse direction, assume that $(A,\mu_A, e_A)$ and $(E, \mu_E, e'_E)$ are two commutative monoids in $(\mathcal{C}, \otimes, 1)$ with $e_E \;\colon\; A  \to E$ a monoid homomorphism. Then $E$ inherits a left $A$-[[module]] structure by

\begin{displaymath}
\rho
    \;\colon\;
  A \otimes E 
    \overset{e_A \otimes id}{\longrightarrow} 
  E \otimes E
    \overset{\mu_E}{\longrightarrow}
  E
  \,.
\end{displaymath}
By commutativity and associativity it follows that $\mu_E$ coequalizes the two induced morphisms $E \otimes A \otimes E \underoverset{\longrightarrow}{\longrightarrow}{\phantom{AA}} E \otimes E$. Hence the [[universal property]] of the [[coequalizer]] gives a factorization through some $\mu_{E/A}\colon E \otimes_A E \longrightarrow E$. This shows that $(E, \mu_{E/A}, e_E)$ is a commutative $A$-algebra.

Finally one checks that these two constructions are inverses to each other, up to isomorphism.

\end{proof}
\hypertarget{day_convolution}{}\paragraph*{{Day convolution}}\label{day_convolution}

\begin{defn}
\label{TopologicalDayConvolutionProduct}\hypertarget{TopologicalDayConvolutionProduct}{}
Let $\mathcal{C}$ be a [[small category|small]] pointed [[topologically enriched category|topological]] [[monoidal category]] (def. \ref{MonoidalCategory}) with [[tensor product]] denoted $\otimes_{\mathcal{C}} \;\colon\; \mathcal{C} \times\mathcal{C} \to \mathcal{C}$.

Then the \textbf{[[Day convolution]] tensor product} on the pointed topological [[enriched functor category]] $[\mathcal{C},Top^{\ast/}_{cg}]$ (def. \ref{PointedTopologicalFunctorCategory}) is the [[functor]]

\begin{displaymath}
\otimes_{Day} 
    \;\colon\; 
  [\mathcal{C},Top^{\ast/}_{cg}] \times [\mathcal{C},Top^{\ast/}_{cg}] 
    \longrightarrow 
  [\mathcal{C},Top^{\ast/}_{cg}]
\end{displaymath}
out of the pointed topological [[product category]] (def. \ref{OppositeAndProductOfPointedTopologicallyEnrichedCategory}) given by the following [[coend]] (def. \ref{EndAndCoendInTopcgSmash})

\begin{displaymath}
X \otimes_{Day} Y
    \;\colon\;
  c 
    \;\mapsto\;
  \overset{(c_1,c_2)\in \mathcal{C}\times \mathcal{C}}{\int}
    \mathcal{C}(c_1 \otimes_{\mathcal{C}} c_2, c) \wedge X(c_1) \wedge Y(c_2)
  \,.
\end{displaymath}
\end{defn}
\begin{example}
\label{}\hypertarget{}{}
Let $Seq$ denote the category with objects the [[natural numbers]], and only the [[zero morphisms]] and [[identity morphisms]] on these objects:

\begin{displaymath}
Seq(n_1,n_2)
  \coloneqq
  \left\{
    \itexarray{
      S^0 & if\; n_1 = n_2
      \\
      \ast & otherwise
    }
  \right.
  \,.
\end{displaymath}
Regard this as a pointed topologically enriched category in the unique way. The operation of addition of natural numbers $\otimes = +$ makes this a monoidal category.

An object $X_\bullet \in [Seq, Top_{cg}^{\ast/}]$ is an $\mathbb{N}$-sequence of pointed topological spaces. Given two such, then their Day convolution according to def. \ref{TopologicalDayConvolutionProduct} is

\begin{displaymath}
\begin{aligned}
    (X \otimes_{Day} Y)_n
    & =
    \overset{(n_1,n_2)}{\int}
     Seq(n_1 + n_2 , n)
     \wedge 
     X_{n_1} \wedge X_{n_2}
    \\
    & = \underset{{n_1+n_2} \atop {= n}}{\coprod} \left(X_{n_1}\wedge X_{n_2}\right)
  \end{aligned} 
  \,.
\end{displaymath}
\end{example}
We observe now that [[Day convolution]] is equivalently a [[left Kan extension]] (def. \ref{TopologicalLeftKanExtensionBCoend}). This will be key for understanding [[monoids]] and [[modules]] with respect to Day convolution.

\begin{defn}
\label{ExternalTensorProduct}\hypertarget{ExternalTensorProduct}{}
Let $\mathcal{C}$ be a [[small category|small]] pointed [[topologically enriched category]] (\href{Introduction+to+Stable+homotopy+theory+--+P#TopEnrichedCategory}{def.}). Its \textbf{[[external tensor product]]} is the pointed [[topologically enriched functor]]

\begin{displaymath}
\overline{\wedge} 
   \;\colon\; 
  [\mathcal{C},Top^{\ast/}_{cg}] 
    \times 
  [\mathcal{C},Top^{\ast/}_{cg}] 
    \longrightarrow 
  [\mathcal{C}\times \mathcal{C}, Top^{\ast/}_{cg}]
\end{displaymath}
given by

\begin{displaymath}
X \overline{\wedge} Y  
    \;\coloneqq\; 
  \wedge \circ (X,Y)
  \,,
\end{displaymath}
i.e.

\begin{displaymath}
(X \overline\wedge Y)(c_1,c_2)
  =
  X(c_1)\wedge X(c_2)
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{DayConvolutionViaKanExtensionOfExternalTensorAlongTensor}\hypertarget{DayConvolutionViaKanExtensionOfExternalTensorAlongTensor}{}
The [[Day convolution]] product (def. \ref{TopologicalDayConvolutionProduct}) of two functors is equivalently the [[left Kan extension]] (def. \ref{TopologicalLeftKanExtensionBCoend}) of their external tensor product (def. \ref{ExternalTensorProduct}) along the tensor product $\otimes_{\mathcal{C}}$: there is a [[natural isomorphism]]

\begin{displaymath}
X \otimes_{Day} Y
  \simeq
  Lan_{\otimes_{\mathcal{C}}} (X \overline{\wedge} Y)
  \,.
\end{displaymath}
Hence the [[adjunction unit]] is a [[natural transformation]] of the form

\begin{displaymath}
\itexarray{
    \mathcal{C} \times \mathcal{C}
    &&
      \overset{X \overline{\wedge} Y}{\longrightarrow}
    &&
      Top^{\ast/}_{cg}
    \\
    & {}^{\mathllap{\otimes}}\searrow 
     &\Downarrow&
    \nearrow_{\mathrlap{X \otimes_{Day} Y}}
    \\
    && \mathcal{C}
  }
  \,.
\end{displaymath}
\end{prop}
This perspective is highlighted in (\hyperlink{MMSS00}{MMSS 00, p. 60}).

\begin{proof}
By prop. \ref{TopologicalLeftKanExtensionBCoend} we may compute the left Kan extension as the following [[coend]]:

\begin{displaymath}
\begin{aligned}
     Lan_{\otimes_{\mathcal{C}}} (X\overline{\wedge} Y)(c)
     &
     \simeq
      \overset{(c_1,c_2)}{\int} 
      \mathcal{C}(c_1 \otimes_{\mathcal{C}} c_2, c )
      \wedge
      (X\overline{\wedge}Y)(c_1,c_2)
     \\
    & =
    \overset{(c_1,c_2)}{\int}
    \mathcal{C}(c_1\otimes c_2) 
    \wedge
     X(c_1)\wedge X(c_2) 
  \end{aligned} 
  \,.
\end{displaymath}
\end{proof}
\begin{cor}
\label{DayConvolutionViaNaturalIsosInvolvingExternalTensorAndTensor}\hypertarget{DayConvolutionViaNaturalIsosInvolvingExternalTensorAndTensor}{}
The [[Day convolution]] $\otimes_{Day}$ (def. \ref{TopologicalDayConvolutionProduct}) is universally characterized by the property that there are [[natural isomorphisms]]

\begin{displaymath}
[\mathcal{C},Top^{\ast/}_{cg}](X \otimes_{Day} Y, Z) 
    \simeq 
  [\mathcal{C}\times \mathcal{C},Top^{\ast/}_{cg}](
    X \overline{\wedge} Y,\; Z \circ \otimes
  )
  \,,
\end{displaymath}
where $\overline{\wedge}$ is the external product of def. \ref{ExternalTensorProduct}.

\end{cor}
Write

\begin{displaymath}
y \;\colon\; \mathcal{C}^{op} \longrightarrow [\mathcal{C}, Top^{\ast/}_{cg}]
\end{displaymath}
for the $Top^{\ast/}_{cg}$-[[Yoneda embedding]], so that for $c\in \mathcal{C}$ any [[object]], $y(c)$ is the [[representable functor|corepresented functor]] $y(c)\colon d \mapsto \mathcal{C}(c,d)$.

\begin{prop}
\label{DayConvolutionYieldsMonoidalCategoryStructure}\hypertarget{DayConvolutionYieldsMonoidalCategoryStructure}{}
For $\mathcal{C}$ a [[small category|small]] pointed [[topologically enriched category|topological]] [[monoidal category]] (def. \ref{MonoidalCategory}), the [[Day convolution]] tensor product $\otimes_{Day}$ of def. \ref{TopologicalDayConvolutionProduct} makes the pointed topologically [[enriched functor category]]

\begin{displaymath}
( [\mathcal{C}, Top^{\ast/}_{cg}], \otimes_{Day}, y(1))
\end{displaymath}
into a pointed topological [[monoidal category]] (def. \ref{MonoidalCategory}) with [[tensor unit]] $y(1)$ [[representable functor|co-represented]] by the tensor unit $1$ of $\mathcal{C}$.

Moreover, if $(\mathcal{C}, \otimes, 1)$ is equipped with a [[braiding]] $\tau^{\mathcal{C}}$ (def. \ref{BraidedMonoidalCategory}), then $( [\mathcal{C}, Top^{\ast/}_{cg}], \otimes_{Day}, y(1))$ becomes itself a [[braided monoidal category]] with braiding given by

\begin{displaymath}
\itexarray{
    (X \otimes_{Day} Y)(c)
    & = &
    \overset{c_1,c_2}{\int} 
     \mathcal{C}(c_1 \otimes c_2) \wedge X(c_1) \wedge Y(c_2)
    \\
    {}^{\mathllap{\tau}_{X,Y}(c)}\downarrow
    &&
    \downarrow^{\mathrlap{\overset{c_1,c_2}{\int} \mathcal{C}(\tau^{\mathcal{C}}_{c_1,c_2}, c  ) \wedge \tau^{Top^{\ast/}}_{X(c(1)), X(c_2)}  }}
    \\
    (Y \otimes_{Day} X)(c)
    & = &
    \overset{c_1,c_2}{\int} 
     \mathcal{C}(c_2 \otimes c_1) \wedge Y(c_2) \wedge X(c_1)   
  }
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
Regarding [[associativity]], observe that

\begin{displaymath}
\begin{aligned}
    (X \otimes_{Day} ( Y \otimes_{Day} Z ))(c)
    & \simeq
    \overset{(c_1,c_2)}{\int}
      \mathcal{C}(c_1 \otimes_{\mathcal{D}} c_2, \,c) 
        \wedge
      X(c_1) 
       \wedge
      \overset{(d_1,d_2)}{\int}
       \mathcal{C}(d_1 \otimes_{\mathcal{C}} d_2, c_2 )
       (Y(d_2) \wedge Z(d_2))
    \\
    &\simeq \overset{c_1, d_1, d_2}{\int}
    \underset{\simeq \mathcal{C}(c_1 \otimes_{\mathcal{C}} d_1 \otimes_{\mathcal{C}} d_2, c )}{
      \underbrace{
       \overset{c_2}{\int} 
         \mathcal{C}(c_1 \otimes_{\mathcal{D}} c_2 , c)
           \wedge
         \mathcal{C}(d_1 \otimes_{\mathcal{C}}d_2, c_2 )
      }
    }
    \wedge X(c_1) \wedge (Y(d_1) \wedge Z(d_2))
    \\
    &\simeq 
    \overset{c_1, d_1, d_2}{\int}
     \mathcal{C}(c_1\otimes_{\mathcal{C}} d_1 \otimes_{\mathcal{C}} d_2, c ) 
       \wedge  
     X(c_1) \wedge (Y(d_1) \wedge Z(d_2))
  \end{aligned}
  \,,
\end{displaymath}
where we used the [[Fubini theorem]] for [[coends]] (prop. \ref{CoendsCommuteWithEachOther}) and then twice the [[co-Yoneda lemma]] (prop. \ref{TopologicalCoYonedaLemma}). An analogous formula follows for $X \otimes_{Day}  (Y \otimes_{Day} Z)))(c)$, and so associativity follows via prop. \ref{DayConvolutionViaKanExtensionOfExternalTensorAlongTensor} from the associativity of the [[smash product]] and of the tensor product $\otimes_{\mathcal{C}}$.

Similarly, if $\mathcal{C}$ is braided then the hexagon identity for the [[braiding]] follows, under the coend, from the hexagon identities for the braidings in $\mathcal{C}$ and $Top^{\ast/}_{cg}$.

To see that $y(1)$ is the tensor unit for $\otimes_{Day}$, use the [[Fubini theorem]] for [[coends]] (prop. \ref{CoendsCommuteWithEachOther}) and then twice the [[co-Yoneda lemma]] (prop. \ref{TopologicalCoYonedaLemma}) to get for any $X \in [\mathcal{C},Top^{\ast/}_{cg}]$ that

\begin{displaymath}
\begin{aligned}
     X \otimes_{Day} y(1)
     & 
     =
     \overset{c_1,c_2 \in \mathcal{C}}{\int}
     \mathcal{C}(c_1\otimes_{\mathcal{D}} c_2,-) 
      \wedge 
     X(c_1) \wedge \mathcal{C}(1,c_2)
     \\
     & \simeq 
     \overset{c_1\in \mathcal{C}}{\int}
      X(c_1) 
        \wedge  
      \overset{c_2 \in \mathcal{C}}{\int}  
        \mathcal{C}(c_1\otimes_{\mathcal{C}} c_2,-)
        \wedge 
        \mathcal{C}(1,c_2) 
    \\
    & \simeq 
      \overset{c_1\in \mathcal{C}}{\int}
      X(c_1) 
         \wedge
      \mathcal{C}(c_1 \otimes_{\mathcal{C}} 1, -)
    \\
    & \simeq 
      \overset{c_1\in \mathcal{C}}{\int}
      X(c_1) 
         \wedge
      \mathcal{C}(c_1, -)
    \\
    & \simeq
    X(-)
    \\
    & \simeq 
    X
  \end{aligned}
  \,.
\end{displaymath}
\end{proof}
\begin{prop}
\label{DayMonoidalStructureIsClosed}\hypertarget{DayMonoidalStructureIsClosed}{}
For $\mathcal{C}$ a [[small category|small]] pointed [[topologically enriched category|topological]] [[monoidal category]] (def. \ref{MonoidalCategory}) with [[tensor product]] denoted $\otimes_{\mathcal{C}} \;\colon\; \mathcal{C} \times\mathcal{C} \to \mathcal{C}$, the [[monoidal category]] with [[Day convolution]] $([\mathcal{C},Top^{\ast/}_{cg}], \otimes_{Day}, y(1))$ from def. \ref{DayConvolutionYieldsMonoidalCategoryStructure} is a [[closed monoidal category]] (def. \ref{ClosedMonoidalCategory}). Its [[internal hom]] $[-,-]_{Day}$ is given by the [[end]] (def. \ref{EndAndCoendInTopcgSmash})

\begin{displaymath}
[X,Y]_{Day}(c)
  \simeq
   \underset{c_1,c_2}{\int}
      Maps\left( 
        \mathcal{C}(c \otimes_{\mathcal{C}} c_1,c_2),
        \;
        Maps(X(c_1) ,  Y(c_2))_\ast 
      \right)_\ast       
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
Using the [[Fubini theorem]] (def. \ref{CoendsCommuteWithEachOther}) and the [[co-Yoneda lemma]] (def. \ref{TopologicalCoYonedaLemma}) and in view of definition \ref{PointedTopologicalFunctorCategory} of the [[enriched functor category]], there is the following sequence of [[natural isomorphisms]]:

\begin{displaymath}
\begin{aligned}
     [\mathcal{C},V]( X, [Y,Z]_{Day} )
       & \simeq
     \underset{c}{\int} 
     Maps\left(
        X(c), 
        \underset{c_1,c_2}{\int}
        Maps\left( 
          \mathcal{C}(c \otimes_{\mathcal{C}} c_1 , c_2),
          Maps(Y(c_1), Z(c_2))_\ast 
        \right)_\ast
     \right)_\ast
     \\
     &
     \simeq
     \underset{c}{\int}
     \underset{c_1,c_2}{\int}
     Maps\left(
       \mathcal{C}(c \otimes_{\mathcal{C}} c_1, c_2)
         \wedge
       X(c)
         \wedge
       Y(c_1)
       ,\;
       Z(c_2)
     \right)_\ast
     \\
     & \simeq
     \underset{c_2}{\int}
     Maps\left(
       \overset{c,c_1}{\int}
       \mathcal{C}(c \otimes_{\mathcal{C}} c_1, c_2)
         \wedge
       X(c)
         \wedge 
       Y(c_1)
       ,\;
       Z(c_2)
     \right)_\ast
     \\
     &\simeq
     \underset{c_2}{\int}
       Maps\left(
         (X \otimes_{Day} Y)(c_2),
         Z(c_2)
       \right)_\ast
     \\
     &\simeq
     [\mathcal{C},V](X \otimes_{Day} Y, Z)
  \end{aligned}
  \,.
\end{displaymath}
\end{proof}
\begin{prop}
\label{}\hypertarget{}{}
In the situation of def. \ref{DayConvolutionYieldsMonoidalCategoryStructure}, the [[Yoneda embedding]] $c\mapsto \mathcal{C}(c,-)$ constitutes a [[strong monoidal functor]]

\begin{displaymath}
(\mathcal{C},\otimes_{\mathcal{C}}, I) \hookrightarrow ([\mathcal{C},V], \otimes_{Day}, y(I))
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
That the [[tensor unit]] is respected is part of prop. \ref{DayConvolutionYieldsMonoidalCategoryStructure}. To see that the [[tensor product]] is respected, apply the [[co-Yoneda lemma]] (prop \ref{TopologicalCoYonedaLemma}) twice to get the following natural isomorphism

\begin{displaymath}
\begin{aligned}
    (y(c_1) \otimes_{Day} y(c_2))(c)
    &
    \simeq
    \overset{d_1, d_2}{\int} 
      \mathcal{C}(d_1 \otimes_{\mathcal{C}} d_2, c )
    \wedge
      \mathcal{C}(c_1,d_1)
    \wedge
      \mathcal{C}(c_2,d_2)
    \\
    & \simeq \mathcal{C}(c_1\otimes_{\mathcal{C}}c_2 , c )
    \\
    & 
    = y(c_1 \otimes_{\mathcal{C}} c_2 )(c)
  \end{aligned}
  \,.
\end{displaymath}
\end{proof}
\hypertarget{FunctorsWithSmashProduct}{}\paragraph*{{Functors with smash product}}\label{FunctorsWithSmashProduct}

\begin{defn}
\label{LaxMonoidalFunctor}\hypertarget{LaxMonoidalFunctor}{}
Let $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ and $(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} )$ be two (pointed) [[topologically enriched category|topologically enriched]] [[monoidal categories]] (def. \ref{MonoidalCategory}). A topologically enriched \textbf{lax monoidal functor} between them is

\begin{enumerate}%
\item a [[topologically enriched functor]]

\begin{displaymath}
F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}
  \,,
\end{displaymath}

\item a morphism

\begin{displaymath}
\epsilon \;\colon\; 1_{\mathcal{D}} \longrightarrow F(1_{\mathcal{C}})
\end{displaymath}

\item a [[natural transformation]]

\begin{displaymath}
\mu_{x,y} 
    \;\colon\; 
  F(x) \otimes_{\mathcal{D}} F(y) 
    \longrightarrow 
  F(x \otimes_{\mathcal{C}} y)
\end{displaymath}
for all $x,y \in \mathcal{C}$



\end{enumerate}
satisfying the following conditions:

\begin{enumerate}%
\item \textbf{([[associativity]])} For all objects $x,y,z \in \mathcal{C}$ the following [[commuting diagram|diagram commutes]]

\begin{displaymath}
\itexarray{
    (F(x) \otimes_{\mathcal{D}} F(y)) \otimes_{\mathcal{D}} F(z)
      &\underoverset{\simeq}{a^{\mathcal{D}}_{F(x),F(y),F(z)}}{\longrightarrow}&
    F(x) \otimes_{\mathcal{D}}( F(y)\otimes_{\mathcal{D}} F(z) )
    \\
    {}^{\mathllap{\mu_{x,y} \otimes id}}\downarrow 
      && 
    \downarrow^{\mathrlap{id\otimes \mu_{y,z}}}
    \\
    F(x \otimes_{\mathcal{C}} y) \otimes_{\mathcal{D}} F(z)
     &&
    F(x) \otimes_{\mathcal{D}} ( F(x \otimes_{\mathcal{C}} y) )
    \\
    {}^{\mathllap{\mu_{x \otimes_{\mathcal{C}} y , z} } }\downarrow 
      && 
    \downarrow^{\mathrlap{\mu_{ x, y \otimes_{\mathcal{C}} z  }}}
    \\
    F( ( x \otimes_{\mathcal{C}} y ) \otimes_{\mathcal{C}} z  )
      &\underset{F(a^{\mathcal{C}}_{x,y,z})}{\longrightarrow}&
    F( x \otimes_{\mathcal{C}} ( y \otimes_{\mathcal{C}} z ) )
  }
  \,,
\end{displaymath}
where $a^{\mathcal{C}}$ and $a^{\mathcal{D}}$ denote the [[associators]] of the monoidal categories;


\item \textbf{([[unitality]])} For all $x \in \mathcal{C}$ the following [[commuting diagram|diagrams commutes]]

\begin{displaymath}
\itexarray{
    1_{\mathcal{D}} \otimes_{\mathcal{D}} F(x)
      &\overset{\epsilon \otimes id}{\longrightarrow}&
    F(1_{\mathcal{C}}) \otimes_{\mathcal{D}} F(x)
    \\
    {}^{\mathllap{\ell^{\mathcal{D}}_{F(x)}}}\downarrow 
      && 
    \downarrow^{\mathrlap{\mu_{1_{\mathcal{C}}, x }}}
    \\
    F(x) 
      &\overset{F(\ell^{\mathcal{C}}_x )}{\longleftarrow}&
    F(1 \otimes_{\mathcal{C}} x  )
  }
\end{displaymath}
and

\begin{displaymath}
\itexarray{
    F(x) \otimes_{\mathcal{D}}  1_{\mathcal{D}}
      &\overset{id \otimes \epsilon }{\longrightarrow}&
    F(x) \otimes_{\mathcal{D}}  F(1_{\mathcal{C}}) 
    \\
    {}^{\mathllap{r^{\mathcal{D}}_{F(x)}}}\downarrow 
      && 
    \downarrow^{\mathrlap{\mu_{x, 1_{\mathcal{C}} }}}
    \\
    F(x) 
      &\overset{F(r^{\mathcal{C}}_x )}{\longleftarrow}&
    F(x \otimes_{\mathcal{C}} 1  )
  }
  \,,
\end{displaymath}
where $\ell^{\mathcal{C}}$, $\ell^{\mathcal{D}}$, $r^{\mathcal{C}}$, $r^{\mathcal{D}}$ denote the left and right [[unitors]] of the two monoidal categories, respectively.



\end{enumerate}
If $\epsilon$ and alll $\mu_{x,y}$ are [[isomorphisms]], then $F$ is called a \textbf{strong monoidal functor}.

If moreover $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ and $(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} )$ are equipped with the structure of [[braided monoidal categories]] (def. \ref{BraidedMonoidalCategory}), then the lax monoidal functor $F$ is called a \textbf{[[braided monoidal functor]]} if in addition the following [[commuting diagram|diagram commutes]] for all objects $x,y \in \mathcal{C}$

\begin{displaymath}
\itexarray{
    F(x) \otimes_{\mathcal{C}} F(y)
      &\overset{\tau^{\mathcal{D}}_{F(x), F(y)}}{\longrightarrow}&
    F(y) \otimes_{\mathcal{D}} F(x)
    \\
    {}^{\mathllap{\mu_{x,y}}}\downarrow 
      && 
    \downarrow^{\mathrlap{\mu_{y,x}}}
    \\
    F(x \otimes_{\mathcal{C}} y )
      &\underset{F(\tau^{\mathcal{C}}_{x,y}  )}{\longrightarrow}&
    F( y \otimes_{\mathcal{C}} x )
  }
  \,.
\end{displaymath}
A [[homomorphism]] $f\;\colon\; (F_1,\mu_1, \epsilon_1) \longrightarrow (F_2, \mu_2, \epsilon_2)$ between two (braided) lax monoidal functors is a \textbf{[[monoidal natural transformation]]}, in that it is

\begin{itemize}%
\item a [[natural transformation]] $f_x \;\colon\; F_1(x) \longrightarrow F_2(x)$ of the underlying functors

\end{itemize}
compatible with the product and the unit in that the following [[commuting diagram|diagrams commute]] for all objects $x,y \in \mathcal{C}$:

\begin{displaymath}
\itexarray{
    F_1(x) \otimes_{\mathcal{D}} F_1(y)
      &\overset{f(x)\otimes_{\mathcal{D}} f(y)}{\longrightarrow}&
    F_2(x) \otimes_{\mathcal{D}} F_2(y)
    \\
    {}^{\mathllap{(\mu_1)_{x,y}}}\downarrow
     &&
    \downarrow^{\mathrlap{(\mu_2)_{x,y}}}
    \\
    F_1(x\otimes_{\mathcal{C}} y)
     &\underset{f(x \otimes_{\mathcal{C}} y ) }{\longrightarrow}&
    F_2(x \otimes_{\mathcal{C}} y)
  }
\end{displaymath}
and

\begin{displaymath}
\itexarray{
    && 1_{\mathcal{D}}
    \\
    & {}^{\mathllap{\epsilon_1}}\swarrow && \searrow^{\mathrlap{\epsilon_2}}
    \\
    F_1(1_{\mathcal{C}})
     &&\underset{f(1_{\mathcal{C}})}{\longrightarrow}&&
    F_2(1_{\mathcal{C}})
  }
  \,.
\end{displaymath}
We write $MonFun(\mathcal{C},\mathcal{D})$ for the resulting [[category]] of lax monoidal functors between monoidal categories $\mathcal{C}$ and $\mathcal{D}$, similarly $BraidMonFun(\mathcal{C},\mathcal{D})$ for the category of braided monoidal functors between [[braided monoidal categories]], and $SymMonFun(\mathcal{C},\mathcal{D})$ for the category of braided monoidal functors between [[symmetric monoidal categories]].

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
In the literature the term ``monoidal functor'' often refers by default to what in def. \ref{LaxMonoidalFunctor} is called a strong monoidal functor. But for the purpose of the discussion of [[functors with smash product]] \hyperlink{FunctorsWithSmashProduct}{below}, it is crucial to admit the generality of lax monoidal functors.

If $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ and $(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} )$ are [[symmetric monoidal categories]] (def. \ref{SymmetricMonoidalCategory}) then a braided monoidal functor (def. \ref{LaxMonoidalFunctor}) between them is often called a \textbf{[[symmetric monoidal functor]]}.

\end{remark}
\begin{defn}
\label{ModuleOverAMonoidalFunctor}\hypertarget{ModuleOverAMonoidalFunctor}{}
Let $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ and $(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} )$ be two (pointed) [[topologically enriched category|topologically enriched]] [[monoidal categories]] (def. \ref{MonoidalCategory}), and let $F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}$ be a [[topologically enriched functor|topologically enriched]] [[lax monoidal functor]] between them, with product operation $\mu$.

Then a left \textbf{[[module over a monoidal functor|module over the lax monoidal functor]]} is

\begin{enumerate}%
\item a [[topologically enriched functor]]

\begin{displaymath}
G \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}
  \,;
\end{displaymath}

\item a [[natural transformation]]

\begin{displaymath}
\rho_{x,y} 
    \;\colon\; 
  F(x) \otimes_{\mathcal{D}} N(y)
   \longrightarrow
 N(x \otimes_{\mathcal{C}} y )
\end{displaymath}


\end{enumerate}
such that

\begin{itemize}%
\item \textbf{(action property)} For all objects $x,y,z \in \mathcal{C}$ the following [[commuting diagram|diagram commutes]]

\begin{displaymath}
\itexarray{
     (F(x) \otimes_{\mathcal{D}} F(y)) \otimes_{\mathcal{D}} G(z)
    &\underoverset{\simeq}{a^{\mathcal{D}}_{F(x),F(y),F(z)}}{\longrightarrow}&
     F(x) \otimes_{\mathcal{D}}( F(y)\otimes_{\mathcal{D}} G(z) )
     \\
     {}^{\mathllap{\mu_{x,y} \otimes id}}\downarrow 
    && 
     \downarrow^{\mathrlap{id\otimes \rho_{y,z}}}
     \\
     F(x \otimes_{\mathcal{C}} y) \otimes_{\mathcal{D}} G(z)
      &&
     F(x) \otimes_{\mathcal{D}} ( G(x \otimes_{\mathcal{C}} y) )
     \\
     {}^{\mathllap{\rho_{x \otimes_{\mathcal{C}} y , z} } }\downarrow 
    && 
     \downarrow^{\mathrlap{\rho_{ x, y \otimes_{\mathcal{C}} z  }}}
     \\
     G( ( x \otimes_{\mathcal{C}} y ) \otimes_{\mathcal{C}} z  )
    &\underset{F(a^{\mathcal{C}}_{x,y,z})}{\longrightarrow}&
     G( x \otimes_{\mathcal{C}} ( y \otimes_{\mathcal{C}} z ) )
   }
   \,,
\end{displaymath}


\end{itemize}
A [[homomorphism]] $f\;\colon\; (G_1, \rho_1) \longrightarrow (G_2,\rho_2)$ between two modules over a monoidal functor $(F,\mu,\epsilon)$ is

\begin{itemize}%
\item a [[natural transformation]] $f_x \;\colon\; N_1(x) \longrightarrow N_2(x)$ of the underlying functors

\end{itemize}
compatible with the action in that the following [[commuting diagram|diagram commute]] for all objects $x,y \in \mathcal{C}$:

\begin{displaymath}
\itexarray{
    F(x) \otimes_{\mathcal{D}} N_1(y)
      &\overset{id \otimes_{\mathcal{D}} f(y)}{\longrightarrow}&
    F(x) \otimes_{\mathcal{D}} N_2(y)
    \\
    {}^{\mathllap{(\rho_1)_{x,y}}}\downarrow
     &&
    \downarrow^{\mathrlap{(\rhi_2)_{x,y}}}
    \\
    N_1(x\otimes_{\mathcal{C}} y)
     &\underset{f(x \otimes_{\mathcal{C}} y ) }{\longrightarrow}&
    N_2(x \otimes_{\mathcal{C}} y)
  }
\end{displaymath}
We write $F Mod$ for the resulting category of modules over the monoidal functor $F$.

\end{defn}
\begin{prop}
\label{DayMonoidsAreLaxMonoidalFunctorsOnTheSite}\hypertarget{DayMonoidsAreLaxMonoidalFunctorsOnTheSite}{}
Let $(\mathcal{C},\otimes I)$ be a pointed [[topologically enriched category]] ([[symmetric monoidal category]]) [[monoidal category]] (def. \ref{MonoidalCategory}). Regard $(Top_{cg}^{\ast/}, \wedge , S^0)$ as a topological [[symmetric monoidal category]] as in example \ref{PointedTopologicalSpacesWithSmashIsSymmetricMonoidalCategory}.

Then ([[commutative monoid in a symmetric monoidal category|commutative]]) [[monoid in a monoidal category|monoids in]] (def. \ref{MonoidsInMonoidalCategory}) the [[Day convolution]] monoidal category $([\mathcal{C}, Top^{\ast/}_{cg}], \otimes_{Day}, y(1_{\mathcal{C}}))$ of prop. \ref{DayConvolutionYieldsMonoidalCategoryStructure} are equivalent to ([[braided monoidal functor|braided]]) [[lax monoidal functors]] (def. \ref{LaxMonoidalFunctor}) of the form

\begin{displaymath}
(\mathcal{C},\otimes, I) \longrightarrow (Top^{\ast}_{cg}, \wedge, S^0)
  \,,
\end{displaymath}
called \textbf{functors with smash products} on $\mathcal{C}$, i.e. there are [[equivalences of categories]] of the form

\begin{displaymath}
\begin{aligned}
    Mon([\mathcal{C},Top^{\ast/}_{cg}], \otimes_{Day}, y(1_{\mathcal{C}}))
      &\underoverset{\simeq}{\phi}{\longrightarrow}
    MonFunc(\mathcal{C},Top^{\ast/}_{cg})
    \\
    CMon([\mathcal{C},Top^{\ast/}_{cg}], \otimes_{Day}, y(1_{\mathcal{C}}))
      &\simeq
    SymMonFunc(\mathcal{C},Top^{\ast/}_{cg})
  \end{aligned}
  \,.
\end{displaymath}
Furthermore, for $A \in Mon([\mathcal{C},Top^{\ast/}_{cg}], \otimes_{Day}, y(1_{\mathcal{C}}))$ a given [[monoid in a monoidal category|monoid object]], then left $A$-[[module objects]] (def. \ref{ModulesInMonoidalCategory}) are equivalent to left [[modules over monoidal functors]] (def. \ref{ModuleOverAMonoidalFunctor}):

\begin{displaymath}
A Mod
   \simeq
  \phi(A) Mod
  \,.
\end{displaymath}
\end{prop}
This is stated in some form in (\href{Day+convolution#Day70}{Day 70, example 3.2.2}). It is highlighted again in (\hyperlink{MMSS00}{MMSS 00, prop. 22.1}).

\begin{proof}
By definition \ref{LaxMonoidalFunctor}, a [[lax monoidal functor]] $F \colon \mathcal{C} \to Top^{\ast/}_{cg}$ is a topologically enriched functor equipped with a morphism of [[pointed topological spaces]] of the form

\begin{displaymath}
S^0 \longrightarrow F(1_{\mathcal{C}})
\end{displaymath}
and equipped with a [[natural transformation|natural]] system of maps of pointed topological spaces of the form

\begin{displaymath}
F(c_1) \wedge F(c_2) \longrightarrow F(c_1 \otimes_{\mathcal{C}} c_2)
\end{displaymath}
for all $c_1,c_2 \in \mathcal{C}$.

Under the [[Yoneda lemma]] (prop. \ref{YonedaReductionTopological}) the first of these is equivalently a morphism in $[\mathcal{C}, Top^{\ast/}_{cg}]$ of the form

\begin{displaymath}
y(S^0) \longrightarrow F
  \,.
\end{displaymath}
Moreover, under the [[natural isomorphism]] of corollary \ref{DayConvolutionViaNaturalIsosInvolvingExternalTensorAndTensor} the second of these is equivalently a morphism in $[\mathcal{C}, Top^{\ast/}_{cg}]$ of the form

\begin{displaymath}
F \otimes_{Day} F \longrightarrow F
  \,.
\end{displaymath}
Translating the conditions of def. \ref{LaxMonoidalFunctor} satisfied by a [[lax monoidal functor]] through these identifications gives precisely the conditions of def. \ref{MonoidsInMonoidalCategory} on a ([[commutative monoid in a symmetric monoidal category|commutative]]) [[monoid in a monoidal category|monoid in]] object $F$ under $\otimes_{Day}$.

Similarly for [[module objects]] and [[modules over monoidal functors]].

\end{proof}
\begin{prop}
\label{PullbackAlongLaxMonoidalFunctorPreservesMonoidsForDayConvolution}\hypertarget{PullbackAlongLaxMonoidalFunctorPreservesMonoidsForDayConvolution}{}
Let $f \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}$ be a [[lax monoidal functor]] (def. \ref{LaxMonoidalFunctor}) between pointed [[topologically enriched category|topologically enriched]] [[monoidal categories]] (def. \ref{MonoidalCategory}). Then the induced functor

\begin{displaymath}
f^\ast
    \;\colon\;
  [\mathcal{D}, Top^{\ast/}_{cg}]
    \longrightarrow
  [\mathcal{C}, Top_{cg}^{\ast}]
\end{displaymath}
given by $(f^\ast X)(c)\coloneqq X(f(c))$ preserves [[monoid in a monoidal category|monoids]] under [[Day convolution]]

\begin{displaymath}
f^\ast
    \;\colon\;
  Mon([\mathcal{D}, Top^{\ast/}_{cg}], \otimes_{Day}, y(1_{\mathcal{D}}))
    \longrightarrow
  Mon([\mathcal{C}, Top_{cg}^{\ast}], \otimes_{Day}, y(1_{\mathcal{C}})
\end{displaymath}
Moreover, if $\mathcal{C}$ and $\mathcal{D}$ are [[symmetric monoidal categories]] (def. \ref{SymmetricMonoidalCategory}) and $f$ is a [[braided monoidal functor]] (def. \ref{LaxMonoidalFunctor}), then $f^\ast$ also preserves [[commutative monoids in a symmetric monoidal category|commutative monoids]]

\begin{displaymath}
f^\ast
    \;\colon\;
  CMon([\mathcal{D}, Top^{\ast/}_{cg}], \otimes_{Day}, y(1_{\mathcal{D}}))
    \longrightarrow
  CMon([\mathcal{C}, Top_{cg}^{\ast}], \otimes_{Day}, y(1_{\mathcal{C}})
  \,.
\end{displaymath}
Similarly, for

\begin{displaymath}
A 
    \in 
   Mon([\mathcal{D}, Top^{\ast/}_{cg}], \otimes_{Day}, y(1_{\mathcal{D}}))
\end{displaymath}
any fixed monoid, then $f^\ast$ sends $A$-[[module object]] to $f^\ast(A)$-modules

\begin{displaymath}
f^\ast
    \;\colon\;
  A Mod(\mathcal{D})
    \longrightarrow
  (f^\ast A)Mod(\mathcal{C})
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
This is an immediate corollary of prop. \ref{DayMonoidsAreLaxMonoidalFunctorsOnTheSite}, since the composite of two (braided) lax monoidal functors is itself canonically a (braided) lax monoidal functor.

\end{proof}
\begin{prop}
\label{}\hypertarget{}{}
Let $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ be a [[topologically enriched category|topologically enriched]] [[monoidal category]] (def. \ref{MonoidalCategory}), and let $A \in Mon([\mathcal{C},Top^{\ast/}_{cg}],\otimes_{Day}, y(1_{\mathcal{C}}))$ be a [[monoid in a monoidal category|monoid in]] (def. \ref{MonoidsInMonoidalCategory}) the pointed topological [[Day convolution]] monoidal category over $\mathcal{C}$ from prop. \ref{DayConvolutionYieldsMonoidalCategoryStructure}.

Then the [[category of modules|category of left A-modules]] (def. \ref{ModulesInMonoidalCategory}) is a pointed topologically [[enriched functor category]] category (\href{Introduction+to+Stable+homotopy+theory+--+P#TopologicallyEnrichedFunctorsToTopk}{exmpl.})

\begin{displaymath}
A Mod
  \;\simeq\;
  [ A Free_{\mathcal{C}}Mod^{op}, \; Top_{cg}^{\ast/} ]
  \,,
\end{displaymath}
over the category of [[free modules]] over $A$ (def. \ref{MonoidModuleOverItself}) on objects in $\mathcal{C}$ (under the [[Yoneda embedding]] $y \colon \mathcal{C}^{op} \to [\mathcal{C}, Top^{\ast/}_{cg}]$). Hence the objects of $A Free_{\mathcal{C}}Mod$ are identified with those of $\mathcal{C}$, and its [[hom-spaces]] are

\begin{displaymath}
A Free_{\mathcal{C}}Mod( c_1, c_2)
  \;=\;
  A Mod( A \otimes_{Day} y(c_1),\; A \otimes_{Day} y(c_2) )
  \,.
\end{displaymath}
\end{prop}
(\hyperlink{MMSS00}{MMSS 00, theorem 2.2})

\begin{proof}
Use the identification from prop. \ref{DayMonoidsAreLaxMonoidalFunctorsOnTheSite} of $A$ with a [[lax monoidal functor]] and of any $A$-[[module object]] $N$ as a functor with the structure of a [[module over a monoidal functor]], given by [[natural transformations]]

\begin{displaymath}
A(c_1)\otimes N(c_2) \overset{\rho_{c_1,c_2}}{\longrightarrow} N(c_1 \otimes c_2)
  \,.
\end{displaymath}
Notice that these transformations have just the same structure as those of the [[enriched functor|enriched functoriality]] of $N$ (\href{Introduction+to+Stable+homotopy+theory+--+P#TopologicallyEnrichedFunctor}{def.}) of the form

\begin{displaymath}
\mathcal{C}(c_1,c_2) \otimes N(c_1) \overset{}{\longrightarrow} N(c_2)
  \,.
\end{displaymath}
Hence we may unify these two kinds of transformations into a single kind of the form

\begin{displaymath}
\mathcal{C}(c_3 \otimes c_1 , c_2) 
        \otimes 
      A(c_3)
      \otimes
    N(c_1)   
      \overset{id \otimes \rho_{c_3,c_1}}{\longrightarrow}
    \mathcal{C}(c_3 \otimes c_1, c_2) 
      \otimes
    N(c_3 \otimes c_2)    
      \longrightarrow
    N(c_2)
\end{displaymath}
and subject to certain identifications.

Now observe that the hom-objects of $A Free_{\mathcal{C}}Mod$ have just this structure:

\begin{displaymath}
\begin{aligned}
    A Free_{\mathcal{C}}Mod(c_2,c_1)
      & =
    A Mod( A \otimes_{Day} y(c_2) , A \otimes_{Day} y(c_1) )
    \\
      & \simeq
    [\mathcal{C},Top^{\ast/}_{cg}](y(c_2), A \otimes_{Day} y(c_1) )
    \\
      & \simeq
    (A \otimes_{Day} y(c_1) )(c_2)
    \\
      & \simeq
     \overset{c_3,c_4}{\int}
       \mathcal{C}(c_3 \otimes c_4,c_2)
         \wedge
       A(c_3) \wedge \mathcal{C}(c_1, c_4)
     \\
     & \simeq 
     \overset{c_3}{\int}
       \mathcal{C}(c_3 \otimes c_1, c_2)
       \wedge  A (c_3)
  \end{aligned}
  \,.
\end{displaymath}
Here we used first the [[free-forgetful adjunction]] of prop. \ref{MonoidModuleOverItself}, then the [[enriched Yoneda lemma]] (prop. \ref{YonedaReductionTopological}), then the [[coend]]-expression for [[Day convolution]] (def. \ref{TopologicalDayConvolutionProduct}) and finally the [[co-Yoneda lemma]] (prop. \ref{TopologicalCoYonedaLemma}).

We claim that under this identification, composition in $A Free_{\mathcal{C}}Mod$ is given by the following composite.

\begin{displaymath}
\begin{aligned}
    A Free_{\mathcal{C}}Mod(c_3, c_2)
      \wedge
    A Free_{\mathcal{C}}Mod(c_2, c_1) 
    & =
    \left(
    \overset{c_5}{\int} 
      \mathcal{C}(c_5 \otimes_{\mathcal{C}} c_2 , c_3 )
      \wedge A(c_5)
    \right)
      \wedge
    \left(
    \overset{c_4}{\int} 
       \mathcal{C}(c_4 \otimes_{\mathcal{C}} c_1, c_2)     
       \wedge
       A(c_4)
    \right)
    \\
    & \simeq
    \overset{c_4, c_5}{\int} 
      \mathcal{C}(c_5 \otimes_{\mathcal{C}} c_2 , c_3)
        \wedge
      \mathcal{C}(c_4 \otimes_{\mathcal{C}} c_1, c_2 )
        \wedge
      A(c_5) \wedge A(c_4)    
    \\
    & \longrightarrow
      \overset{c_4,c_5}{\int} 
      \mathcal{C}(c_5 \otimes_{\mathcal{C}} c_2 , c_3)
        \wedge
      \mathcal{C}(c_5 \otimes_{\mathcal{C}} c_4 \otimes_{\mathcal{C}} c_1, c_5 \otimes_{\mathcal{C}} c_2 )
        \wedge
       A(c_5  \otimes_{\mathcal{C}}  c_4   )     
     \\
     & \longrightarrow
      \overset{c_4, c_5}{\int} 
      \mathcal{C}(c_5 \otimes_{\mathcal{C}} c_4 \otimes_{\mathcal{C}} c_1, c_5 \otimes_{\mathcal{C}} c_2 )
        \wedge
       A(c_5  \otimes_{\mathcal{C}} c_4 )     
     \\
     & \longrightarrow
      \overset{c_4}{\int} 
      \mathcal{C}(c_4 \otimes_{\mathcal{C}} c_1  , c_3)
        \otimes_V
      A(c_4 )
  \end{aligned}
  \,,
\end{displaymath}
where

\begin{enumerate}%
\item the equivalence is [[braiding]] in the integrand (and the [[Fubini theorem]], prop. \ref{CoendsCommuteWithEachOther});


\item the first morphism is, in the integrand, the smash product of

\begin{enumerate}%
\item forming the tensor product of hom-objects of $\mathcal{C}$ with the identity morphism on $c_5$;


\item the monoidal functor incarnation $A(c_5) \wedge A(c_4)\longrightarrow A(c_5 \otimes_{\mathcal{C}} c_4 )$ of the monoid structure on $A$;



\end{enumerate}

\item the second morphism is, in the integrand, given by composition in $\mathcal{C}$;


\item the last morphism is the morphism induced on [[coends]] by regarding [[extranatural transformation|extranaturality]] in $c_4$ and $c_5$ separately as a special case of extranaturality in $c_6 \coloneqq c_4 \otimes c_5$ (and then renaming).



\end{enumerate}
It is fairly straightforward to see that, under the above identifications, functoriality under this composition is equivalently functoriality in $\mathcal{C}$ together with the action property over $A$.

\end{proof}
\hypertarget{OnPreExcisiveFunctors}{}\paragraph*{{Pre-Excisive functors}}\label{OnPreExcisiveFunctors}

\begin{defn}
\label{FinitePointedCWComplexes}\hypertarget{FinitePointedCWComplexes}{}
Write

\begin{displaymath}
\iota_{fin}\;\colon\; Top^{\ast/}_{cg,fin} \hookrightarrow Top^{\ast/}_{cg}
\end{displaymath}
for the [[full subcategory]] of [[pointed topological spaces|pointed]] [[compactly generated topological spaces]] (\href{Introduction+to+Stable+homotopy+theory+--+P#Top}{def.}) on those that admit the structure of a [[finite CW-complex]] (a [[CW-complex]] (\href{Introduction+to+Stable+homotopy+theory+--+P#TopologicalCellComplex}{def.}) with a [[finite number]] of cells).

We say that the pointed topological [[enriched functor category]] (def. \ref{PointedTopologicalFunctorCategory})

\begin{displaymath}
Exc(Top_{cg})
   \coloneqq 
  [Top^{\ast/}_{cg,fin}, Top^{\ast/}_{cg}]
\end{displaymath}
is the category of \textbf{[[pre-excisive functors]]}.

Write

\begin{displaymath}
\mathbb{S}_{exc}
    \coloneqq
  y(S^0)
  \coloneqq
  Top^{\ast/}_{cg,fin}(S^0,-)
\end{displaymath}
for the [[representable functor|functor co-represented]] by [[0-sphere]]. This is equivalently the inclusion $\iota_{fin}$ itself:

\begin{displaymath}
\mathbb{S}_{exc} = \iota_{fin} 
    \;\colon\;
  K \mapsto K
  \,.
\end{displaymath}
We call this the standard incarnation of the \textbf{[[sphere spectrum]]} as a pre-excisive functor.

By prop. \ref{DayConvolutionYieldsMonoidalCategoryStructure} the [[smash product]] of [[pointed topological spaces|pointed]] [[compactly generated topological spaces]] induces the structure of a [[closed monoidal category|closed]] (def. \ref{ClosedMonoidalCategory}) [[symmetric monoidal category]] (def. \ref{SymmetricMonoidalCategory})

\begin{displaymath}
\left( 
    Exc(Top_{cg})
    ,\;
    \wedge_{Day}
    ,\; 
   \mathbb{S}_{exc}
  \right)
\end{displaymath}
with

\begin{enumerate}%
\item [[tensor unit]] the [[sphere spectrum]] $\mathbb{S}_{exc}$;


\item [[tensor product]] the [[Day convolution product]] $\otimes_{Day}$ from def. \ref{TopologicalDayConvolutionProduct},

called the \textbf{[[symmetric monoidal smash product of spectra]]} for the model of pre-excisive functors;


\item [[internal hom]] the dual operation $[-,-]_{Day}$ from prop. \ref{DayMonoidalStructureIsClosed},

called the \textbf{[[mapping spectrum]]} construction for pre-excisive functors.



\end{enumerate}
\end{defn}
\begin{remark}
\label{EveryPreExcisiveFunctorIsSModule}\hypertarget{EveryPreExcisiveFunctorIsSModule}{}
By example \ref{MonoidGivenByTensorUnit} the [[sphere spectrum]] incarnated as a pre-excisive functor $\mathbb{S}_{exc}$ (according to def. \ref{FinitePointedCWComplexes}) is canonically a [[commutative monoid in a symmetric monoidal category|commutative monoid in]] the category of pre-excisive functors (def. \ref{MonoidsInMonoidalCategory})

Moreover, by example \ref{EveryObjectIsModuleOverTensorUnit}, every object of $Exc(Top_{cg})$ (def. \ref{FinitePointedCWComplexes}) is canonically a [[module object]] over $\mathbb{S}_{exc}$. We may therefore tautologically identify the category of pre-excisive functors with the [[module category]] over the sphere spectrum:

\begin{displaymath}
Exc(Top_{cg})
    \simeq
  \mathbb{S}_{exc}Mod
  \,.
\end{displaymath}
\end{remark}
We now consider restricting the domain of the pre-excisive functors of def. \ref{FinitePointedCWComplexes}.

\begin{defn}
\label{TopologicalDiagramCategoriesForSpectra}\hypertarget{TopologicalDiagramCategoriesForSpectra}{}
Define the following [[pointed topologically enriched categories|pointed topologically enriched]] (\href{Introduction+to+Stable+homotopy+theory+--+P#TopEnrichedCategory}{def.}) [[symmetric monoidal categories]] (def. \ref{SymmetricMonoidalCategory}):

\begin{enumerate}%
\item $Seq$ is the category whose objects are the [[natural numbers]] and which has only identity morphisms and [[zero morphisms]] on these objects, hence the [[hom-spaces]] are

\begin{displaymath}
Seq(n_1,n_2) = 
  \left\{
    \itexarray{
    S^0 & for\; n_1 = n_2
    \\
    \ast & otherwise
    }
 \right.
\end{displaymath}
The tensor product is the addition of natural numbers, $\otimes = +$, and the [[tensor unit]] is 0. The [[braiding]] is, necessarily, the identity.


\item $Sym$ is the standard [[skeletal category|skeleton]] of the [[core]] of [[FinSet]] with [[zero morphisms]] adjoined: its [[objects]] are the [[finite sets]] $\overline{n} \coloneqq \{1, \cdots,n\}$ for $n \in \mathbb{N}$, all non-[[zero morphism|zero]] morphisms are [[automorphisms]] and the [[automorphism group]] of $\{1,\cdots,n\}$ is the [[symmetric group]] $\Sigma_n$, hence the [[hom-spaces]] are the following [[discrete topological spaces]]:

\begin{displaymath}
Sym(n_1, n_2) =
  \left\{
    \itexarray{
    (\Sigma_{n_1})_+ & for \; n_1 = n_2
    \\
    \ast & otherwise
    }
  \right.
\end{displaymath}
The [[tensor product]] is the [[disjoint union]] of sets, tensor unit is the [[empty set]]. The [[braiding]]

\begin{displaymath}
\tau_{n_1 , n_2}
  \;\colon\;
  \overline{n_1} \cup \overline{n_2}
    \overset{}{\longrightarrow}
  \overline{n_2} \cup \overline{n_1}
\end{displaymath}
is given by the canonical [[permutation]] in $\Sigma_{n_1+n_2}$ that [[shuffle|shuffles]] the first $n_1$ elements past the remaining $n_2$ elements.


\item $Orth$ has as objects finite dimenional real linear [[inner product spaces]] $(V, \langle -,-\rangle)$ and as non-zero morphisms the [[linear map|linear]] [[isometry|isometric]] [[isomorphisms]] between these; hence the [[automorphism group]] of the object $(V, \langle -,-\rangle)$ is the [[orthogonal group]] $O(V)$; the monoidal product is [[direct sum]] of linear spaces, the tensor unit is the 0-vector space; again we turn this into a $Top^{\ast/}$-enriched category by adjoining a basepoint to the hom-spaces;

\begin{displaymath}
Orth(V_1,V_2) 
  \simeq
  \left\{
     \itexarray{
    O(V_1)_+ & for \; dim(V_1) = dim(V_2)
    \\
    \ast & otherwise
     }
  \right.
\end{displaymath}
The [[tensor product]] is the [[direct sum]] of linear inner product spaces, tensor unit is the 0-vector space. The [[braiding]] is that of $Sym$, under the canonical embedding $\Sigma_{n_1+n_2} \hookrightarrow O(n_1+n_2)$ of the [[symmetric group]] into the [[orthogonal group]].



\end{enumerate}
There is a sequence of canonical [[faithful functor|faithful]] pointed topological [[subcategory]] inclusions

\begin{displaymath}
\itexarray{
   Seq 
     &\stackrel{seq}{\hookrightarrow}& 
   Sym 
    &\stackrel{sym}{\hookrightarrow}& 
   Orth 
     &\stackrel{orth}{\hookrightarrow}& 
   Top_{cg,fin}^{\ast/}
   \\
   n 
    &\mapsto& 
   \{1,\cdots, n\} 
     &\mapsto& 
   \mathbb{R}^n 
    &\mapsto& 
   S^n
   \\
    && 
    && 
   V
    &\mapsto& 
   S^V
  }
  \,,
\end{displaymath}
into the pointed topological category of pointed compactly generated topological spaces of finite CW-type (def. \ref{FinitePointedCWComplexes}).

Here $S^V$ denotes the [[one-point compactification]] of $V$. On morphisms $sym \colon (\Sigma_n)_+ \hookrightarrow (O(n))_+$ is the canonical inclusion of [[permutation]] matrices into [[orthogonal group|orthogonal]] matrices and $orth \colon O(V)_+ \hookrightarrow Aut(S^V)$ is on $O(V)$ the [[topological subspace]] inclusions of the pointed [[homeomorphisms]] $S^V \to S^V$ that are induced under forming [[one-point compactification]] from linear isometries of $V$ (``[[representation spheres]]'').

Consider the sequence of restrictions of topological diagram categories, according to prop. \ref{PullbackAlongLaxMonoidalFunctorPreservesMonoidsForDayConvolution} along the above inclusions:

\begin{displaymath}
Exc(Top_{cg})
    \overset{orth^\ast}{\longrightarrow}
  [Orth,Top^{\ast/}_{cg}]
    \overset{sym^\ast}{\longrightarrow}
  [Sym,Top^{\ast/}_{cg}]
    \overset{seq^\ast}{\longrightarrow}
  [Seq,Top^{\ast/}_{cg}]
  \,.
\end{displaymath}
Write

\begin{displaymath}
\mathbb{S}_{orth} \coloneqq orth^\ast \mathbb{S}_{exc} 
  \,,
  \;\;\;\;\;\;\;\;
  \mathbb{S}_{sym} \coloneqq sym^\ast \mathbb{S}_{orth} 
  \,,
  \;\;\;\;\;\;\;\;
  \mathbb{S}_{seq} \coloneqq seq^\ast \mathbb{S}_{sym}
\end{displaymath}
for the restriction of the excisive functor incarnation of the [[sphere spectrum]] (from def. \ref{FinitePointedCWComplexes}) along these inclusions.

\end{defn}
\begin{remark}
\label{RestrictionsOfExcisiveSphere}\hypertarget{RestrictionsOfExcisiveSphere}{}
Since $\mathbb{S}_{exc}$ is the [[tensor unit]] with repect to the [[Day convolution]] product on pre-excisive functors, and since it is therefore canonically a [[commutative monoid]], by example \ref{MonoidGivenByTensorUnit}, prop. \ref{PullbackAlongLaxMonoidalFunctorPreservesMonoidsForDayConvolution} says that all these restricted sphere spectra are still [[monoid object|monoids]], and that under restriction every [[pre-excisive functor]], regarded as a $\mathbb{S}_{exc}$-[[module object|module]] via remark \ref{EveryPreExcisiveFunctorIsSModule}, canonically becomes a [[module object|module]] under the restricted sphere spectrum:

\begin{displaymath}
\begin{aligned}
    orth^\ast
    & \colon\;
  Exc(Top_{cg})
  \simeq
  \mathbb{S}_{exc} Mod
   \longrightarrow
  \mathbb{S}_{orth} Mod
  \\
  sym^\ast
   &\colon\;
  Exc(Top_{cg})
  \simeq
  \mathbb{S}_{exc} Mod
   \longrightarrow
  \mathbb{S}_{sym} Mod
  \\
  seq^\ast
   &\colon\;
  Exc(Top_{cg})
  \simeq
  \mathbb{S}_{exc} Mod
   \longrightarrow
  \mathbb{S}_{seq} Mod
  \end{aligned}
  \,.
\end{displaymath}
However, while $orth$ and $sym$ are [[braided monoidal functors]], the functor $seq$ is not braided, hence $\mathbb{S}_{orth}$ and $\mathbb{S}_{sym}$ are commutative monoids, but $\mathbb{S}_{Seq}$ is not commutative. Hence prop. \ref{PullbackAlongLaxMonoidalFunctorPreservesMonoidsForDayConvolution} gives the following situation

\begin{tabular}{l|l|l|l|l}
[[sphere spectrum]]&$\mathbb{S}_{exc}$&$\mathbb{S}_{orth}$&$\mathbb{S}_{sym}$&$\mathbb{S}_{seq}$\\
\hline 
[[monoid in a monoidal category&monoid]]&yes&yes&yes\\
[[commutative monoid in a symmetric monoidal category&commutative monoid]]&yes&yes&yes\\
[[tensor unit]]&yes&no&no&no\\
\end{tabular}

\end{remark}
\begin{prop}
\label{SseqModulesAreSequentialSpectra}\hypertarget{SseqModulesAreSequentialSpectra}{}
There is an [[equivalence of categories]]

\begin{displaymath}
(-)^{seq}
  \;\colon\;
  \mathbb{S}_{seq} Mod
    \overset{}{\longrightarrow}
  SeqSpec(Top_{cg})
\end{displaymath}
which identifies the [[category of modules]] (def. \ref{ModulesInMonoidalCategory}) over the [[monoid object|monoid]] $\mathbb{S}_{seq}$ (remark \ref{RestrictionsOfExcisiveSphere}) in the [[Day convolution]] monoidal structure (prop. \ref{DayConvolutionYieldsMonoidalCategoryStructure}) over the topological functor category $[Seq,Top^{\ast/}_{cg}]$ from def. \ref{TopologicalDiagramCategoriesForSpectra} with the category of [[sequential spectra]] (\href{Introduction+to+Stable+homotopy+theory+--+1#SequentialSpectra}{def.})

Under this equivalence, an $\mathbb{S}_{seq}$-module $X$ is taken to the sequential pre-spectrum $X^{seq}$ whose component spaces are the values of the [[pre-excisive functor]] $X$ on the standard [[n-sphere]] $S^n = (S^1)^{\wedge n}$

\begin{displaymath}
(X^{seq})_n \coloneqq X(seq(n)) = X(S^n)
\end{displaymath}
and whose structure maps are the images of the action morphisms

\begin{displaymath}
\mathbb{S}_{seq} \otimes_{Day} X
    \longrightarrow
  X
\end{displaymath}
under the isomorphism of corollary \ref{DayConvolutionViaNaturalIsosInvolvingExternalTensorAndTensor}

\begin{displaymath}
\mathbb{S}_{seq}(n_1) \wedge X(n_1) \longrightarrow X_{n_1 + n_2}
\end{displaymath}
evaluated at $n_1 = 1$

\begin{displaymath}
\itexarray{
    \mathbb{S}_{seq}(1) \wedge X(n) 
      &\longrightarrow& 
    X_{n+1}
    \\
    {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\simeq}}
    \\
    S^1 \wedge X_n &\longrightarrow& X_{n+1}
  }
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
After unwinding the definitions, the only point to observe is that due to the action property,

\begin{displaymath}
\itexarray{
    \mathbb{S}_{seq} \otimes_{Day} \mathbb{S}_{seq} \otimes_{Day} X
      &\overset{id \otimes_{Day} \rho}{\longrightarrow}&
    \mathbb{S}_{seq} \otimes_{Day} X
    \\
    {}^{\mathllap{\mu \otimes_{Day} id } }\downarrow 
      && 
    \downarrow^{\mathrlap{\rho}}
    \\
    \mathbb{S}_{seq} \otimes_{Day} X
      &\underset{\rho}{\longrightarrow}&
    X
  }
\end{displaymath}
any $\mathbb{S}_{seq}$-action

\begin{displaymath}
\rho
  \;\colon\;
  \mathbb{S}_{seq} \otimes_{Day} X \longrightarrow X
\end{displaymath}
is indeed uniquely fixed by the components of the form

\begin{displaymath}
\mathbb{S}_{seq}(1) \wedge X(n) \longrightarrow X(n)
  \,.
\end{displaymath}
This is because under corollary \ref{DayConvolutionViaNaturalIsosInvolvingExternalTensorAndTensor} the action property is identified with the componentwise property

\begin{displaymath}
\itexarray{
    S^{n_1} \wedge S^{n_2} \wedge X_{n_3}
      &\overset{id \wedge \rho_{n_2,n_3}}{\longrightarrow}&
    S^{n_1} \wedge X_{n_2 + n_3}
    \\
    {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\rho_{n_1,n_2+n_3}}}
    \\
    S^{n_1 + n_2} \wedge X_{n_3}
     &\underset{\rho_{n_1+n_2,n_3}}{\longrightarrow}&
    X_{n_1 + n_2 + n_3}
  }
  \,,
\end{displaymath}
where the left vertical morphism is an isomorphism by the nature of $\mathbb{S}_{seq}$. Hence this fixes the components $\rho_{n',n}$ to be the $n'$-fold composition of the structure maps $\sigma_n \coloneqq \rho(1,n)$.

\end{proof}
However, since, by remark \ref{SseqModulesAreSequentialSpectra}, $\mathbb{S}_{seq}$ is not commutative, there is no tensor product induced on $SeqSpec(Top_{cg})$ under the identification in prop. \ref{SseqModulesAreSequentialSpectra}. But since $\mathbb{S}_{orth}$ and $\mathbb{S}_{sym}$ are commutative monoids by remark \ref{SseqModulesAreSequentialSpectra}, it makes sense to consider the following definition.

\begin{defn}
\label{SsymModuleSymmetricSpectra}\hypertarget{SsymModuleSymmetricSpectra}{}
In the terminology of remark \ref{RestrictionsOfExcisiveSphere} we say that

\begin{displaymath}
OrthSpec(Top_{cg})
   \coloneqq
  \mathbb{S}_{orth} Mod
\end{displaymath}
is the \textbf{[[category]] of [[orthogonal spectra]]}; and that

\begin{displaymath}
SymSpec(Top_{cg})
   \coloneqq
  \mathbb{S}_{sym} Mod
\end{displaymath}
is the \textbf{[[category]] of [[symmetric spectra]]}.

By remark \ref{RestrictionsOfExcisiveSphere} and by prop. \ref{MonoidalCategoryOfModules} these categories canonically carry a [[symmetric monoidal category|symmetric monoidal]] [[tensor product]] $\otimes_{\mathbb{S}_{orth}}$ and $\otimes_{\mathbb{S}_{seq}}$, respectively. This we call the \textbf{[[symmetric monoidal smash product of spectra]]}. We usually just write for short

\begin{displaymath}
\wedge 
    \coloneqq 
  \otimes_{\mathbb{S}_{orth}}   
  \;\colon\;
  OrthSpec(Top_{cg}) \times OrthSpec(Top_{cg})
   \longrightarrow
  OrthSpec(Top_{cg})
\end{displaymath}
and

\begin{displaymath}
\wedge 
    \coloneqq 
  \otimes_{\mathbb{S}_{sym}}   
  \;\colon\;
  SymSpec(Top_{cg}) \times SymSpec(Top_{cg})
   \longrightarrow
  SymSpec(Top_{cg})
\end{displaymath}
\end{defn}
In the next section we work out what these symmetric monoidal categories of orthogonal and of symmetric spectra look like more explicitly.

\hypertarget{ForSymmetricAndOrthogonalSpectra}{}\subsubsection*{{For symmetric and orthogonal spectra}}\label{ForSymmetricAndOrthogonalSpectra}

We now define [[symmetric spectra]] and [[orthogonal spectra]] and their symmetric monoidal smash product. We proceed by giving the explicit definitions and then checking that these are equivalent to the abstract definition \ref{SsymModuleSymmetricSpectra} from above.

\textbf{Literature.} ( \hyperlink{HoveyShipleySmith00}{Hovey-Shipley-Smith 00, section 1, section 2}, \hyperlink{Schwede12}{Schwede 12, chapter I})

$\,$

\begin{defn}
\label{SymmetricSpectrum}\hypertarget{SymmetricSpectrum}{}
A topological \textbf{[[symmetric spectrum]]} $X$ is

\begin{enumerate}%
\item a sequence $\{X_n  \in Top_{cg}^{\ast/}\;\vert\; n \in \mathbb{N}\}$ of [[pointed topological space|pointed]] [[compactly generated topological spaces]];


\item a basepoint preserving continuous right [[action]] of the [[symmetric group]] $\Sigma(n)$ on $X_n$;


\item a sequence of morphisms $\sigma_n \colon S^1 \wedge X_n  \longrightarrow X_{n+1}$



\end{enumerate}
such that

\begin{itemize}%
\item for all $n, k \in \mathbb{N}$ the [[composition|composite]]

\begin{displaymath}
S^{k} \wedge X_n 
   \simeq
  S^{k-1} \wedge S^1 \wedge X_n
    \stackrel{id \wedge \sigma_n }{\longrightarrow} 
  S^{k-1} \wedge X_{n+1} 
    \simeq
  S^{k-2}\wedge S^1 \wedge X_{n+2}
    \stackrel{id \wedge \sigma_{n+1}}{\longrightarrow} 
  \cdots 
  \stackrel{\sigma_{n+k-1}}{\longrightarrow} X_{n+k}
\end{displaymath}
[[intertwiner|intertwines]] the $\Sigma(n) \times \Sigma(k)$-[[action]].



\end{itemize}
A [[homomorphism]] of symmetric spectra $f\colon X \longrightarrow Y$ is

\begin{itemize}%
\item a sequence of maps $f_n \colon X_n \longrightarrow Y_n$

\end{itemize}
such that

\begin{enumerate}%
\item each $f_n$ [[intertwiner|intetwines]] the $\Sigma(n)$-[[action]];


\item the following [[commuting diagram|diagrams commute]]

\begin{displaymath}
\itexarray{
     S^1 \wedge X_n  
    &\stackrel{f_n \wedge id}{\longrightarrow}& 
     S^1 \wedge Y_n 
     \\
     \downarrow^{\mathrlap{\sigma^X_n}} && \downarrow^{\mathrlap{\sigma^Y_n}}
     \\
     X_{n+1} &\stackrel{f_{n+1}}{\longrightarrow}& Y_{n+1}
  }
  \,.
\end{displaymath}


\end{enumerate}
We write $SymSpec(Top_{cg})$ for the resulting [[category]] of symmetric spectra.

\end{defn}
(\hyperlink{HoveyShipleySmith00}{Hovey-Shipley-Smith 00, def. 1.2.2}, \hyperlink{Schwede12}{Schwede 12, def. 1.1})

The definition of orthogonal spectra has the same structure, just with the [[symmetric groups]] replaced by the [[orthogonal groups]].

\begin{defn}
\label{OrthogonalSpectrum}\hypertarget{OrthogonalSpectrum}{}
A topological \textbf{[[orthogonal spectrum]]} $X$ is

\begin{enumerate}%
\item a sequence $\{X_n  \in Top_{cg}^{\ast/}\;\vert\; n \in \mathbb{N}\}$ of [[pointed topological space|pointed]] [[compactly generated topological spaces]];


\item a basepoint preserving continuous right [[action]] of the [[orthogonal group]] $O(n)$ on $X_n$;


\item a sequence of morphisms $\sigma_n \colon S^1 \wedge X_n  \longrightarrow X_{n+1}$



\end{enumerate}
such that

\begin{itemize}%
\item for all $n, k \in \mathbb{N}$ the [[composition|composite]]

\begin{displaymath}
S^{k} \wedge X_n 
   \simeq
  S^{k-1} \wedge S^1 \wedge X_n
    \stackrel{id \wedge \sigma_n }{\longrightarrow} 
  S^{k-1} \wedge X_{n+1} 
    \simeq
  S^{k-2}\wedge S^1 \wedge X_{n+2}
    \stackrel{id \wedge \sigma_{n+1}}{\longrightarrow} 
  \cdots 
  \stackrel{\sigma_{n+k-1}}{\longrightarrow} X_{n+k}
\end{displaymath}
[[intertwiner|intertwines]] the $O(n) \times Ok()$-[[action]].



\end{itemize}
A [[homomorphism]] of orthogonal spectra $f\colon X \longrightarrow Y$ is

\begin{itemize}%
\item a sequence of maps $f_n \colon X_n \longrightarrow Y_n$

\end{itemize}
such that

\begin{enumerate}%
\item each $f_n$ [[intertwiner|intetwines]] the $O(n)$-[[action]];


\item the following [[commuting diagram|diagrams commute]]

\begin{displaymath}
\itexarray{
     S^1 \wedge X_n  
    &\stackrel{f_n \wedge id}{\longrightarrow}& 
     S^1 \wedge Y_n 
     \\
     \downarrow^{\mathrlap{\sigma^X_n}} && \downarrow^{\mathrlap{\sigma^Y_n}}
     \\
     X_{n+1} &\stackrel{f_{n+1}}{\longrightarrow}& Y_{n+1}
  }
  \,.
\end{displaymath}


\end{enumerate}
We write $OrthSpec(Top_{cg})$ for the resulting [[category]] of orthogonal spectra.

\end{defn}
\begin{prop}
\label{DiagramSpectraGiveSymmetricAndOrthogonalSpectra}\hypertarget{DiagramSpectraGiveSymmetricAndOrthogonalSpectra}{}
Definitions \ref{SymmetricSpectrum} and \ref{OrthogonalSpectrum} are indeed equivalent to def. \ref{SsymModuleSymmetricSpectra}:

orthogonal spectra are euqivalently the [[module objects]] over the incarnation $\mathbb{S}_{orth}$ of the sphere spectrum

\begin{displaymath}
OrthSpec(Top_{cg})
   \simeq
  \mathbb{S}_{orth} Mod
\end{displaymath}
and symmetric spectra sre equivalently the module objects over the incarnation $\mathbb{S}_{sym}$ of the sphere spectrum

\begin{displaymath}
SymSpec(Top_{cg})
   \simeq
  \mathbb{S}_{sym} Mod
  \,.
\end{displaymath}
\end{prop}
(\hyperlink{HoveyShipleySmith00}{Hovey-Shipley-Smith 00, prop. 2.2.1})

\begin{proof}
We discuss this for symmetric spectra. The proof for orthogonal spectra is of the same form.

First of all, (by \href{Introduction+to+Stable+homotopy+theory+--+1-2#CoendGivesQuotientByDiagonalGroupAction}{this example}) an object in $[Sym, Top^{\ast/}_{cg}]$ is equivalently a ``symmetric sequence'', namely a sequence of pointed topological spaces $X_k$, for $k \in \mathbb{N}$, equipped with an [[action]] of $\Sigma(k)$ (def. \ref{TopologicalDiagramCategoriesForSpectra}).

By corollary \ref{DayConvolutionViaNaturalIsosInvolvingExternalTensorAndTensor} and \href{Introduction+to+Stable+homotopy+theory+--+1-2#FSPStructuredSphereSpectra}{this lemma}, the structure morphism of an $\mathbb{S}_{sym}$-[[module object]] on $X$

\begin{displaymath}
\mathbb{S}_{sym} \otimes_{Day} X \longrightarrow X
\end{displaymath}
is equivalently (as a [[functor with smash products]]) a natural transformation

\begin{displaymath}
S^{n_1} \wedge X_{n_2} \longrightarrow X_{n_1 + n_2}
\end{displaymath}
over $Sym \times Sym$. This means equivalently that there is such a morphism for all $n_1, n_2 \in \mathbb{N}$ and that it is $\Sigma(n_1) \times \Sigma(n_2)$-equivariant.

Hence it only remains to see that these natural transformations are uniquely fixed once the one for $n_1 = 1$ is given. To that end, observe that \href{Introduction+to+Stable+homotopy+theory+--+1-2#FSPStructuredSphereSpectra}{this lemma} says that in the following [[commuting squares]] (exhibiting the action property on the level of functors with smash product, where we are notationally suppressing the [[associators]]) the left vertical morphisms are [[isomorphisms]]: a

\begin{displaymath}
\itexarray{
    S^{n_1}\wedge S^{n_2} \wedge X_{n_3} 
      &\longrightarrow&
    S^{n_1} \wedge X_{n_2 + n_3}
    \\
    {}^{\mathllap{\simeq}}\downarrow
      &&
    \downarrow
    \\
    S^{n_1+ n_2} \wedge X_{n_3}
      &\longrightarrow&
    X_{n_1 + n_2 + n_3}
  }
  \,.
\end{displaymath}
This says exactly that the action of $S^{n_1 + n_2}$ has to be the composite of the actions of $S^{n_2}$ followed by that of $S^{n_1}$. Hence the statement follows by [[induction]].

Finally, the definition of [[homomorphisms]] on both sides of the equivalence are just so as to preserve precisely this structure, hence they conincide under this identification.

\end{proof}
\begin{defn}
\label{SmashProductOfSymmetricSpectra}\hypertarget{SmashProductOfSymmetricSpectra}{}
Given $X,Y \in SymSpec(Top_{cg})$ two [[symmetric spectra]], def. \ref{SymmetricSpectrum}, then their \textbf{[[smash product of spectra]]} is the symmetric spectrum

\begin{displaymath}
X \wedge Y 
   \; \in SymSpec(Top_{cg})
\end{displaymath}
with component spaces the [[coequalizer]]

\begin{displaymath}
\underset{p+1+q = n}{\bigvee}
  \Sigma(p+1+q)_+ 
    \underset{\Sigma_p \times \Sigma_1 \times \Sigma_q}{\wedge}
  X_p \wedge S^1 \wedge Y_q
  \underoverset
   {\underset{r}{\longrightarrow}}
   {\overset{\ell}{\longrightarrow}}
   {\phantom{AAAA}}
  \underset{p+q=n}{\bigvee}
   \Sigma(p+q)_+
   \underset{\Sigma_p \times \Sigma_q}{\wedge}
   X_p \wedge Y_q
  \overset{coeq}{\longrightarrow}
  (X \wedge Y)(n)
\end{displaymath}
where $\ell$ has components given by the structure maps

\begin{displaymath}
X_p \wedge S^1 \wedge Y_q
    \overset{id \wedge \sigma_{q}}{\longrightarrow}
  X_p \wedge Y_q
\end{displaymath}
while $r$ has components given by the structure maps conjugated by the [[braiding]] in $Top^{\ast/}_{cg}$ and the [[permutation]] [[action]] $\chi_{p,1}$ (that [[shuffle|shuffles]] the element on the right to the left)

\begin{displaymath}
X_p \wedge S^1 \wedge X_q
    \overset{\tau^{Top^{\ast/}_{cg}}_{X_p,S^1} \wedge id}{\longrightarrow}
  S^1 \wedge X_p \wedge X_q
    \overset{\sigma_p\wedge id}{\longrightarrow}
  X_{p+1} \wedge X_q
    \overset{\chi_{p,1} \wedge id}{\longrightarrow}
  X_{1+p} \wedge X_q
  \,.
\end{displaymath}
The structure maps of $X \wedge Y$ are those induced under the coequalizer by

\begin{displaymath}
X_p \wedge Y_q \wedge S^1
    \overset{id \wedge \sigma_{p}}{\longrightarrow}
  X_{p} \wedge Y_{q+1}
  \,.
\end{displaymath}
Analogously for orthogonal spectra.

\end{defn}
(\hyperlink{Schwede12}{Schwede 12, p. 82})

\begin{prop}
\label{AbstractFormulaGivesSmashProductOfSymmetricSpectra}\hypertarget{AbstractFormulaGivesSmashProductOfSymmetricSpectra}{}
Under the identification of prop. \ref{DiagramSpectraGiveSymmetricAndOrthogonalSpectra}, the explicit [[smash product of spectra]] in def. \ref{SmashProductOfSymmetricSpectra} is equivalent to the abstractly defined tensor product in def. \ref{SsymModuleSymmetricSpectra}:

in the case of [[symmetric spectra]]:

\begin{displaymath}
\wedge \simeq \otimes_{\mathbb{S}_{sym}}
\end{displaymath}
in the case of [[orthogonal spectra]]:

\begin{displaymath}
\wedge \simeq \otimes_{\mathbb{S}_{orth}}
  \,.
\end{displaymath}
\end{prop}
(\hyperlink{Schwede12}{Schwede 12, E.1.16})

\begin{proof}
By def. \ref{TensorProductOfModulesOverCommutativeMonoidObject} the abstractly defined tensor product of two $\mathbb{S}_{sym}$-modules $X$ and $Y$ is the [[coequalizer]]

\begin{displaymath}
X \otimes_{Day} \mathbb{S}_{sym} \otimes_{Day} Y
  \underoverset
    {\underset{\rho_{1}\circ (\tau^{Day}_{X, \mathbb{S}_{sym}} \otimes id)}{\longrightarrow}}
    {\overset{X \otimes \rho_2}{\longrightarrow}}
    {\phantom{AAAA}}
  X \otimes Y
    \overset{coeq}{\longrightarrow}
  X \otimes_{\mathbb{S}_{sym}} Y
  \,.
\end{displaymath}
The [[Day convolution]] product appearing here is over the category $Sym$ from def. \ref{TopologicalDiagramCategoriesForSpectra}. By \href{Introduction+to+Stable+homotopy+theory+--+1-2#CoendGivesQuotientByDiagonalGroupAction}{this example} and unwinding the definitions, this is for any two symmetric spectra $A$ and $B$ given degreewise by the [[wedge sum]] of component spaces summing to that total degree, smashed with the symmetric group with basepoint adjoined and then quotiented by the diagonal action of the symmetric group acting on the degrees separately:

\begin{displaymath}
\begin{aligned}
    (A \otimes_{Day} B)(n)
    & =
    \overset{n_1,n_2}{\int}
     \underset{
      = \left\{
          \itexarray{
             \Sigma(n_1 + n_2,n)_+ & if \; n_1+n_2 = n 
             \\
             \ast & otherwise
           }
      \right.
     }{
       \underbrace{\Sigma(n_1 + n_2, n)}
     }_+
      \wedge
     A_{n_1}
      \wedge
     B_{n_1}
    \\
    & \simeq
    \underset{n_1 + n_2 = n}{\bigvee}
     \Sigma(n_1+n_2)_+
     \underset{O(n_1) \times O(n_2) }{\wedge}     
     \left(
       A_{n_1}
         \wedge
       B_{n_2}
     \right)
  \end{aligned}
  \,.
\end{displaymath}
This establishes the form of the coequalizer diagram. It remains to see that under this identification the two abstractly defined morphisms are the ones given in def. \ref{SmashProductOfSymmetricSpectra}.

To see this, we apply the adjunction isomorphism between the [[Day convolution product]] and the [[external tensor product]] (cor. \ref{DayConvolutionViaNaturalIsosInvolvingExternalTensorAndTensor}) twice, to find the following sequence of equivalent incarnations of morphisms:

\begin{displaymath}
\itexarray{
  \arrayopts{\rowlines{solid}}
    (X \otimes_{Day} ( \mathbb{S}_{orth} \otimes_{Day} Y ))(n)
      &\longrightarrow&
    (X \otimes_{Day} Y)(n)
      &\longrightarrow&
    Z_n
    \\
    X_{n_1} \wedge (\mathbb{S}_{sym} \otimes_{Day} Y)(n'_2)
      &\longrightarrow&
    X_{n_1}\wedge Y(n'_2) 
      &\longrightarrow&
    Z_{n_1 + n'_2}
    \\
    (\mathbb{S}_{sym} \otimes_{Day} Y)(n'_2) 
      &\longrightarrow&
    Y(n'_2)
      &\longrightarrow&
    Maps(X_{n_1}, Z_{n_1 + n'_2})
    \\
    S^{n_2} \wedge Y_{n_3}
      &\longrightarrow&
    Y_{n_2 + n_3}
      &\longrightarrow&
    Maps(X_{n_1}, Z_{n_1 + n_2 + n_3})
    \\
    X_{n_1} \wedge S^{n_2} \wedge Y_{n_3}
      &\longrightarrow&
    X_{n_1} \wedge Y_{n_2 + n_3}
      &\longrightarrow&
    Z_{n_1 + n_2 + n_3}
  }
  \,.
\end{displaymath}
This establishes the form of the morphism $\ell$. By the same reasoning as in the proof of prop. \ref{DiagramSpectraGiveSymmetricAndOrthogonalSpectra}, we may restrict the coequalizer to $n_2 = 1$ without changing it.

The form of the morphism $r$ is obtained by the analogous sequence of identifications of morphisms, now with the parenthesis to the left. That it involves $\tau^{Top^{\ast/}_{cg}}$ and the permutation action $\tau^{sym}$ as shown \hyperlink{ShuffleActionInCoequalizerForSmashProductOfSpectra}{above} follows from the formula for the braiding of the Day convolution tensor product from the proof of prop. \ref{DayConvolutionYieldsMonoidalCategoryStructure}:

\begin{displaymath}
\tau^{Day}_{A,B}(n)
  =
  \overset{n_1,n_2}{\int}
   Sym( \tau^{Sym}_{n_1,n_2}, n )
    \wedge 
   \tau^{Top^{\ast/}_{cg}}_{A_{n_1}, B_{n_2}}
\end{displaymath}
by translating it to the components of the precomposition

\begin{displaymath}
X \otimes_{Day} \mathbb{S}_{sym}
   \overset{\tau^{Day}_{X,\mathbb{S}_{sym}}}{\longrightarrow}
  \mathbb{S}_{sym} \otimes_{Day} X
  \overset{}{\longrightarrow}
  X
\end{displaymath}
via the formula from the proof of prop. \ref{TopologicalLeftKanExtensionBCoend} for the [[left Kan extension]] $A \otimes_{Day} B \simeq Lan_{\otimes} A \overline{\wedge} B$ (prop. \ref{DayConvolutionViaKanExtensionOfExternalTensorAlongTensor}):

\begin{displaymath}
\begin{aligned}
    [Sym, Top^{\ast/}_{cg}]( \tau^{Day}_{X,\mathbb{S}_{sym}}, X)
    &
    \simeq
    \underset{n}{\int}
    Maps( 
      \overset{n_1, n_2}{\int}
      Sym( \tau^{sym}_{n_1,n_2}, n )
       \wedge 
      \tau^{Top^{\ast/}_{cg}}_{X_{n_1}, S^{n_2}}
      ,
      X(n)
    )_\ast
    \\
    &
    \simeq
    \underset{n_1,n_2}{\int}
    Maps(
      \tau_{X_{n_1}, S^{n_2} }^{Top^{\ast/}_{cg}}
      ,
      X( \tau^{sym}_{n_1,n_2} )
    )_\ast
  \end{aligned}
  \,.
\end{displaymath}
\end{proof}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[mapping spectrum]]


\item [[functor with smash product]]



\end{itemize}
[[model structure on spectra]]

\begin{itemize}%
\item [[symmetric spectrum]], [[model structure on symmetric spectra]]


\item [[orthogonal spectrum]], [[model structure on orthogonal spectra]]


\item [[S-module]], [[model structure on S-modules]]



\end{itemize}
Also the

\begin{itemize}%
\item [[model structure for excisive functors]]

\end{itemize}
carries a symmetric monoida smash product.

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

\hypertarget{original_sources}{}\subsubsection*{{Original sources}}\label{original_sources}

The original no-go theorem for a well-behave smash product of spectra is

\begin{itemize}%
\item [[Gaunce Lewis]], \emph{Is there a conveinient category of spectra?}, Journal of Pure and Applied Algebra Volume 73, Issue 3, 30 August 1991, Pages 233--246

\end{itemize}
In the mid-1990s, several categories of spectra with nice smash products were discovered, and simultaneously, model categories experienced a major renaissance.

The definition of [[S-modules]] and their theory originates in

\begin{itemize}%
\item [[Anthony Elmendorf]], [[Igor Kriz]], [[Michael Mandell]], [[Peter May]], \emph{Rings, modules and algebras in stable homotopy theory}, AMS 1997, 2014 (aka ``EKMM'')

\end{itemize}
and around 1993 [[Jeff Smith]] gave the first talks about symmetric spectra; the details of the model structure were later worked out and written up in

\begin{itemize}%
\item [[Mark Hovey]], [[Brooke Shipley]], [[Jeff Smith]], \emph{Symmetric spectra}, J. Amer. Math. Soc. 13 (2000), 149-208.

\end{itemize}
Discussion that makes the [[Day convolution]] structure on the symmetric smash product of spectra manifest is in

\begin{itemize}%
\item [[Michael Mandell]], [[Peter May]], [[Stefan Schwede]], [[Brooke Shipley]], part III of \emph{[[Model categories of diagram spectra]]}, Proceedings London Mathematical Society Volume 82, Issue 2, 2000 (\href{http://www.math.uchicago.edu/~may/PAPERS/mmssLMSDec30.pdf}{pdf}, \href{http://plms.oxfordjournals.org/content/82/2/441.short?rss=1&ssource=mfc}{publisher})

\end{itemize}
\hypertarget{reviews_and_introductions}{}\subsubsection*{{Reviews and introductions}}\label{reviews_and_introductions}

Surveys of the history are in

\begin{itemize}%
\item [[Anthony Elmendorf]], [[Igor Kriz]], [[Peter May]], \emph{[[Modern foundations for stable homotopy theory]]},1995 (\href{http://hopf.math.purdue.edu/Elmendorf-Kriz-May/modern_foundations.pdf}{pdf})


\item [[Stefan Schwede]], p.214-216 of \emph{[[Symmetric spectra]]} (2012)



\end{itemize}
A textbook account of the theory of symmetric spectra is

\begin{itemize}%
\item [[Stefan Schwede]], \emph{Symmetric spectra} (\href{http://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf}{pdf})

\end{itemize}
Seminar notes on symmetric spectra are in

\begin{itemize}%
\item [[Sander Kupers]], \emph{[[SymmetricSpectra.pdf:file]]}

\end{itemize}
See also

\begin{itemize}%
\item wikipedia \href{http://en.wikipedia.org/wiki/Highly_structured_ring_spectrum}{highly structured ring spectrum}

\end{itemize}
[[!redirects symmetric monoidal category of spectra]] [[!redirects symmetric monoidal categories of spectra]]

[[!redirects structured ring spectrum]] [[!redirects structured ring spectra]]

[[!redirects symmetric smash products of spectra]]

[[!redirects symmetric monoidal smash product of spectra]]

[[!redirects symmetric smash product on spectra]]



\end{document}