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\begin{document}

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\section*{model structure for excisive functors}

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

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

[[!include model category theory - contents]]

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

[[!include stable homotopy theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{TheUnderlyingCategories}{The underlying categories}\dotfill \pageref*{TheUnderlyingCategories} \linebreak
\noindent\hyperlink{TheModelStructures}{The model structures}\dotfill \pageref*{TheModelStructures} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{ModelStructureForSpectra}{Relation to BF-model structure on sequential spectra}\dotfill \pageref*{ModelStructureForSpectra} \linebreak
\noindent\hyperlink{SmashProduct}{Symmetric monoidal smash product}\dotfill \pageref*{SmashProduct} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A [[model category]] structure for [[excisive functors]] on [[functors]] from ([[finite homotopy type|finite]]) pointed simplicial sets to pointed simplicial sets (\hyperlink{Lydakis98}{Lydakis 98, theorem 9.2}, \hyperlink{BiedermannChornyRondings06}{Biedermann-Chorny-R\"o{}ndings 06, section 9}), hence (see \href{excisive+%28∞%2C1%29-functor#SpectrumObjects}{here}) a [[model structure for spectra]] (\hyperlink{Lydakis98}{Lydakis 98, theorem 11.3}).

Special case of a [[model structure for n-excisive functors]].

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

\hypertarget{TheUnderlyingCategories}{}\subsubsection*{{The underlying categories}}\label{TheUnderlyingCategories}

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

\begin{itemize}%
\item [[sSet]] for the [[category]] of [[simplicial sets]];


\item $sSet^{\ast/}$ for the category of [[pointed object|pointed]] simplicial sets;


\item $sSet_{fin}^{\ast/}\simeq s(FinSet)^{\ast/} \hookrightarrow sSet^{\ast/}$ for the [[full subcategory]] of [[pointed object|pointed]] [[simplicial object|simplicial]] [[finite sets]].



\end{itemize}
Write

\begin{displaymath}
sSet^{\ast/}
  \stackrel{\overset{(-)_+}{\longleftarrow}}{\underset{u}{\longrightarrow}}
  sSet
\end{displaymath}
for the [[free-forgetful adjunction]], where the [[left adjoint]] functor $(-)_+$ freely adjoins a base point.

Write

\begin{displaymath}
\wedge \colon sSet^{\ast/} \times sSet^{\ast/} \longrightarrow sSet^{\ast/}
\end{displaymath}
for the [[smash product]] of [[pointed object|pointed]] [[simplicial sets]], similarly for its restriction to $sSet_{fin}^{\ast}$:

\begin{displaymath}
X \wedge Y 
    \coloneqq 
  cofib\left(
    \; 
    \left(\,
      (u(X),\ast) \sqcup (\ast, u(Y)) 
    \,\right) 
      \longrightarrow 
     u(X) \times u(Y) 
    \;
  \right)
  \,.
\end{displaymath}
This gives $sSet^{\ast/}$ and $sSet^{\ast/}_{fin}$ the structure of a [[closed monoidal category]] and we write

\begin{displaymath}
[-,-]_\ast \;\colon\; (sSet^{\ast/})^{op} \times sSet^{\ast/} \longrightarrow sSet^{\ast/}
\end{displaymath}
for the corresponding [[internal hom]], the pointed [[function complex]] functor.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
For $X,Y\in sSet^{\ast/}$, the internal hom $[X,Y] \in sSet^{\ast/}$ is the simplicial set

\begin{displaymath}
[X,Y]_n = Hom_{sSet^{\ast/}}(X \wedge \Delta[n]_+, Y)
\end{displaymath}
regarded as pointed by the [[zero morphism]] (the one that factors through the base point), and the [[composition]] morphism

\begin{displaymath}
\circ \:\colon\; [X,Y] \wedge [Y,Z] \longrightarrow [X,Z]
\end{displaymath}
is given by

\begin{displaymath}
\begin{aligned}
    \circ_n 
      & \;\colon\; 
    ( 
      X \wedge \Delta[n]_+ \stackrel{f}{\longrightarrow} Y \;,\; Y \wedge \Delta[n]_+ \stackrel{g}{\longrightarrow} Z 
    )
    \\
    &
    \mapsto
    (
      X \wedge \Delta[n]_+
        \stackrel{X \wedge (diag_{\Delta[n]})_+}{\longrightarrow}
      X \wedge (\Delta[n] \times \Delta[n])_+
        \simeq
      X \wedge \Delta[n]_+ \wedge \Delta[n]_+
        \stackrel{f \wedge id}{\longrightarrow}
      Y \wedge \Delta[n]_+
        \stackrel{g}{\longrightarrow}
      Z
    )
  \end{aligned}
  \,.
\end{displaymath}
\end{remark}
We regard all the categories in def. \ref{SimplicialSetsPointedAndFinite} canonically as [[simplicially enriched categories]], and in fact regard $sSet^{\ast/}$ and $sSet^{\ast/}_{fin}$ as $sSet^{\ast/}$-[[enriched categories]].

\begin{remark}
\label{}\hypertarget{}{}
The [[smash product]] as an $sSet^{\ast/}$-[[enriched functor]] takes

\begin{displaymath}
\wedge \;\colon\; [X_1,Y_1] \wedge [X_2, Y_2] \longrightarrow [X_1 \wedge X_2, Y_1\wedge Y_2]
\end{displaymath}
by

\begin{displaymath}
\begin{aligned}
    \wedge_n 
    & \colon
    ( 
      X_1 \wedge \Delta[n]_+ \stackrel{f_1}{\longrightarrow} Y_1 \;,\;
      X_2 \wedge \Delta[n]_+ \stackrel{f_2}{\longrightarrow} Y_2
    )
    \\
    & \mapsto
    X_1 \wedge X_2 \wedge \Delta[n]_+
      \stackrel{id \wedge diag_+}{\longrightarrow}
    X_1 \wedge X_2 \wedge (\Delta[n] \times \Delta[n])_+
      \simeq
    X_1 \wedge X_2 \wedge \Delta[n]_+ \wedge \Delta[n]_+
      \simeq
    (X_1 \wedge \Delta[n]_+) \wedge (X_2 \wedge \Delta[n]_+)
      \stackrel{(f_1)_n \wedge (f_2)_n}{\longrightarrow}
    Y_1 \wedge Y_2
  \end{aligned}
\end{displaymath}
\end{remark}
The category that is discussed \hyperlink{TheModelStructures}{below} to support a model structure for [[excisive functors]] is the $sSet^{\ast/}$-[[enriched functor category]]

\begin{displaymath}
[sSet^{\ast/}_{fin}, sSet^{\ast/}]
  \,.
\end{displaymath}
(\hyperlink{Lydakis98}{Lydakis 98, example 3.8, def. 4.4})

In order to compare this to model structures for [[sequential spectra]] we consider also the following variant.

\begin{defn}
\label{CategoriesOfStandardSpheres}\hypertarget{CategoriesOfStandardSpheres}{}
Write $S^1_{std} \coloneqq \Delta[1]/\partial\Delta[1]\in sSet^{\ast/}$ for the standard minimal pointed simplicial [[1-sphere]].

Write

\begin{displaymath}
\iota \;\colon\; StdSpheres \longrightarrow sSet^{\ast/}_{fin}
\end{displaymath}
for the non-full $sSet^{\ast/}$-[[enriched category|enriched]] [[subcategory]] of pointed [[simplicial object|simplicial]] [[finite sets]], def. \ref{SimplicialSetsPointedAndFinite} whose

\begin{itemize}%
\item [[objects]] are the [[smash product]] powers $S^n_{std} \coloneqq (S^1_{std})^{\wedge^n}$ (the standard minimal simplicial [[n-spheres]]);


\item [[hom-objects]] are

\begin{displaymath}
[S^{n}_{std}, S^{n+k}_{std}]_{StdSpheres}
  \coloneqq
  \left\{
    \itexarray{
      \ast & for & k \lt 0
      \\
      im(S^{k}_{std} \stackrel{}{\to} [S^n_{std}, S^{n+k}_{std}]_{sSet^{\ast/}_{fin}})
      & otherwise
    }
  \right.
\end{displaymath}


\end{itemize}
\end{defn}
(\hyperlink{Lydakis98}{Lydakis 98, def. 4.2})

\begin{prop}
\label{}\hypertarget{}{}
There is an $sSet^{\ast/}$-[[enriched functor]]

\begin{displaymath}
(-)^seq
  \;\colon\;
  [StdSpheres,sSet^{\ast/}]
    \longrightarrow
  SeqPreSpec(sSet)
\end{displaymath}
(from the category of $sSet^{\ast/}$-[[enriched presheaves|enriched copresheaves]] on the categories of standard simplicial spheres of def. \ref{CategoriesOfStandardSpheres} to the category of [[sequential prespectra]] in [[sSet]]) given on objects by sending $X \in [StdSpheres,sSet^{\ast/}]$ to the sequential prespectrum $X^{seq}$ with components

\begin{displaymath}
X^{seq}_n \coloneqq X(S^n_{std})
\end{displaymath}
and with structure maps

\begin{displaymath}
\frac{S^1_{std} \wedge X^{seq}_n \stackrel{\sigma_n}{\longrightarrow} X^{seq}_n}{S^1_{std} \longrightarrow [X^{seq}_n, X^{seq}_{n+1}]}
\end{displaymath}
given by

\begin{displaymath}
S^1_{std} 
    \stackrel{\widetilde{id}}{\longrightarrow}
  [S^n_{std}, S^{n+1}_{std}]
    \stackrel{X_{S^n_{std}, S^{n+1}_{std}}}{\longrightarrow}
  [X^{seq}_n, X^{seq}_{n+1}]
  \,.
\end{displaymath}
This is an $sSet^{\ast/}$ [[enriched category theory|enriched]] [[equivalence of categories]].

\end{prop}
(\hyperlink{Lydakis98}{Lydakis 98, prop. 4.3})

\hypertarget{TheModelStructures}{}\subsubsection*{{The model structures}}\label{TheModelStructures}

Consider the $sSet^{\ast/}$-[[enriched functor category]] $[sSet^{\ast/}_{fin}, sSet^{\ast/}]$ from \hyperlink{TheUnderlyingCategories}{above}.

With $S^1_{std} \coloneqq \Delta[1]/\partial\Delta[1] \in sSet^{\ast/}$ we take [[looping and delooping]] $(\Sigma \dashv \Omega)$ to mean concretely the operation on [[smash product]] and pointed [[exponential]] with this particular $S^1_{std}$:

\begin{displaymath}
(\Sigma \dashv \Omega) \coloneqq ( S^1_{std}\wedge(-) \dashv [S^1_{std},-] ) \colon sSet^{\ast/} \longrightarrow sSet^{\ast/}
  \,.-
\end{displaymath}
These operations extend objectwise to $[sSet^{\ast/}_{fin}, sSet^{\ast/}]$, where we denote them by the same symbols.

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

\begin{displaymath}
T 
    \;\colon\; 
  [sSet^{\ast/}_{fin}, sSet^{\ast/}] 
    \longrightarrow 
  [sSet^{\ast/}_{fin}, sSet^{\ast/}]
\end{displaymath}
for the [[functor]] given on $X$ by

\begin{displaymath}
T X \colon K \mapsto \Omega X(\Sigma K)
  \,.
\end{displaymath}
Write

\begin{displaymath}
\tau \;\colon\; id \longrightarrow T
\end{displaymath}
for the [[natural transformation]] whose component $\tau_{X}(K) \;\colon\; X(K) \to \Omega (X(\Sigma K))$ is the $(\Sigma \dashv \Omega)$-[[adjunct]] of the canonical morphism $\Sigma X(K) \longrightarrow X(\Sigma K)$ induced from

\begin{displaymath}
X
  \left(
  \itexarray{
    K & \longrightarrow & \ast
    \\
    \downarrow &\swArrow& \downarrow
    \\
    \ast &\longrightarrow& \Sigma K
  }
  \right)
  \;\;\;\;
    =
  \;\;\;\;
  \itexarray{
     X(K) &&\longrightarrow&& \ast
     \\
     \downarrow &&& \swarrow & \downarrow
     \\
     \downarrow && \Sigma X(K) && \downarrow
     \\
     \downarrow & \nearrow && \searrow^{\mathrlap{\tau_{X}(K)}}& \downarrow
     \\
     \ast &&\longrightarrow &&  X(\Sigma K)
  }
  \,.
\end{displaymath}
Write

\begin{displaymath}
T^\infty
    \;\colon\; 
  [sSet^{\ast/}_{fin}, sSet^{\ast/}] 
    \longrightarrow 
  [sSet^{\ast/}_{fin}, sSet^{\ast/}]
\end{displaymath}
for the functor given by $X$ by the [[sequential colimit]]

\begin{displaymath}
T^\infty X 
    \coloneqq 
  \underset{\longrightarrow}{\lim}
  \left(
    X 
      \stackrel{\tau_X}{\longrightarrow} 
    T X
      \stackrel{T(\tau_X)}{\longrightarrow}
    T (T X)
       \stackrel{}{\longrightarrow}
      \simeq 
  \right)
  \,.
\end{displaymath}
Write $Fib \colon sSet^{\ast} \to sSet^{\ast}$ for any [[Kan fibrant replacement]] functor.

Say that the \emph{stabilization} ([[spectrification]]) of $X$ is

\begin{displaymath}
stab(X) \coloneqq T^\infty (Fib(Lan X(Fib(-))))
  \,,
\end{displaymath}
where $Lan X \colon sSet^{\ast/} \to sSet^{\ast}$ is the [[left Kan extension]] of $X$ along the inclusion $sSet^{\ast/}_{fin} \hookrightarrow sSet^{\ast/}$.

\end{defn}
\begin{defn}
\label{ClassesOfMorphismsForModelStructureForExcisiveFunctors}\hypertarget{ClassesOfMorphismsForModelStructureForExcisiveFunctors}{}
Say that a morphism $f \colon X \to Y$ in $[sSet^{\ast/}_{fin}, sSet^{\ast/}]$ is

\begin{itemize}%
\item a \emph{stable weak equivalence} if its stabilization, def. \ref{Tinfinity}, takes value on each $K \in sSet^{\ast/}$ in [[weak homotopy equivalences]] in $sSet^{\ast/}$;


\item a \emph{stable cofibration} if it has the [[left lifting property]] against those morphisms whose value on every $K \in sSet^{\ast/}$ is a [[Kan fibration]].



\end{itemize}
\end{defn}
(\hyperlink{Lydakis98}{Lydakis 98, def. 9.1, def. 7.1})

\begin{prop}
\label{LydakisModelStructure}\hypertarget{LydakisModelStructure}{}
The classes of morphisms of def. \ref{ClassesOfMorphismsForModelStructureForExcisiveFunctors}, define a [[model category]] structure

\begin{displaymath}
[sSet^{\ast/}_{fin}, sSet^{\ast/}]_{Ly}
  \,.
\end{displaymath}
\end{prop}
(\hyperlink{Lydakis98}{Lydakis 98, theorem 9.2})

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

\hypertarget{ModelStructureForSpectra}{}\subsubsection*{{Relation to BF-model structure on sequential spectra}}\label{ModelStructureForSpectra}

There is a [[Quillen equivalence]] between the [[Bousfield-Friedlander model structure]] on [[sequential spectra]] and the Lydakis model structure $[sSet^{\ast/}_{fin}, sSet^{\ast/}]_{Ly}$ from prop. \ref{LydakisModelStructure}.

\begin{prop}
\label{SequentialSpectraAsSimplicialFunctorsOnStandardSpheres}\hypertarget{SequentialSpectraAsSimplicialFunctorsOnStandardSpheres}{}
There is an $sSet^{\ast/}$-[[enriched functor]]

\begin{displaymath}
(-)^seq
  \;\colon\;
  [StdSpheres,sSet^{\ast/}]
    \longrightarrow
  SeqPreSpec(sSet)
\end{displaymath}
(from the category of $sSet^{\ast/}$-[[enriched presheaves|enriched copresheaves]] on the categories of standard simplicial spheres of def. \ref{CategoriesOfStandardSpheres} to the category of sequential prespectra in [[sSet]]) given on objects by sending $X \in [StdSpheres,sSet^{\ast/}]$ to the sequential prespectrum $X^{seq}$ with components

\begin{displaymath}
X^{seq}_n \coloneqq X(S^n_{std})
\end{displaymath}
and with structure maps

\begin{displaymath}
\frac{S^1_{std} \wedge X^{seq}_n \stackrel{\sigma_n}{\longrightarrow} X^{seq}_n}{S^1_{std} \longrightarrow [X^{seq}_n, X^{seq}_{n+1}]}
\end{displaymath}
given by

\begin{displaymath}
S^1_{std} 
    \stackrel{\widetilde{id}}{\longrightarrow}
  [S^n_{std}, S^{n+1}_{std}]
    \stackrel{X_{S^n_{std}, S^{n+1}_{std}}}{\longrightarrow}
  [X^{seq}_n, X^{seq}_{n+1}]
  \,.
\end{displaymath}
This is an $sSet^{\ast/}$ [[enriched category theory|enriched]] [[equivalence of categories]].

\end{prop}
(\hyperlink{Lydakis98}{Lydakis 98, prop. 4.3})

\begin{prop}
\label{QuillenEquivalenceBetweenSequentialSpectraAndExcisiveFunctors}\hypertarget{QuillenEquivalenceBetweenSequentialSpectraAndExcisiveFunctors}{}
The [[adjunction]]

\begin{displaymath}
(\iota_\ast \dashv \iota^\ast)
  \;\colon\;
  [sSet^{\ast/}_{fin}, sSet^{\ast/}]_{Ly}
    \stackrel{\overset{\iota_\ast}{\longleftarrow}}{\underset{\iota^\ast}{\longrightarrow}}
  [StdSpheres, sSet^{\ast/}]
    \underoverset{\simeq}{(-)^{seq}}{\longrightarrow}
  SeqPreSpec(sSet)_{BF}
\end{displaymath}
(given by restriction $\iota^\ast$ along the defining inclusion $\iota$ of def. \ref{CategoriesOfStandardSpheres} and by left [[Kan extension]] $\iota_\ast$ along $\iota$, and combined with the equivalence $(-)^{seq}$ of prop. \ref{SequentialSpectraAsSimplicialFunctorsOnStandardSpheres}) is a [[Quillen adjunction]] and in fact a [[Quillen equivalence]] between the [[Bousfield-Friedlander model structure]] on sequential prespectra and Lydakis' model structure for excisive functors, prop. \ref{LydakisModelStructure}.

\end{prop}
\hypertarget{SmashProduct}{}\subsubsection*{{Symmetric monoidal smash product}}\label{SmashProduct}

The [[excisive functors]] naturally carry a [[smash product]] (\hyperlink{Lydakis98}{Lydakis 98, def. 5.1}) making the model structure for 1-excisive functors a [[symmetric monoidal model category]] (\hyperlink{Lydakis98}{Lydakis 98, section 12}). Via the translation to [[sequential spectra]] of prop. \ref{LydakisModelStructure} this is a model for the [[smash product of spectra]] (\hyperlink{Lydakis98}{Lydakis 98, theorem 12.5}); hence it is a [[symmetric smash product on spectra]].

A [[monoid]] with respect to this smash product (hence a [[ring spectrum]]) is equivalently a [[functor with smash products]] (``FSP'') as earlier considered in (\hyperlink{B&#246;kstedt86}{B\"o{}kstedt 86}).

\begin{defn}
\label{DayConvolutionStructureOnExcisiveFunctors}\hypertarget{DayConvolutionStructureOnExcisiveFunctors}{}
Since $(sSet^{\ast/}, \wedge)$ (def. \ref{SimplicialSetsPointedAndFinite}) is a [[symmetric monoidal category]], $[sSet^{\ast}_{fin}, sSet^{\ast/}]$ canonically becomes symmetric monoidal itself via the induced [[Day convolution product]]. We write

\begin{displaymath}
\left(\, [sSet^{\ast/}_{fin}, sSet^{\ast/}], \; \wedge_{Say} \right)
\end{displaymath}
for this [[symmetric monoidal category]].

\end{defn}
\begin{prop}
\label{LydakisTensorProductIsDayConvolution}\hypertarget{LydakisTensorProductIsDayConvolution}{}
The smash product on $[sSet^{\ast/}_{fin}, sSet^{\ast/}]$ considered (\hyperlink{Lydakis98}{Lydakis 98, def. 5.1}) coincides with the [[Day convolution product]] of def. \ref{DayConvolutionStructureOnExcisiveFunctors}.

\end{prop}
\begin{proof}
The [[Day convolution]] product is characterized (see \href{Day+convolution#DayConvolutionViaNaturalIsosInvolvingExternalTensorAndTensor}{this proposition}) by making a [[natural isomorphism]] of the form

\begin{displaymath}
[sSet^{\ast/}_{fin}, sSet^{\ast/}](X \wedge Y, Z) \simeq [sSet^{\ast/}_{fin} \times sSet^{\ast/}_{fin}, sSet^{\ast/}](X \tilde{\wedge} Y, Z \circ \wedge)
\end{displaymath}
where the \emph{external smash product} $\tilde {\wedge}$ on the right is defined by $X \tilde{\wedge} Y  \coloneqq  \wedge \circ (X,Y)$. Now, (\hyperlink{Lydakis98}{Lydakis 98, def. 5.1}) sets

\begin{displaymath}
X \wedge Y \coloneqq \wedge_\ast (X \tilde{\wedge} Y)
\end{displaymath}
where, by (\hyperlink{Lydakis98}{Lydakis 98, prop. 3.23}), $\wedge_\ast$ is the [[left adjoint]] to $\wedge^\ast(-) \coloneqq (-)\circ \wedge$. Hence the [[adjunction]] isomorphism gives the above characterization.

\end{proof}
\begin{prop}
\label{SmashProductCompatibleWithTheOneOnSpectra}\hypertarget{SmashProductCompatibleWithTheOneOnSpectra}{}
Under the Quillen equivalence of prop. \ref{QuillenEquivalenceBetweenSequentialSpectraAndExcisiveFunctors} the symmetric monoidal [[Day convolution product]] on excisive simplicial functors (prop. \ref{LydakisTensorProductIsDayConvolution}) is identified with the proper [[smash product of spectra]] realized on sequential spectra by the \href{sequential+spectrum#SmashProductByDoublingDegrees}{standard formula}.

\end{prop}
(\hyperlink{Lydakis98}{Lydakis 98, theorem 12.5})

This implies that any incarnation of the [[sphere spectrum]] in $[sSet^{\ast}_{fin}, sSet^{\ast/}]$, possibly suitably replaced acts as the [[tensor unit]] up to stable weak equivalence. The following says that the canonical incarnation of the sphere spectrum actually is the genuine (1-categorical) tensor unit:

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

\begin{displaymath}
\mathbb{S}_{std}\in [sSet^{\ast/}_{fin}, sSet^{\ast/}]
\end{displaymath}
for the canonical inclusion $sSet^{\ast/}_{fin} \hookrightarrow sSet^{\ast/}$.

(the standard incarnation of the [[sphere spectrum]] in the model structure for excisive functors).

\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
The object $\mathbb{S}_{std}$ of def. \ref{StandardSphereSpectrum} is (up to [[isomorphism]]) the [[tensor unit]] in $([sSet^{\ast/}_{fin}, sSet^{\ast}], \wedge_{Day})$.

\end{prop}
\begin{proof}
This is (\hyperlink{Lydakis98}{Lydakis 98, theorem 5.9}), but it is immediate with prop. \ref{LydakisTensorProductIsDayConvolution}, using that the tensor unit for Day convolution is the functor [[representable functor|represented]] by the tensor unit in the underlying site (\href{Day+convolution#DayConvolutionYieldsMonoidalCategoryStructure}{this proposition}).

\end{proof}
\begin{prop}
\label{LydakisModelCategoryIsMonoidal}\hypertarget{LydakisModelCategoryIsMonoidal}{}
Equipped with the [[Day convolution]] tensor product (prop. \ref{LydakisTensorProductIsDayConvolution}) the Lydakis model category of prop. \ref{LydakisModelStructure} becomes a [[monoidal model category]]

\begin{displaymath}
\left(
    [sSet^{\ast/}_{fin}, sSet^{\ast/}]_{Ly}, \; \wedge_{Day}
  \right)
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
The [[pushout product axiom]] is (\hyperlink{Lydakis98}{Lydakis 98, theorem 12.3}). Moreover (\hyperlink{Lydakis98}{Lydakis 98, theorem 12.4}), shows that tensoring with cofibrant objects preserves all stable weak equivalences, hence in particular preserves cofibrant resolution of the tensor unit.

\end{proof}
This means that ([[commutative monoid|commutative]]) [[monoids]] in the monoidal Lydakis model structure $\left( [sSet^{\ast/}_{fin}, sSet^{\ast/}]_{Ly}, \; \wedge_{Day} \right)$ are good models for [[ring spectra]] ([[E-infinity rings]]/[[A-infinity rings]]).

\begin{prop}
\label{MonoidsInLydakisModelStructureAreFSP}\hypertarget{MonoidsInLydakisModelStructureAreFSP}{}
Monoids (commutative monoids) in the Lydakis monoidal model category $\left( [sSet^{\ast/}_{fin}, sSet^{\ast/}]_{Ly}, \; \wedge_{Day} \right)$ of prop. \ref{LydakisModelCategoryIsMonoidal} are equivalently ([[symmetric monoidal functor|symmetric]]) [[lax monoidal functors]] of the form

\begin{displaymath}
sSet^{\ast/}_{fin} \longrightarrow sSet^{\ast/}
\end{displaymath}
also known as ``[[functors with smash product]]'' ([[FSP]]s).

\end{prop}
(\hyperlink{Lydakis98}{Lydakis 98, remark 5.12})

\begin{proof}
Since the tensor product is [[Day convolution]] of the [[smash product]] on $sSet^{\ast/}_{fin}$, def. \ref{DayConvolutionStructureOnExcisiveFunctors}, this is a special casse of a general property of [[Day convolution]], see \href{Day+convolution#DayMonoidsAreLaxMonoidalFunctorsOnTheSite}{this proposition}.

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

[[model structure on functors]]

[[model structure on spectra]]

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

\end{itemize}
with [[symmetric monoidal smash product of 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]]


\item [[excisive functor]], \textbf{model structure on excisive functors}


\item [[Model categories of diagram spectra]]



\end{itemize}
[[model structure for n-excisive functors]]

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

Model structure for [[excisive functors]] on [[simplicial sets]] (hence also a [[model structure for spectra]]) is discussed in:

\begin{itemize}%
\item Lydakis, \emph{Simplicial functors and stable homotopy theory} Preprint, available via Hopf archive, 1998 (\href{http://hopf.math.purdue.edu/Lydakis/s_functors.pdf}{pdf})

\end{itemize}
A similar model structure on functors on topological spaces was given in

\begin{itemize}%
\item [[William Dwyer]], \emph{Localizations}, In \emph{Axiomatic, enriched and motivic homotopy theory}, volume 131 of NATO Sci. Ser. II Math. Phys. Chem., pages 3--28. Kluwer Acad. Publ., Dordrecht, 2004

\end{itemize}
and also excisive functors modeled on topological spaces are the $\mathcal{W}$-spectra in

\begin{itemize}%
\item [[Michael Mandell]], [[Peter May]], [[Stefan Schwede]], [[Brooke Shipley]], \emph{[[Model categories of diagram spectra]]}, Proceedings of the London Mathematical Society, 82 (2001), 441-512 (\href{http://www.math.uchicago.edu/~may/PAPERS/mmssLMSDec30.pdf}{pdf})

\end{itemize}
Discussion of the restriction from excisive functors to [[symmetric spectra]] includes

\begin{itemize}%
\item [[Stefan Schwede]], chapter I, section 7.3 of \emph{[[Symmetric spectra]]} (2012)

\end{itemize}
The [[functors with smash products]] (``FSP''s) appearing in (\hyperlink{Lydakis98}{Lydakis 98, remark 5.12}) had earlier been considered in

\begin{itemize}%
\item [[Marcel Bökstedt]], \emph{Topological Hochschild homology}. Preprint, Bielefeld, 1986

\end{itemize}
Further generalization of the model structure for excisive functor, in particular to [[enriched functors]] and to a [[model structure for n-excisive functors]] for $n \geq 1$ is discussed in

\begin{itemize}%
\item [[Georg Biedermann]], [[Boris Chorny]], Oliver R\"o{}ndigs, \emph{Calculus of functors and model categories}, Advances in Mathematics 214 (2007) 92-115 (\href{http://arxiv.org/abs/math/0601221}{arXiv:math/0601221})


\item [[Georg Biedermann]], Oliver R\"o{}ndigs, \emph{Calculus of functors and model categories II} (\href{http://arxiv.org/abs/1305.2834v2}{arXiv:1305.2834v2})



\end{itemize}
[[!redirects model structure for excisive functors]] [[!redirects model structures for excisive functors]]

[[!redirects model structure on excisive functors]] [[!redirects model structures on excisive functors]]



\end{document}