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\begin{document}

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\section*{Wirthmüller context}

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

\hypertarget{geometry}{}\paragraph*{{Geometry}}\label{geometry}

[[!include higher geometry - contents]]

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

\hypertarget{duality}{}\paragraph*{{Duality}}\label{duality}

[[!include duality - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak
\noindent\hyperlink{TheComparisonMaps}{The comparison maps}\dotfill \pageref*{TheComparisonMaps} \linebreak
\noindent\hyperlink{ComparisonOfPushForwardsAndWirthmuelleriso}{Comparison of push-forwards and Wirthm\"u{}ller isomorphism}\dotfill \pageref*{ComparisonOfPushForwardsAndWirthmuelleriso} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{CartesianWirthmuellerContexts}{Cartesian Wirthm\"u{}ller contexts in toposes}\dotfill \pageref*{CartesianWirthmuellerContexts} \linebreak
\noindent\hyperlink{PointedObjects}{Pointed objects with smash product}\dotfill \pageref*{PointedObjects} \linebreak
\noindent\hyperlink{bundles_of_modules}{Bundles of modules}\dotfill \pageref*{bundles_of_modules} \linebreak
\noindent\hyperlink{bundles_of_modules_2}{Bundles of $\infty$-modules}\dotfill \pageref*{bundles_of_modules_2} \linebreak
\noindent\hyperlink{beckergottlieb_transfer}{Becker-Gottlieb transfer}\dotfill \pageref*{beckergottlieb_transfer} \linebreak
\noindent\hyperlink{in_equivariant_stable_homotopy_theory}{In equivariant stable homotopy theory}\dotfill \pageref*{in_equivariant_stable_homotopy_theory} \linebreak
\noindent\hyperlink{ForQuasicoherentSheaves}{For quasicoherent sheaves (in $E_\infty$-geometry)}\dotfill \pageref*{ForQuasicoherentSheaves} \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 \emph{Wirthm\"u{}ller context} is a pair of two [[symmetric monoidal category|symmetric]] [[closed monoidal categories]] $(\mathcal{X}, \otimes_X, 1_{X})$, $(\mathcal{Y}, \otimes_Y, 1_Y)$ which are connected by an [[adjoint triple]] of [[functors]] such that the middle one is a [[closed monoidal functor]].

This is the variant/special case of the [[yoga of six operations]] consisting of two [[adjoint pairs]] $(f_! \dashv f^!)$ and $(f^\ast \dashv f_\ast)$ and the [[tensor product]]/[[internal hom]] [[adjunctions]] $((-)\otimes B \dashv [B,-])$, specialized to the case that $f^! \simeq f^\ast$:

\begin{displaymath}
f_! \dashv (f^! = f^\ast) \dashv f_\ast
  \;\colon\;
  \mathcal{X}
    \stackrel{\overset{f_!}{\longrightarrow}}{\stackrel{\overset{f^! = f^\ast }{\leftarrow}}{\underset{f_\ast}{\longrightarrow}}}
  \mathcal{Y}
  \,.
\end{displaymath}
(The other specialization of the [[six operations]] to $f_\ast \simeq f_!$ is called the \emph{[[Grothendieck context]]}).

Often one is interested in the case that there is an [[object]] $C \in \mathcal{X}$ and an [[equivalence]]

\begin{displaymath}
f_\ast 1_{X} \simeq f_! C
  \,.
\end{displaymath}
If this induces a [[natural equivalence]]

\begin{displaymath}
f_\ast A  \simeq f_!(A \otimes_X C)
\end{displaymath}
for $A \in \mathcal{X}$, then one says this is a \emph{Wirthm\"u{}ller isomorphism}, following (\hyperlink{Wirthmueller74}{Wirthmueller 74}). In particular there is a canonical [[natural transformation]]

\begin{displaymath}
f_\ast A  \longrightarrow f_!(A \otimes_X C)
\end{displaymath}
and one can ask this to be an equivalence, hence a Wirthm\"u{}ller isomorphism (\hyperlink{May05}{May 05}).

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

\begin{defn}
\label{WirthmullerContext}\hypertarget{WirthmullerContext}{}
Let $(\mathcal{X}, \otimes_X, 1_{X})$, $(\mathcal{Y}, \otimes_Y, 1_Y)$ be two [[symmetric monoidal category|symmetric]] [[closed monoidal categories]] and let

\begin{displaymath}
f_! \dashv (f^! = f^\ast) \dashv f_\ast
  \;\colon\;
  \mathcal{X}
    \stackrel{\overset{f_!}{\longrightarrow}}{\stackrel{\overset{f^! = f^\ast }{\leftarrow}}{\underset{f_\ast}{\longrightarrow}}}
  \mathcal{Y}
\end{displaymath}
be an [[adjoint triple]] of [[functors]] between them. We call this setup

\begin{itemize}%
\item a \emph{pre-Wirthm\"u{}ller context} if $f^\ast$ is a [[strong monoidal functor]]:


\item a \emph{Wirthm\"u{}ller context} if $f^\ast$ is in addition a [[strong closed functor]], hence a strongly [[closed monoidal functor]].



\end{itemize}
\end{defn}
(\hyperlink{May05}{May 05, def. 2.12})

\begin{defn}
\label{}\hypertarget{}{}
We write $[-,-]$ for the [[internal hom]] functors. For $A \in \mathcal{X}$ we write

\begin{displaymath}
\mathbb{D}A \coloneqq [A,1_X]
\end{displaymath}
for the [[internal hom]] from $A$ into the [[unit object]], hence for dual of $A$ with respect to the [[closed category]] structure, its [[dual object in a closed category]].

We say ``$A$ is dualizable'' to mean that it is a [[dualizable object]] with respect, insead, to the (symmetric) [[monoidal category]] structure $\otimes_X$. If $A$ is dualizable we write $A^\vee$ for its monoidal [[dual object]].

Similarly for $B \in \mathcal{Y}$.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
If all objects in $\mathcal{X}$ and $\mathcal{Y}$ are [[dualizable object|dualizable]], hence if they are [[compact closed categories]], then they are in particular also [[star-autonomous categories]] with [[dualizing object]] the [[tensor unit]]. As such their [[internal logic]] is \emph{[[linear logic]]} and their [[type theory]] is \emph{[[linear type theory]]}. In terms of this a Wirthm\"u{}ller morphism, def. \ref{WirthmullerContext}, is the linear analog of a [[context extension]] morphism in a [[hyperdoctrine]]: it defines a \emph{[[dependent linear type theory]]}. See there for more.

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

\hypertarget{TheComparisonMaps}{}\subsubsection*{{The comparison maps}}\label{TheComparisonMaps}

\begin{remark}
\label{QuasiMonoidalnessOfLeftAdjoint}\hypertarget{QuasiMonoidalnessOfLeftAdjoint}{}
In a pre-Wirthm\"u{}ller context, def. \ref{WirthmullerContext}, there is a canonical [[natural transformation]]

\begin{displaymath}
f_!(A \otimes_X B)
    \longrightarrow
  (f_! A) \otimes_Y (f_! B)
  \,,
\end{displaymath}
(not necessarily an [[equivalence]]) being the [[adjunct]] of the composite

\begin{displaymath}
A \otimes_X B
  \stackrel{}{\longrightarrow}
  (f^\ast f_! A) \otimes_X (f^\ast f_! B)
  \stackrel{\simeq}{\longrightarrow}
  f^\ast ( (f_! A) \otimes_Y (f_! B) )
  \,,
\end{displaymath}
where the first morphism is the [[tensor product]] of two copies of the [[unit of an adjunction|adjunction unit]] and where the second is the [[equivalence]] that exhibits $f^\ast$ as a [[strong monoidal functor]].

\end{remark}
\begin{defn}
\label{ComparisonMaps}\hypertarget{ComparisonMaps}{}
In a pre-Wirthm\"u{}ller context, def. \ref{WirthmullerContext}, write $\overline {\pi}$ for the [[natural transformation]]

\begin{displaymath}
\overline{\pi}
  \;\colon\;
  f_! ((f^\ast B) \otimes A) 
  \longrightarrow 
  B \otimes f_! A
\end{displaymath}
given as the composite

\begin{displaymath}
\overline{\pi}
  \;\colon\;
  f_! ((f^\ast B) \otimes A) 
   \stackrel{}{\longrightarrow}
  (f_! f^\ast B) \otimes_Y (f_! A)
   \longrightarrow 
  B \otimes f_! A
  \,,
\end{displaymath}
where the first morphism is that of remark \ref{QuasiMonoidalnessOfLeftAdjoint} and where the second is the $(f_! \dashv f^\ast)$ [[counit of an adjunction|counit]] (tensored with an identity).

Also write

\begin{displaymath}
\overline{\gamma}
    \;\colon\;
  [f_! A, B]
   \longrightarrow
  f_\ast [A, f^\ast B]
\end{displaymath}
for the $(f^\ast \dashv f_\ast)$ [[adjunct]] of the [[natural transformation]] given as the composite

\begin{displaymath}
f^\ast [f_! A, B]
   \stackrel{\simeq}{\longrightarrow}
  [f^\ast f_! A, \, f^\ast B]
   \longrightarrow
  [A, f^\ast B]  
  \,,
\end{displaymath}
where the first map exhibits $f^\ast$ as a [[closed monoidal functor]] and where the second is the $(f_! \dashv f^\ast)$-[[unit of an adjunction|unit]] (under the [[internal hom]]).

\end{defn}
see (\hyperlink{May05}{May 05, prop. 2.11})

\begin{prop}
\label{ComparisonIsEquivalenceOnDualizables}\hypertarget{ComparisonIsEquivalenceOnDualizables}{}
In a pre-Wirthm\"u{}ller context, def. \ref{WirthmullerContext}, the comparison maps of def. \ref{ComparisonMaps} are [[equivalences]] when the argument $B \in (\mathcal{Y}, \otimes_Y, 1_Y)$ is a [[dualizable object]].

If either of the two happens to be a [[natural equivalence]] (hence an equivalence for all arguments), then so is the other.

\end{prop}
(\hyperlink{May05}{May 05, prop. 2.8 and prop. 2.11})

\begin{prop}
\label{InWirthmuellerContextProjectionIsEquivalence}\hypertarget{InWirthmuellerContextProjectionIsEquivalence}{}
Precisely if the pre-Wirthm\"u{}ller context is a Wirthm\"u{}ller context, def. \ref{WirthmullerContext}, are both comparison maps of def. \ref{ComparisonMaps} are natural equivalences.

\end{prop}
\begin{proof}
For all $A \in \mathcal{X}$ and $B,C \in \mathcal{Y}$ we have by the $(f_! \dashv f^\ast)$-[[adjunction]] and the tensor$\dashv$hom-adjunction a [[commuting diagram]] of the form

\begin{displaymath}
\itexarray{
    \mathcal{Y}(B \otimes f_! A, \, C  )
     &
      \stackrel{
        \mathcal{Y}(\overline{\pi}(A,B), C)
      }{
        \longrightarrow
     }
     &
    \mathcal{Y}(f_! ((f^\ast B) \otimes A),\, C)
    \\
    \downarrow^{\mathrlap{\simeq}}
     &&
    \downarrow^{\mathrlap{\simeq}}
    \\
    \mathcal{X}(A, f^\ast [B,C])
    &\stackrel{}{\longrightarrow}&
    \mathcal{X}(A, [(f^\ast B), (f^\ast C)])
  }
  \,.
\end{displaymath}
By naturality in $A$ and by the [[Yoneda lemma]] this shows that $\overline{\pi}$ is an equivalence precisey if $f^\ast$ is strong closed.

For $\overline{\gamma}$ the same statement follows from this with prop. \ref{ComparisonIsEquivalenceOnDualizables}.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
The first [[natural equivalence]] of prop. \ref{InWirthmuellerContextProjectionIsEquivalence}

\begin{displaymath}
f_!(f^\ast A \otimes B) \simeq A \otimes f_!(B)
\end{displaymath}
is often called the \emph{[[projection formula]]}. In [[representation theory]] this is also sometimes called \emph{[[Frobenius reciprocity]]}, though mostly that term is used for (just) the existence of the $(f_! \dashv f^\ast)$-[[adjunction]], where in representation theory the [[left adjoint]] $f_!$ forms [[induced representations]].

\end{remark}
\hypertarget{ComparisonOfPushForwardsAndWirthmuelleriso}{}\subsubsection*{{Comparison of push-forwards and Wirthm\"u{}ller isomorphism}}\label{ComparisonOfPushForwardsAndWirthmuelleriso}

\begin{cor}
\label{PushforwardsIntertwinedByDuality}\hypertarget{PushforwardsIntertwinedByDuality}{}
In a pre-Wirthm\"u{}ller context, def. \ref{WirthmullerContext}, the functors $f_!$ and $f_\ast$ are intertwined by dualization, in that there is a [[natural equivalence]]

\begin{displaymath}
\mathbb{D}(f_! A) \simeq f_\ast(\mathbb{D} A)
  \,.
\end{displaymath}
\end{cor}
\begin{proof}
This is the special case of the property of $\overline{\gamma}$ in prop. \ref{ComparisonIsEquivalenceOnDualizables} for $B = 1_Y$:

\begin{displaymath}
\mathbb{D}(f_! A)
  =  
  [f_! A, 1_Y]
   \underoverset{\simeq}{\overline{\gamma}}{\longrightarrow}
  f_\ast [A, f^\ast 1_Y]
  \simeq
  f_\ast [A, 1_Y]
  =
  f_\ast \mathbb{D} A
  \,.
\end{displaymath}
\end{proof}
\begin{remark}
\label{LinearDeMorganDualityForPushForwards}\hypertarget{LinearDeMorganDualityForPushForwards}{}
With a Wirthm\"u{}ller context regarded as [[categorical semantics]] for [[dependent linear type theory]] (see there), then the statement of cor. \ref{PushforwardsIntertwinedByDuality} is an instance of [[de Morgan duality]] where linear [[dual object|dualization]] intertwines linear [[dependent sum]] $\sum$ and [[dependent product]] $\prod$

\begin{displaymath}
\prod_f \mathbb{D} \simeq \mathbb{D} \sum_f
  \,.
\end{displaymath}
For more on this see also ([[schreiber:Quantization via Linear homotopy types|Schreiber 14, section 3.3]]).

\end{remark}
\begin{example}
\label{}\hypertarget{}{}
In a pre-Wirthm\"u{}ller context

\begin{displaymath}
f_\ast 1_X \simeq \mathbb{D}(f_! 1_X)
  \,.
\end{displaymath}
\end{example}
\begin{proof}
By cor. \ref{PushforwardsIntertwinedByDuality}, since $\mathbb{D} 1_X \simeq 1_X$.

\end{proof}
\begin{prop}
\label{AbstractWirthmuellerIso}\hypertarget{AbstractWirthmuellerIso}{}
\textbf{(Wirthm\"u{}ller isomorphism)}

In a Wirthm\"u{}ller context, def. \ref{WirthmullerContext} if $f_! 1_X$ is a [[dualizable object]] with dual $f_! C$, then there is a [[natural equivalence]]

\begin{displaymath}
\omega
   \;\colon\;
  f_\ast f^\ast A \stackrel{\simeq}{\longrightarrow} f_!((f^\ast A) \otimes C)
  \,.
\end{displaymath}
\end{prop}
(\hyperlink{May05}{May 05, prop. 4.13}).

\begin{remark}
\label{MonadCommutesWithDualityUpTotwist}\hypertarget{MonadCommutesWithDualityUpTotwist}{}
In particular if there is $D \in \mathcal{Y}$ with

\begin{displaymath}
\mathbb{D}(f_! f^\ast 1_Y) \simeq f_! f^\ast D
\end{displaymath}
(hence if $C \simeq f^\ast D$ in the notation of prop. \ref{AbstractWirthmuellerIso}) and using that by cor. \ref{PushforwardsIntertwinedByDuality}, $f_\ast f^\ast  \mathbb{D}B
  \simeq
  f_\ast \mathbb{D} f^\ast B
  \simeq
  \mathbb{D}(f_! f^\ast B)$ then prop. \ref{AbstractWirthmuellerIso} gives a natural equivalence of the form

\begin{displaymath}
\mathbb{D}(f_! f^\ast B)
    \stackrel{\simeq}{\longrightarrow}
   f_! f^\ast ((\mathbb{D}B) \otimes D)
  \,,
\end{displaymath}
saying that the [[comonad]] $f_! f^\ast$ commutes with dualization up to a ``twist'' given by tensoring with $D$.

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

\hypertarget{CartesianWirthmuellerContexts}{}\subsubsection*{{Cartesian Wirthm\"u{}ller contexts in toposes}}\label{CartesianWirthmuellerContexts}

For $\mathbf{H}$ a [[topos]] and $f \colon X \longrightarrow Y$ any [[morphism]], then in the induced [[base change]] [[etale geometric morphism]]

\begin{displaymath}
(\sum_f \dashv f^\ast \dashv \prod_f)
  \;\colon\;
  \mathbf{H}_{/X}
  \longrightarrow
  \mathbf{H}_{/Y}
\end{displaymath}
the [[inverse image]]/[[context extension]] is a [[cartesian closed functor]] (see there for the proof). Therefore any base change of [[toposes]] constitutes a cartesian Wirthm\"u{}ller context.

Conversely, this means that systems of Wirthm\"u{}ller contexts are generalizations of [[categorical logic]] ([[hyperdoctrines]]) to non-cartesian contexts (see at \emph{[[dependent linear type theory]]}).

Notice that in a cartesian Wirthm\"u{}ller context duality is trivial, in that $\mathbb{D}X \simeq 1$ for all objects $X$. Therefore to the extent that the [[six operations]] yoga involves duality, it is interesting only the more non-cartesian (non-classical) the ambient Wirthm\"u{}ller context is.

For instance the [[projection formula]] $\overline{\gamma}$ in def. \ref{ComparisonMaps} for base change along a pointed connected type $\mathbf{B}G \to \ast$ equivalently says that genuine ([[Bredon cohomology|Bredon]]) [[equivariant cohomology]] reduces to [[Borel equivariant cohomology]] when the action on the coefficients is trivial. See at \emph{[[equivariant cohomology]]} for more on this.

\hypertarget{PointedObjects}{}\subsubsection*{{Pointed objects with smash product}}\label{PointedObjects}

A first step away from the Cartesian example \hyperlink{CartesianWirthmuellerContexts}{above} is the following.

Let $\mathbf{H}$ be a [[topos]]. For $X \in \mathbf{H}$ any [[object]], write

\begin{displaymath}
\mathcal{C}_X \coloneqq \mathbf{H}_{/X}^{X/}
\end{displaymath}
for the [[category of pointed objects]] in the [[slice topos]] $\mathbf{H}_{/X}$. Equipped with the [[smash product]] $\wedge_X$ this is a [[closed monoidal category|closed]] [[symmetric monoidal category]] $(\mathcal{C}_X, \wedge_X, X \coprod X)$.

\begin{prop}
\label{}\hypertarget{}{}
For $f \colon X \longrightarrow Y$ any [[morphism]] in $\mathbf{H}$, the [[base change]] [[inverse image]] $f^\ast$ restricts to a functor $f^\ast \colon \mathcal{C}_Y \longrightarrow \mathcal{C}_X$ which is a Wirthm\"u{}ller context.

\end{prop}
This appears as (\hyperlink{Shulman07}{Shulman 07, examples 12.13 and 13.7}) and (\hyperlink{Shulman12}{Shulman 12, example 2.33}).

\begin{proof}
For $f \colon X \longrightarrow Y$ any [[morphism]] in $\mathbf{H}$ then the [[base change]] [[inverse image]] $f^\ast \colon \mathbf{H}_{/Y} \longrightarrow \mathbf{H}_{/X}$ preserves pointedness, and the [[pushout]] functor $f_! \colon \mathbf{H}^{X/} \longrightarrow \mathbf{H}^{/Y}$ preserves co-pointedness. These two functors hence form an [[adjoint pair]] $(f_1 \dashv f^\ast) \colon \mathcal{C}_X \longrightarrow \mathcal{C}_Y$. Moreover, since [[colimits]] in the under-over category $\mathbf{H}_{/X}^{X/}$ are computed as colimits in $\mathbf{H}$ of [[diagrams]] with an [[initial object]] adjoined, and since by the [[Giraud axioms]] in the [[topos]] $\mathbf{H}$ [[pullback]] preserves these colimits, it follows that $f^\ast \colon \mathcal{C}_Y \to \mathcal{C}_X$ preserves colimits. Finally by the discussion at \emph{[[category of pointed objects]]} we have that $\mathcal{C}_X$ and $\mathcal{C}_Y$ are [[locally presentable categories]], so that by the [[adjoint functor theorem]] it follows that $f^\ast$ has also a [[right adjoint]] $f_\ast \colon \mathcal{C}_X \to \mathcal{C}_Y$.

To see that $f^\ast$ is a [[strong monoidal functor]] observe that the [[smash product]] is, by the discussion there, given by a [[pushout]] over [[coproducts]] and [[products]] in the [[slice topos]]. As above these are all preserved by [[pullback]]. Finally to see that $f^\ast$ is also a [[strong closed functor]] observe that the [[internal hom]] on [[pointed objects]] is, by the discussion there, a [[fiber product]] of cartesian internal homs. These are preserved by \hyperlink{CartesianWirthmuellerContexts}{the above case}, and the fiber product is preserved since $f^\ast$ preserves all limits. Hence $f^\ast$ preserves also the internal homs of pointed objects.

\end{proof}
\hypertarget{bundles_of_modules}{}\subsubsection*{{Bundles of modules}}\label{bundles_of_modules}

For $R$ a [[ring]], $R Mod$ its [[category of modules]], there is a [[functor]]

\begin{displaymath}
[-, R Mod] \;\colon\; Set^{op} \longrightarrow ClMonCat
\end{displaymath}
which sends a [[set]] $X$ to the [[closed monoidal category]] of $R$-[[modules]] parameterized over $X$.

This takes values in Wirthm\"u{}ller morphisms. (\hyperlink{Shulman12}{Shulman 12, example 2.2, 2.17}).

\hypertarget{bundles_of_modules_2}{}\subsubsection*{{Bundles of $\infty$-modules}}\label{bundles_of_modules_2}

for [[(infinity,1)-module bundles]]: ([[schreiber:master thesis Nuiten|Nuiten 13]], \hyperlink{HopkinsLurie}{Hopkins-Lurie}, [[schreiber:Homotopy-type semantics for quantization|Schreiber 14]])

\hypertarget{beckergottlieb_transfer}{}\subsubsection*{{Becker-Gottlieb transfer}}\label{beckergottlieb_transfer}

A \emph{[[transfer context]]} is a Wirthm\"u{}ller context in which also $f_\ast$ satisfies its [[projection formula]]. In this context there is an abstract concept of \emph{[[Becker-Gottlieb transfer]]}. See there for more.

\hypertarget{in_equivariant_stable_homotopy_theory}{}\subsubsection*{{In equivariant stable homotopy theory}}\label{in_equivariant_stable_homotopy_theory}

In [[equivariant stable homotopy theory]], see (\hyperlink{May05b}{May 05b}).

\hypertarget{ForQuasicoherentSheaves}{}\subsubsection*{{For quasicoherent sheaves (in $E_\infty$-geometry)}}\label{ForQuasicoherentSheaves}

Pull-push of [[quasicoherent sheaves]] is usually discussed as a [[Grothendieck context]] of [[six operations]], but under some conditions it also becomes a Wirthm\"u{}ller context.

Using results of Lurie this follows in the full generality of [[E-∞ geometry]] ([[spectral geometry]]).

Consider quasi-compact and quasi-separated [[E-∞ algebraic spaces]] ([[spectral algebraic spaces]]). (This includes precisely those [[spectral Deligne-Mumford stacks]] which have a [[scallop decomposition]], see \href{derived+Deligne-Mumford+stack#RelationToDerivedAlgebraicSpaces}{here}.)

If $f \;\colon\; X \longrightarrow Y$ is a map between these which is

\begin{enumerate}%
\item locally almost of finite presentation;


\item strongly proper;


\item has finite [[Tor-amplitude]]



\end{enumerate}
then the left adjoint to pullback of [[quasicoherent sheaves]] exists

\begin{displaymath}
(f_! \dashv f^\ast)
    \;\colon\;
  QCoh(X)
    \stackrel{\overset{f_!}{\longrightarrow}}{\underset{f^\ast}{\longleftarrow}}
  QCoh(Y)
  \,.
\end{displaymath}
([[Proper Morphisms, Completions, and the Grothendieck Existence Theorem|LurieProper, proposition 3.3.23]])

If $f$ is

\begin{itemize}%
\item quasi-affine

\end{itemize}
then the right adjoint exists

\begin{displaymath}
(f^\ast \dashv f_\ast)
    \;\colon\;
  QCoh(X)
    \stackrel{\overset{f^\ast}{\longleftarrow}}{\underset{f_\ast}{\longrightarrow}}
  QCoh(Y)
  \,.
\end{displaymath}
([[Quasi-Coherent Sheaves and Tannaka Duality Theorems|LurieQC, prop. 2.5.12]], [[Proper Morphisms, Completions, and the Grothendieck Existence Theorem|LurieProper, proposition 2.5.12]])

The [[projection formula]] in the dual form

\begin{displaymath}
f_\ast A \otimes B \longrightarrow f_\ast (A\otimes f^\ast B)
\end{displaymath}
for $f$ quasi-compact and quasi-separated appears as ([[Proper Morphisms, Completions, and the Grothendieck Existence Theorem|LurieProper, remark 1.3.14]]).

Now if all the conditions on $f$ hold, so that $(f_! \dashv f^\ast \dashv f_\ast) \;\colon\; QCoh(X) \longrightarrow QCoh(Y)$, then passing to opposite categories $QCoh(X)^{op} \longrightarrow QCoh(Y)^{op}$ exchanges the roles of $f_!$ and $f_\ast$, makes the projection formula be as in the above discussion and hence yields a Wirthm\"u{}ller context.

The existence of [[dualizing modules]] $K$

\begin{displaymath}
D X = [X,K]
\end{displaymath}
is discussed in ([[Representability theorems|Lurie, Representability theorems, section 4.2]].)

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

\begin{itemize}%
\item [[Verdier-Grothendieck context]]


\item [[Grothendieck context]]


\item [[Beck-Chevalley condition]]


\item [[ambidextrous adjunction]]



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

The Wirthmüller isomorphism in [[equivariant stable homotopy theory]] goes back to and is named after

\begin{itemize}%
\item [[Klaus Wirthmüller]], \emph{Equivariant homology and duality}. Manuscripta Math. 11(1974), 373-390

\end{itemize}
see Theorem II.6.2 in

\begin{itemize}%
\item [[L. Gaunce Lewis]], [[Peter May]], and Mark Steinberger (with contributions by J.E. McClure), \emph{Equivariant stable homotopy theory}, Springer Lecture Notes in Mathematics Vol.1213. 1986 (\href{http://www.math.uchicago.edu/~may/BOOKS/equi.pdf}{pdf})

\end{itemize}
see also

\begin{itemize}%
\item [[Denis Nardin]], section 2.6 and A.3 of \emph{Stability and distributivity over orbital ∞-categories}, 2012 (\href{http://hdl.handle.net/1721.1/112895}{hdl.handle.net/1721.1/112895}, \href{https://www.math.univ-paris13.fr/~nardin/thesis.pdf}{pdf})

\end{itemize}
A clear discussion of axioms of [[six operations]] and their consequences, with emphasis on the Wirthm\"u{}ller isomorphisms, is in

\begin{itemize}%
\item [[Halvard Fausk]], P. Hu, [[Peter May]], \emph{Isomorphisms between left and right adjoints}, Theory and Applications of Categories , Vol. 11, 2003, No. 4, pp 107-131. (\href{http://www.tac.mta.ca/tac/volumes/11/4/11-04abs.html}{TAC}, \href{http://www.math.uiuc.edu/K-theory/0573/FormalFeb16.pdf}{pdf})


\item [[Paul Balmer]], [[Ivo Dell'Ambrogio]], [[Beren Sanders]], \emph{Grothendieck-Neeman duality and the Wirthm\"u{}ller isomorphism}, Compositio Mathematica 152.8 (2016): 1740-1776 (\href{http://arxiv.org/abs/1501.01999}{arXiv:1501.01999})



\end{itemize}
Discussion of the original Wirthm\"u{}ller isomorphism in [[equivariant stable homotopy theory]], based on this, is in

\begin{itemize}%
\item [[Peter May]], \emph{The Wirthm\"u{}ller isomorphism revisited}, Theory and Applications of Categories 11.5 (2003): 132-142 (\href{http://www.tac.mta.ca/tac/volumes/11/5/11-05abs.html}{TAC:11/5/11-05}, \href{http://www.math.uiuc.edu/K-theory/0574/WirthRev.pdf}{K-theory:0574})

\end{itemize}
More elaboration of the Wirthm\"u{}ller context is in

\begin{itemize}%
\item Baptiste Calm\`e{}s, Jens Hornbostel, section 4 of \emph{Tensor-triangulated categories and dualities}, Theory and Applications of Categories, Vol. 22, 2009, No. 6, pp 136-198 (\href{http://www.tac.mta.ca/tac/volumes/22/6/22-06abs.html}{TAC}, \href{http://arxiv.org/abs/0806.0569}{arXiv:0806.0569})

\end{itemize}
Discussion in the context of [[pure motives]] includes

\begin{itemize}%
\item [[Frédéric Déglise]], around prop. 1.34 of \emph{Finite correspondences and transfers over a regular base}, \href{http://www.math.uiuc.edu/K-theory/0765/regular_base.pdf}{pdf}.

\end{itemize}
Wirthm\"u{}ller morphisms between pointed objects and between bundles of modules are discussed in

\begin{itemize}%
\item [[Mike Shulman]], \emph{Framed bicategories and monoidal fibrations}, in Theory and Applications of Categories, Vol. 20, 2008, No. 18, pp 650-738. (\href{http://arxiv.org/abs/0706.1286}{arXiv:0706.1286}, \href{http://www.tac.mta.ca/tac/volumes/20/18/20-18abs.html}{TAC})


\item [[Mike Shulman]], \emph{Enriched indexed categories} (\href{http://arxiv.org/abs/1212.3914}{arXiv:1212.3914})



\end{itemize}
Discussion in [[E-∞ geometry]] is in

\begin{itemize}%
\item [[Jacob Lurie]], section 3.3. of \emph{[[Proper Morphisms, Completions, and the Grothendieck Existence Theorem]]}


\item [[Jacob Lurie]], section 4.2 of \emph{[[Representability theorems]]}


\item [[Michael Hopkins]], [[Jacob Lurie]], \emph{[[Ambidexterity in K(n)-Local Stable Homotopy Theory]]}



\end{itemize}
[[!redirects Wirthmüller contexts]]

[[!redirects Wirthmueller context]] [[!redirects Wirthmueller contexts]]

[[!redirects Wirthmuller context]] [[!redirects Wirthmuller contexts]]

[[!redirects Wirthmüller isomorphism]] [[!redirects Wirthmüller isomorphisms]]

[[!redirects Wirthmueller isomorphism]] [[!redirects Wirthmueller isomorphisms]]

[[!redirects Wirthmuller isomorphism]] [[!redirects Wirthmuller isomorphisms]]



\end{document}