nLab Wirthmüller context

Idea

A Wirthmüller context is a pair of two 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 $f^! \simeq f^\ast$:

(The other specialization of the six operations to $f_\ast \simeq f_!$ is called the Grothendieck context).

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

$$f_\ast 1_{X} \simeq f_! C \,.$$

If this induces a natural equivalence

$$f_\ast A \simeq f_!(A \otimes_X C)$$

for $A \in \mathcal{X}$, then one says this is a Wirthmüller isomorphism, following (Wirthmueller 74). In particular there is a canonical natural transformation

$$f_\ast A \longrightarrow f_!(A \otimes_X C)$$

and one can ask this to be an equivalence, hence a Wirthmüller isomorphism (May 05).

Definition

Defininition

Let $(\mathcal{X}, \otimes_X, 1_{X})$, $(\mathcal{Y}, \otimes_Y, 1_Y)$ be two symmetric closed monoidal categories and let

be an adjoint triple of functors between them. We call this setup

• a pre-Wirthmüller context if $f^\ast$ is a strong monoidal functor:

• a Wirthmüller context if $f^\ast$ is in addition a strong closed functor, hence a strongly closed monoidal functor.

(May 05, def. 2.12)

Definition (Notation)

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

$$\mathbb{D}A \coloneqq [A,1_X]$$

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, instead, to the (symmetric) monoidal category structure $\otimes_X$. If $A$ is dualizable we write $A^\vee$ for its monoidal dual object.

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

Remark

If all objects in $\mathcal{X}$ and $\mathcal{Y}$ are 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 linear logic and their type theory is linear type theory. In terms of this a Wirthmüller morphism, def. , is the linear analog of a context extension morphism in a hyperdoctrine: it defines a dependent linear type theory. See there for more.

Properties

The comparison maps

Remark

In a pre-Wirthmüller context, def. , there is a canonical natural transformation

$$f_!(A \otimes_X B) \longrightarrow (f_! A) \otimes_Y (f_! B) \,,$$

(not necessarily an equivalence) being the adjunct of the composite

$$A \otimes_X B \overset { \eta_A \otimes \eta_B } {\longrightarrow} (f^\ast f_! A) \otimes_X (f^\ast f_! B) \overset {\simeq} {\longrightarrow} f^\ast ( (f_! A) \otimes_Y (f_! B) ) \,,$$

where the first morphism is the tensor product of two copies of the adjunction unit and where the second is the equivalence that exhibits $f^\ast$ as a strong monoidal functor.

Definition

In a pre-Wirthmüller context, def. , write $\overline {\pi}$ for the natural transformation

$$\overline{\pi} \;\colon\; f_! ((f^\ast B) \otimes A) \longrightarrow B \otimes f_! A$$

given as the composite

$$\overline{\pi} \;\colon\; f_! \big( (f^\ast B) \otimes A \big) \overset { \widetilde{ \eta_{f^\ast B} \otimes \eta_A } } {\longrightarrow} (f_! f^\ast B) \otimes_Y (f_! A) \overset { \epsilon_B \otimes_Y id } {\longrightarrow} B \otimes f_! A \,,$$

where

• the first morphism is that of remark

• the second is the $(f_! \dashv f^\ast)$ counit (tensored with an identity morphism).

Also write

$$\overline{\gamma} \;\colon\; [f_! A ,\, B] \longrightarrow f_\ast [A ,\, f^\ast B]$$

for the $(f^\ast \dashv f_\ast)$ adjunct of the natural transformation given as the composite

(1)$$f^\ast [f_! A ,\, B] \stackrel{\simeq}{\longrightarrow} [f^\ast f_! A ,\, f^\ast B] \longrightarrow [A ,\, f^\ast B] \,,$$

where the first map exhibits $f^\ast$ as a closed monoidal functor and where the second is the $(f_! \dashv f^\ast)$-unit (under the internal hom).

see (May 05, prop. 2.11)

Proposition

In a pre-Wirthmüller context, def. , the comparison maps of def. 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.

(May 05, prop. 2.8 and prop. 2.11)

Proposition

Precisely when the pre-Wirthmüller context is a Wirthmüller context, def. , both comparison maps of def. are natural equivalences.

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

$$\array{ \mathcal{Y}(B \otimes f_! A, \, C ) & \stackrel{ \mathcal{Y}(\overline{\pi}(A,B), C) }{ \longrightarrow } & \mathcal{Y}(f_! ((f^\ast B) \otimes A),\, C) \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ \mathcal{X}(A, f^\ast [B,C]) &\stackrel{}{\longrightarrow}& \mathcal{X}(A, [(f^\ast B), (f^\ast C)]) } \,.$$

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. .

Remark

The first natural equivalence of prop.

$$f_!(f^\ast A \otimes B) \simeq A \otimes f_!(B)$$

is often called the projection formula. In representation theory this is also sometimes called 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.

Consequences

Assuming a Wirthmüller context, the projection formula has the following implications.

Proposition

In the special case that both of its variables are in the image of $f^\ast$, then composing the projection formula-map from Def. with the adjunction counit in the first variable equals the adjunction unit of the full term, in that the following diagram commutes (where we are leaving the structure isomorphism of the strong monoidal functor $f^\ast$ notationally implicit):

$$\array{ f_! \big( (f^\ast A) \otimes (f^\ast B) \big) &\overset{ \overline{\pi} }{\longrightarrow}& (f_! f^\ast A) \otimes B \\ \mathllap{{}^{\epsilon_{A \otimes B}}}\Big\downarrow && \Big\downarrow \mathrlap{ {}^{ \epsilon \otimes id } } \\ A \otimes B &=& A \otimes B }$$

Proof

The adjunct of the composite of the top and the right morphism is the composite of

• the adjunct of the top morphism, which by Def. , is

  $f^\ast A \otimes f^\ast B \overset{\eta_{f^\ast A} \otimes \eta_{f^\ast B}}{\longrightarrow} f^\ast f_! f^\ast A \otimes f^\ast f_! f^\ast B \overset{id \otimes f^\ast \epsilon_B}{\longrightarrow} f^\ast f_! f^\ast A \otimes f^\ast B$

• the image under $f^\ast$ of the right morphism, which is $f^\ast f_! f^ \ast A \otimes f^\ast B \overset{f^\ast \epsilon_A \otimes id}{\longrightarrow} f^\ast A \otimes f^\ast B$.

That composite is the identity by the triangle identity in each of the two variables.

In conclusion, the left vertical morphism is the adjunct of the identity, and hence the counit as claimed.

Proposition

The $(f_! \dashv f^\ast)$-adjunction counit on any object is, up to isomorphism given by the tensor product with the adjunction counit on the tensor unit:

$$\epsilon_{\mathscr{V}} \,\colon\, f_! f^\ast \mathscr{V} \,\simeq\, \mathscr{V} \otimes f_! f^\ast(\mathscr{V}) \overset{ id_{\mathscr{V}} \,\otimes\, \epsilon_{\mathbb{1}} }{\longrightarrow} \mathscr{V} \,\otimes\, \mathbb{1} \,\simeq\, \mathscr{V}$$

Proof

The identification is given by the following commuting diagram:

Here

1. the top square is the naturality square of the adjunction counit on the right unitor,

2. the bottom square is that from Prop. .

Comparison of push-forwards and Wirthmüller isomorphism

Corollary

In a pre-Wirthmüller context, def. , the functors $f_!$ and $f_\ast$ are intertwined by dualization, in that there is a natural equivalence

$$\mathbb{D}(f_! A) \simeq f_\ast(\mathbb{D} A) \,.$$

Proof

This is the special case of the property of $\overline{\gamma}$ in prop. for $B = 1_Y$:

$$\mathbb{D}(f_! A) = [f_! A, 1_Y] \underoverset{\simeq}{\overline{\gamma}}{\longrightarrow} f_\ast [A, f^\ast 1_Y] \simeq f_\ast [A, 1_Y] = f_\ast \mathbb{D} A \,.$$

Remark

With a Wirthmüller context regarded as categorical semantics for dependent linear type theory (see there), then the statement of cor. is an instance of de Morgan duality where linear dualization intertwines linear dependent sum $\sum$ and dependent product $\prod$

$$\prod_f \mathbb{D} \simeq \mathbb{D} \sum_f \,.$$

For more on this see also (Schreiber 14, section 3.3).

Example

In a pre-Wirthmüller context

$$f_\ast 1_X \simeq \mathbb{D}(f_! 1_X) \,.$$

Proof

By cor. , since $\mathbb{D} 1_X \simeq 1_X$.

Proposition

(Wirthmüller isomorphism)

In a Wirthmüller context, def. if $f_! 1_X$ is a dualizable object with dual $f_! C$, then there is a natural equivalence

$$\omega \;\colon\; f_\ast f^\ast A \stackrel{\simeq}{\longrightarrow} f_!((f^\ast A) \otimes C) \,.$$

(May 05, prop. 4.13).

Remark

In particular if there is $D \in \mathcal{Y}$ with

$$\mathbb{D}(f_! f^\ast 1_Y) \simeq f_! f^\ast D$$

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

$$\mathbb{D}(f_! f^\ast B) \stackrel{\simeq}{\longrightarrow} f_! f^\ast ((\mathbb{D}B) \otimes D) \,,$$

saying that the comonad $f_! f^\ast$ commutes with dualization up to a “twist” given by tensoring with $D$.

Examples

Cartesian Wirthmüller contexts in toposes

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

$$(\sum_f \dashv f^\ast \dashv \prod_f) \;\colon\; \mathbf{H}_{/X} \longrightarrow \mathbf{H}_{/Y}$$

the inverse image/context extension is a cartesian closed functor (see there for the proof). Therefore any base change of toposes constitutes a cartesian Wirthmüller context.

Conversely, this means that systems of Wirthmüller contexts are generalizations of categorical logic (hyperdoctrines) to non-cartesian contexts (see at dependent linear type theory).

Notice that in a cartesian Wirthmüller 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 Wirthmüller context is.

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

Pointed objects with smash product

A first step away from the Cartesian example above is the following.

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

$$\mathcal{C}_X \coloneqq \mathbf{H}_{/X}^{X/}$$

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

Proposition

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 Wirthmüller context.

This appears as (Shulman 07, examples 12.13 and 13.7) and (Shulman 12, example 2.33).

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 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 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.

Bundles of modules

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

$$[-, R Mod] \;\colon\; Set^{op} \longrightarrow ClMonCat$$

which sends a set $X$ to the closed monoidal category of $R$-modules parameterized over $X$.

This takes values in Wirthmüller morphisms. (Shulman 12, example 2.2, 2.17).

Bundles of $\infty$-modules

for (infinity,1)-module bundles: (Nuiten 13, Hopkins-Lurie, Schreiber 14)

Becker-Gottlieb transfer

A transfer context is a Wirthmüller context in which also $f_\ast$ satisfies its projection formula. In this context there is an abstract concept of Becker-Gottlieb transfer. See there for more.

In equivariant stable homotopy theory

In equivariant stable homotopy theory, see (May 05b).

For quasicoherent sheaves (in $E_\infty$-geometry)

Pull-push of quasicoherent sheaves is usually discussed as a Grothendieck context of six operations, but under some conditions it also becomes a Wirthmüller 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 here.)

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

1. locally almost of finite presentation;

2. strongly proper;

3. has finite Tor-amplitude?

then the left adjoint to pullback of quasicoherent sheaves exists

$$(f_! \dashv f^\ast) \;\colon\; QCoh(X) \stackrel{\overset{f_!}{\longrightarrow}}{\underset{f^\ast}{\longleftarrow}} QCoh(Y) \,.$$

(LurieProper, proposition 3.3.23)

If $f$ is

• quasi-affine

then the right adjoint exists

$$(f^\ast \dashv f_\ast) \;\colon\; QCoh(X) \stackrel{\overset{f^\ast}{\longleftarrow}}{\underset{f_\ast}{\longrightarrow}} QCoh(Y) \,.$$

(LurieQC, prop. 2.5.12, LurieProper, proposition 2.5.12)

The projection formula in the dual form

$$f_\ast A \otimes B \longrightarrow f_\ast (A\otimes f^\ast B)$$

for $f$ quasi-compact and quasi-separated appears as (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 Wirthmüller context.

The existence of dualizing modules $K$

$$D X = [X,K]$$

is discussed in (Lurie, Representability theorems, section 4.2.)

Related concepts

• Verdier-Grothendieck context

• Grothendieck context

• Beck-Chevalley condition

• ambidextrous adjunction

References

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

• Klaus Wirthmüller, Equivariant homology and duality. Manuscripta Math. 11(1974), 373-390

see Theorem II.6.2 in

• L. Gaunce Lewis, Peter May, and Mark Steinberger (with contributions by J.E. McClure), Equivariant stable homotopy theory, Springer Lecture Notes in Mathematics Vol.1213. 1986 (pdf)

see also

• Denis Nardin, section 2.6 and A.3 of Stability and distributivity over orbital ∞-categories, 2012 (hdl.handle.net/1721.1/112895, pdf)

A clear discussion of axioms of six operations and their consequences, with emphasis on the Wirthmüller isomorphisms, is in

• Halvard Fausk, Po Hu, Peter May, Isomorphisms between left and right adjoints, Theory and Applications of Categories, 11 4 (2003) 107-131 [tac:11-04, pdf]

• Paul Balmer, Ivo Dell'Ambrogio, Beren Sanders, Grothendieck-Neeman duality and the Wirthmüller isomorphism, Compositio Mathematica 152.8 (2016): 1740-1776 (arXiv:1501.01999)

Discussion of the original Wirthmüller isomorphism in equivariant stable homotopy theory, based on this, is in

• Peter May, The Wirthmüller isomorphism revisited, Theory and Applications of Categories 11.5 (2003): 132-142 (TAC:11/5/11-05, K-theory:0574)

More elaboration of the Wirthmüller context is in

• Baptiste Calmès, Jens Hornbostel, section 4 of Tensor-triangulated categories and dualities, Theory and Applications of Categories, Vol. 22, 2009, No. 6, pp 136-198 (TAC, arXiv:0806.0569)

Discussion in the context of pure motives includes

• Frédéric Déglise, around prop. 1.34 of Finite correspondences and transfers over a regular base, pdf.

Wirthmüller morphisms between pointed objects and between bundles of modules are discussed in

• Mike Shulman, Framed bicategories and monoidal fibrations, in Theory and Applications of Categories, Vol. 20, 2008, No. 18, pp 650-738. (arXiv:0706.1286, TAC)

• Mike Shulman, Enriched indexed categories (arXiv:1212.3914)

A twisted version of the Wirthmüller isomorphism is discussed in

• Bastiaan Cnossen, Twisted ambidexterity in equivariant homotopy theory (arXiv:2303.00736)

Discussion in E-∞ geometry is in

• Jacob Lurie, section 3.3. of Proper Morphisms, Completions, and the Grothendieck Existence Theorem

• Jacob Lurie, section 4.2 of Representability theorems

• Michael Hopkins, Jacob Lurie, Ambidexterity in K(n)-Local Stable Homotopy Theory