Special and general types
A Wirthmüller context is a pair of two symmetric closed monoidal categories , 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 and and the tensor product/internal hom adjunctions , specialized to the case that :
(The other specialization of the six operations to is called the Grothendieck context).
Often one is interested in the case that there is an object and an equivalence
If this induces a natural equivalence
for , then one says this is a Wirthmüller isomorphism, following (Wirthmueller 74). In particular there is a canonical natural transformation
and one can ask this to be an equivalence, hence a Wirthmüller isomorphism (May 05).
Let , be two symmetric closed monoidal categories and let
be an adjoint triple of functors between them. We call this setup
(May 05, def. 2.12)
The comparison maps
In a pre-Wirthmüller context, def. 1, write for the natural transformation
given as the composite
where the first morphism is that of remark 2 and where the second is the counit (tensored with an identity).
for the adjunct of the natural transformation given as the composite
where the first map exhibits as a closed monoidal functor and where the second is the -unit (under the internal hom).
see (May 05, prop. 2.11)
In a pre-Wirthmüller context, def. 1, the comparison maps of def. 3 are equivalences when the argument 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)
Precisely if the pre-Wirthmüller context is a Wirthmüller context, def. 1, are both comparison maps of def. 3 are natural equivalences.
For all and we have by the -adjunction and the tensorhom-adjunction a commuting diagram of the form
By naturality in and by the Yoneda lemma this shows that is an equivalence precisey if is strong closed.
For the same statement follows from this with prop. 1.
Comparison of push-forwards and Wirthmüller isomorphism
In a pre-Wirthmüller context, def. 1, the functors and are intertwined by dualization, in that there is a natural equivalence
This is the special case of the property of in prop. 1 for :
In a pre-Wirthmüller context
By cor. 1, since .
In a Wirthmüller context, def. 1 if is a dualizable object with dual , then there is a natural equivalence
(May 05, prop. 4.13).
Cartesian Wirthmüller contexts in toposes
For a topos and any morphism, then in the induced base change etale geometric morphism
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 for all objects . 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 in def. 3 for base change along a pointed connected type 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 be a topos. For any object, write
for the category of pointed objects in the slice topos . Equipped with the smash product this is a closed symmetric monoidal category .
For any morphism in , the base change inverse image restricts to a functor which is a Wirthmüller context.
This appears as (Shulman 07, examples 12.13 and 13.7) and (Shulman 12, example 2.33).
For any morphism in then the base change inverse image preserves pointedness, and the pushout functor preserves co-pointedness. These two functors hence form an adjoint pair . Moreover, since colimits in the under-over category are computed as colimits in of diagrams with an initial object adjoined, and since by the Giraud axioms in the topos pullback preserves these colimits, it follows that preserves colimits. Finally by the discussion at category of pointed objects we have that and are locally presentable categories, so that by the adjoint functor theorem it follows that has also a right adjoint .
To see that 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 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 preserves all limits. Hence preserves also the internal homs of pointed objects.
Bundles of modules
For a ring, its category of modules, there is a functor
which sends a set to the closed monoidal category of -modules parameterized over .
This takes values in Wirthmüller morphisms. (Shulman 12, example 2.2, 2.17).
Bundles of -modules
for (infinity,1)-module bundles: (Nuiten 13, Hopkins-Lurie, Schreiber 14)
A transfer context is a Wirthmüller context in which also 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 -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 is a map between these which is
locally almost of finite presentation;
has finite Tor-amplitude?
then the left adjoint to pullback of quasicoherent sheaves exists
(LurieProper, proposition 3.3.23)
then the right adjoint exists
(LurieQC, prop. 2.5.12, LurieProper, proposition 2.5.12)
The projection formula in the dual form
for quasi-compact and quasi-separated appears as (LurieProper, remark 1.3.14).
Now if all the conditions on hold, so that , then passing to opposite categories exchanges the roles of and , makes the projection formula be as in the above discussion and hence yields a Wirthmüller context.
The existence of dualizing modules
is discussed in (Lurie, Representability theorems, section 4.2.)
The construction goes back to and is named after
- Klaus Wirthmüller, Equivariant homology and duality. Manuscripta Math. 11(1974), 373-390
A clear discussion of axioms of six operations and their consequences, with emphasis on the Wirthmüller isomorphisms, is in
Halvard Fausk, P. Hu, Peter May, Isomorphisms between left and right adjoints, Theory and Applications of Categories , Vol. 11, 2003, No. 4, pp 107-131. (TAC, pdf)
Paul Balmer, Ivo Dell'Ambrogio, Beren Sanders, Grothendieck-Neeman duality and the Wirthmüller isomorphism, arXiv:1501.01999.
Discussion of the Wirthmüller isomorphism in equivariant stable homotopy theory, based on this, is in
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
Discussion in E-∞ geometry is in