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created brief entries Wirthmüller context and Grothendieck context, following Peter May’s terminology for the two special cases where four of the Grothendieck six operations specialize to an adjoint triple.
The main thing I’d like to record is lists of classes of examples that realize either of these contexts. But haven’t gotten around to that yet.
added to Grothendieck context pointers to the original articles by Grothendieck, Deligne, Verdier and the more recent nice one by Neeman on $f^!$ along proper maps of schemes.
added to Wirthmüller context a pointer to prop. 3.3.23 Lurie’s, Proper Morphisms, Completions, and the Grothendieck Existence Theorem which discusses the left adjoint $f_!$ along suitable morphisms of spaces in E-infinity geometry.
(Thanks to Joost Nuiten for pointers.)
What I am after is more of that $E_\infty$-geometric (“spectral geometry”) six operations yoga. Can anyone help me with further pointers or insights?
added discussion of some basic properties to Wirthmüller context, following May’s text (in particular the abstract statement of the Wirthmüller isomorphism) in a new section Properties
I have expanded a bit more at Wirthmüller context with the intention to make it clearer what depends on which assumption. To that end I say now for an adjoint triple $(f_! \dashv f^\ast \dashv f_\ast)$ between closed symmetric monoidal cats
in addition to May’s
This to better disentangle where the closedness condition appears and what it is equivalent to.
disentable = disentangle I guess.
Yes, thanks, fixed.
Is there a nice nPOV story on what the six operations are for? Perhaps that’s what’s being asked for in #2. Is $E_\infty$-geometric the most general setting?
Is there a nice nPOV story on what the six operations are for?
Yes, so that’s the reason for my questions in the other thread.
Suppose we are in a higher geometric context which pairs some geometry with $E_\infty$-geometry (such as smooth E-∞-groupoids).
Then for $X$ any geometric object and $E$ an $E_\infty$-ring, then a line bundle ($\mathbb{G}_m$-principal bundle) on $X \times Spec(E)$ corresponds to a $E$-(infinity,1)-module bundle over $X$.
Under this identification, one finds that definition 4.1.24 in Joost Nuiten’s thesis, used for “cohomological quantization by pull-push”, is essentially about the Wirthmüller isomorphism in a Wirthmüller context in such smooth $E_\infty$-geometry. This is secretly the content of remark 2 at Wirthmüller context.
Generally, six-operations Yoga is at its heart about characterizing how in the presence of two different pullbacks $f^!$, $f^\ast$ or in the presence of two different push-forwards $f_\ast$, $f_!$, one may differ from the other only by a composite of dualizing and tensoring with a correction factor. For instance as in that remark 2.
Now interpreted in $E_\infty$-geometry as above, this becomes identified with another famous construction: the dualization becomes the Pontryagin-duality map from generalized $E$-cohomology to $E$-homology which is used to push-forward cohomology, and the tensoring with the correction factor becomes the cohomological twist induced by the Atiyah dual Thom spectrum arising from possible non-$E$-orientability.
Hence when interpreted in smooth $E_\infty$-geometry then the prescription of “motivic quantization” becomes equivalently an exercise in six operations yoga.
The $n$Lab used to state the fact that the inverse image of base change between toposes is cartesian closed only at cartesian closed functor. I have now added there the remark that this means that every base change of toposes constitutes a cartesian Wirthmüller context. Then I added brief remarks on both these statements to a few related entries, such as closed functor, base change, etale geometric morphism and made it a brief Examples-section at Wirthmüller context.
Have written out a further class of examples at Wirthmüller context – Examples – On pointed objects.
I’d like to add to the list of examples at Wirthmüller context
(and maybe more generally smooth maps, not sure yet about the degree twists pan out in that case).
I still need to go through this in more detail to confirm. But I gather that if $f \;\colon\; X \longrightarrow Y$ is an étale map of (irreducible?) algebraic varieties, then there is an adjoint triple
$(f_! \dashv f^\ast \dashv f_\ast) \;\colon\; D_{coh}(X) \longrightarrow D_{coh}(Y) \,.$Moreover, generally the projection formula holds in the form $A \otimes f_\ast B \stackrel{\simeq}{\longrightarrow} f_\ast ((f^\ast A) \otimes B)$.
This should be an “op-Wirthmüller context”, in that it becomes one after passing to opposite categories and opposite functors.
Zhen Lin seems to be energetic about these kinds of questions at the moment. Any help from him or anyone in confirming and/or ironing out subtleties will be appreciated. (I wish I had time to look into this in more leisure at the moment…)
I added the reference
to Wirthmüller context, Grothendieck context, and six operations. Though I haven’t read the paper so I don’t know if it actually fits on all these pages.
By the way, Urs, you are probably aware, but Gaitsgory has written about the six operations in the dg-setting, for quasi-coherent sheaves, ind-coherent sheaves, and D-modules. It’s not quite the spectral/E_infinity setting but it’s the only thing in that direction that I’m aware of.
Thanks for adding this reference! Yes, I am aware that Gaitsgory has notes on this, it was pointed out to me when I spoke about this in Paris in November. But I didn’t find time yet to look into it. If you have pointer to relevant document and page easily available, maybe you could add a further link to the $n$Lab entry? Thanks!!
I added a reference to the page six operations.
Now I realize that I never checked the following, which however would be the first thing to check:
A Wirthmüller isomorphism in the generalized sense of Fausk-Hu-May 05 induces operations of pull-push through spans, along the lines discussed at indexed monoidal (infinity,1)-category.
If we specialize this pull-push construction to the Wirthmüller isomorphism in the original and special sense, concerning left and right adjoints to the restriction-of-groups-functor in equivariant stable homotopy theory, we get from genuine equivariant spectra a pull-push of their fixed point spectra through spans in the orbit category.
This special case of the general pull-push construction induced by generalized Wirthmüller isomorphisms ought to give just the spectral Mackey functor corresponding to the given genuine $G$-spectrum.
Has this been made explicit anywhere?
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