Generalities on sheaves - Where can I find the technology that can make this “proof” of Atiyah duality precise?

Fix $R$ an $E_{\infty}$ ring spectrum which admits a "six functor formalism" over a suitable class of spaces (by which I mean a context in which what I'm about to say can be made correct).

Let $X$ be a manifold and consider the projection $X \to pt$. Now replace the point by the homotopically equivalent embedding $j: X \to \mathbb{R}^n$ for some $n$. Denote by $i:U \to \mathbb{R}^n$ the inclusion of the complement. Now consider the exact triangle in local cohomology for the constant sheaf $R$ on $\mathbb{R}^n$. We get:

$$j_{!}j^! R \to R \to i_*i^*R$$

The LHS (I hope) may be interpreted as the Thom spectra of the normal bundle of the embedding via a tubular neighborhood argument. If everything so far makes sense an $R$-orientation on the normal bundle is precisely choosing a trivilization $D_R(X):=j^! R\cong R_X$ which is precisely what's needed for Poincare duality for cohomology with coefficients in $R$. The right hand side of this triangle is confusing and I have no idea how it relates to story of orientations and Thom spectra (I mean, it looks like cohomology of the unit sphere bundle with coefficients in R but that's not really what I need...).

The rest of the story is also blurred for me. Almost entirely due to the fact that I have almost no idea about any of six functors and I'm basically flying blind. Is there any modern source treating the theory of (locally constant) sheaves of spectra in an $\infty$ categorical way?

• If you want to work with "six functors" you shouldn't use locally constant sheaves of spectra. Of the functors you write down, only $i^\ast$ preserves the property of being locally constant. – Dan Petersen Jun 29 '17 at 13:49
• I asked a related question a while ago: mathoverflow.net/questions/170319 . My understanding now is: yes, there 'should' exist a fully fledged six functors formalism for sheaves of spectra, generalizing the usual six functors formalism for sheaves of abelian groups. But no part of it exists in the literature. – Dan Petersen Jun 29 '17 at 13:56
• @DanPetersen Thanks for the comment. By "locally constant" I meant define the category of sheaves to be the category of $\infty$ functors $Fun(X,Sepctra)$ sorry if this was unclear (I should have said presheaves probably). There's the book by may and Sigurdsson about this but it doesn't use higher categories so it's a bit difficult to decipher. Do you think that in this general setting one would have an upper shriek $f^!$ without any restrictions on $f$? It seems that some kind of fiberwise finiteness condition should be necessary... I hope you're wrong about the non-existence. – Saal Hardali Jun 29 '17 at 15:27
• Right - what I'm saying is that $\mathrm{Fun}(X,\mathrm{Spectra})$ morally corresponds to locally constant sheaves, not to arbitrary sheaves of abelian groups. And if you want a six functors formalism satisfying the usual properties (like a distinguished triangle for local cohomology) then you had better work with arbitrary sheaves of spectra, ie not the setting of the book of May-Sigurdsson. This is articulated in Ben Wieland's very nice answer to the question I linked. – Dan Petersen Jun 29 '17 at 15:38