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In May and Sigurdsson's Parameterized Homotopy Theory, Proposition 2.2.11, four isomorphisms of functors are given. For a pullback square of base spaces $C=holim(A\overset{f}\to B\overset{j}\leftarrow D)$, with induced maps $C\overset{g}\to D$ and $C\overset{i}\to A$, there are natural isomorphisms:

$$(1)~~j^\ast f_!\cong g_!i^\ast~~~(2)~~f^\ast j_\ast \cong i_\ast g^\ast~~(3)~~f^\ast j_!\cong i_!g^\ast~~~(4)~~ j^*f_\ast\cong g_*i^*$$

Where for a map $f$ of base spaces, $f^\ast$ is the usual pullback of a parameterized space, $f_\ast$ is its right adjoint, and $f_!$ is its left adjoint.

Do these same natural equivalences hold for parameterized spectra over a pullback diagram of spaces? In particular, I am interested in equivalence $(3)$ above. And even more specifically, I am interested in comparing the two sides when $A$ above is a contractible space. In other words, $C$ is the homotopy fiber of the map $D\to B$. In that case, for a bundle of sphere spectra over $B$, we have that $i_!g^\ast$ yields the Thom spectrum of the pullback. How then should we think of $f^\ast j_!$ of the same bundle of sphere spectra?

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Yes, the Beck-Chevalley condition holds for parameterized spectra. A proof appears as prop.4.3.3 in

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

Some chat related to this is also around example 5.6 in my note "Quantization via Linear homotopy types" (arXiv:1402.7041)

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