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?


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)


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.