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Extending the push–pull framework to derived smooth stacks

Construct a well-behaved representing object Q^⊗ for derived smooth stacks that determines the 2- and 3-morphisms and admits pushforwards, thereby enabling the push–pull spans construction (Theorem 1.2/2.7) in derived differential geometry.

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Background

While the paper develops the theory in derived algebraic geometry, the authors note that derived differential geometry (derived smooth manifolds) is another natural setting for Rozansky–Witten theory. Their main construction applies provided one has an appropriate choice of Q⊗ with pushforwards to serve as the local system representing object.

The authors explicitly leave the production of such a Q⊗ in the smooth setting to future work, identifying it as a concrete step needed to generalize their push–pull (∞,3)-category framework beyond the algebraic context.

References

However, we want to stress that Theorem 3.12 is general enough that it can be applied to derived smooth stacks, given a well-behaved choice of Q⊗ — the representing object determining the 2- and 3-morphisms. We leave the task of producing such a Q⊗ to future work.

— Higher categories of push-pull spans, I: Construction and applications (2404.14597 - Riva, 22 Apr 2024) in Section 4 (Applications of push–pull spans), Subsection 4.1 (Background on derived algebraic geometry)