Using six-functor sheaf theory, the Bauer-Furuta invariant is defined as the proper pushforward f_* f^!(1), with f^!(1) computed as the Thom spectrum of the family index.
Global spaces and the homotopy theory of stacks
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abstract
We show that the $\infty$-category of global spaces is equivalent to the homotopy localization of the $\infty$-category of sheaves on the site of separated differentiable stacks, following a philosophy proposed by Gepner-Henriques. We further prove that this $\infty$-category of sheaves is a cohesive $\infty$-topos and that it fully faithfully contains the singular-cohesive $\infty$-topos of Sati-Schreiber.
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On the Bauer--Furuta construction
Using six-functor sheaf theory, the Bauer-Furuta invariant is defined as the proper pushforward f_* f^!(1), with f^!(1) computed as the Thom spectrum of the family index.