Global spaces and the homotopy theory of stacks

Adrian Clough; Bastiaan Cnossen; Sil Linskens

Global spaces and the homotopy theory of stacks

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arxiv 2407.06877 v3 pith:OKPEL4TU submitted 2024-07-09 math.AT

Adrian Clough , Bastiaan Cnossen , Sil Linskens This is my paper · ORCID

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keywords inftycategoryglobalhomotopysheavesspacesstackstopos

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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
math.AT 2024-12 conditional novelty 7.0 of 10

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.

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