Description:

Title: Elements of Parametrized Sheaf Theory

Abstract: This dissertation develops the foundations for a particular approach to parameterized sheaf theory. We introduce a generalization of sheaf theory that allows the coefficient category to be a local system on the underlying geometric structure.

Given a monodromic ∞-topos X and a functor A from its shape Π∞(X) to PrL, we define the category of twisted sheaves on X valued in A. In particular, this allows us to define twisted sheaves on any locally contractible topological space. This construction recovers ordinary sheaf theory when the coefficient functor A is constant. Furthermore, its subcategory of locally constant objects recovers the category of twisted spectra of Douglas and Hedelund-Moulinos.

Finally, we establish the basic functorial properties of this construction and, to demonstrate that it behaves as expected, we prove the monodromy and exodromy equivalences for twisted sheaves.

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