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I am wondering about some basic issues in higher geometry and specifically in E-infinity geometry (βspectral geometryβ), maybe somebody has a hint for me:
in Structured Spaces we get the very general formulation of what it means to have a structure sheaf πͺX. In Deformation Theory we get the very general formulation of what it means to be a module. One would hope that in this generality there is a definition of quasicoherent πͺX-modules, too. But at least in Quasi-Coherent Sheaves and Tannaka Duality Theorems the definition of QCoh is given in a more traditional way. Is there anything on QCoh away from Eβ-geometry in a more general context as in Structured Spaces?
What can we say about sufficient conditions for morphisms f:XβY in Eβ-geoemtry to induce closed monoidal f*:QCoh(Y)βQCoh(X)? I donβt care for the moment so much about X and /or Y being regular (schemes, algebraic spaces, DM-stacks,β¦) but am looking for answers for X,Y generally β-presheaves on CRingopβ.
for E an Eβ-algebra and X an β-groupoid, Pic(QCoh(Xβ SpecE)) is closely related to the twisted generalized E-cohomology of the homotopy type of X. Joost and I are currently playing with rephrasing the constructions in his thesis in terms of six operations yoga in Eβ-geometry under this identification. Part of this is at least implicit in A Survey of Elliptic Cohomology. Has this been developed/spelled out further anywhere?
I donβt know the answers, but I would like you just to have in mind that in generalized algebraic geometry of Nikolai Durov thesis where the local model is based on a finitary monad in Set (very fundamental and basic setup, everything commutative), the categories of quasicoherent sheaves of πͺ-modules are not forming an abelian category, hence the infinity analogue would not form a stable (infinity,1)-category. Sufficiently general treatment should have this in mind.
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