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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeDec 5th 2013
   • (edited Dec 9th 2013)
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   Author: Urs
   Format: MarkdownItexI am wondering about some basic issues in higher geometry and specifically in [[E-infinity geometry]] ("spectral geometry"), 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 $\mathcal{O}_X$. 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 $\mathcal{O}_X$-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 $E_\infty$-geometry in a more general context as in _[[Structured Spaces]]_? * What can we say about sufficient conditions for morphisms $f : X \to Y$ in $E_\infty$-geoemtry to induce closed monoidal $f^\ast : QCoh(Y) \to QCoh(X)$? I don't care for the moment so much about $X$ and /or $Y$ being regular (schemes, algebraic spaces, DM-stacks,...) but am looking for answers for $X,Y$ generally $\infty$-presheaves on $CRing_\infty^{op}$. * for $E$ an $E_\infty$-algebra and $X$ an $\infty$-groupoid, $Pic(QCoh(X \cdot Spec E))$ 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 [[schreiber:master thesis Nuiten|his thesis]] in terms of six operations yoga in $E_\infty$-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 am wondering about some basic issues in higher geometry and specifically in E-infinity geometry (“spectral geometry”), 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 $\mathcal{O}_X$. 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 $\mathcal{O}_X$-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 $E_\infty$-geometry in a more general context as in Structured Spaces?

   • What can we say about sufficient conditions for morphisms $f : X \to Y$ in $E_\infty$-geoemtry to induce closed monoidal $f^\ast : QCoh(Y) \to QCoh(X)$? I don’t care for the moment so much about $X$ and /or $Y$ being regular (schemes, algebraic spaces, DM-stacks,…) but am looking for answers for $X,Y$ generally $\infty$-presheaves on $CRing_\infty^{op}$.

   • for $E$ an $E_\infty$-algebra and $X$ an $\infty$-groupoid, $Pic(QCoh(X \cdot Spec E))$ 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 $E_\infty$-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?

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeDec 5th 2013
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   Author: zskoda
   Format: MarkdownItexI don't 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 $\mathcal{O}$-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.

   I don’t 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 $\mathcal{O}$-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.