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      10
      votes
      0answers
      684 views

      Visualization and new geometry in higher stacks

      I am trying to develop a geometrical intuition for "higher spaces", i.e. both in the sense of higher dimensional spaces (more than three dimensions) and in the sense of abstractions beyond manifolds ...
      6
      votes
      1answer
      268 views

      Sheaves over a sheaf

      Everything I write I mean in the in the sense of Lurie's HTT. Suppose that $ \mathcal{C}$ be a site and let $ F \in Fun( \mathcal{C}^{op} , \mathcal{S})$. Is it always/ever true that $ Sh(\mathcal{C}...
      4
      votes
      0answers
      344 views

      The lisse-etale site and derived algebraic geometry

      If one reads say Olsson's book on algebraic stacks or Laumon-Moret-Bailly. The lisse-etale topology is used to define quasi-coherent sheaves and the cotangent complex (or rather cutoff's of the ...
      1
      vote
      0answers
      102 views

      Localization of a 2-category

      I am looking for a basic reference about localization of 2-categories, possibly avoiding the full formalism of n-categories.
      6
      votes
      0answers
      121 views

      Dense (∞,1)-subsites

      So if $C$ is a 1-site and $D$ is a subsite (with the induced coverage), there are some conditions that ensure that the pre-composition and right Kan extension functors yield an equivalence of ...
      3
      votes
      0answers
      182 views

      What is a proper n-etale morphism?

      Let $Y$ be a complex algebraic variety, and let $n\in \mathbb Z_{\geq 1}\cup \{\infty\}$. How do I think about a proper $n$-etale morphism $X\to Y$? If $n=1$, I think this should be a finite etale ...
      6
      votes
      0answers
      141 views

      How to realize the descent data of Qcoh as a (pseudo)-limit in Cat?

      It is well-know that $Qcoh$ is a fibered category on $Sch$. In more details let $\mathcal{C}$ be the category $(Sch/S)$ of schemes over a fixed base scheme S. For each scheme $U$ we define $Qcoh(U)$ ...
      7
      votes
      0answers
      374 views

      Derived Functors, Projection Formula and Base Change in the Derived $\infty$-Category

      Let $X$ be a smooth stack and $\mathcal O_X$ the ring of smooth functions on $X$, i.e. for any smooth $M \to X$, $\mathcal O_X(M \to X) = C^\infty(M)$. In HigherAlgebra, the derived category $\...
      23
      votes
      3answers
      4k views

      Conjectures in Grothendieck's “Pursuing stacks”

      I read on the nLab that in "Pursuing stacks" Grothendieck made several interesting conjectures, some of which have been proved since then. For example, as David Roberts wrote in answer to this ...
      6
      votes
      1answer
      375 views

      Why are sheaves not preserved in this case?

      Suppose that $C$ is a Grothendieck site, and $\mathscr{X}$ is a stack over $C$ (which is NOT equivalent to a sheaf). Let $$\pi_{\mathscr{X}}:\int_{C} \mathscr{X}\to C$$ denote the associated fibered ...
      8
      votes
      1answer
      373 views

      Separation condition for higher Deligne-Mumford stacks

      Let $X$ be a stack of $n$-groupoids on the site of affine schemes over a fixed base, with the etale topology. If $n=1$ then for $X$ to be Deligne-Mumford, aside from having an etale atlas from an ...
      2
      votes
      1answer
      354 views

      Classification of principal G-bundles over a differentiable stack

      According to "Notes on differentiable stacks" by Heinloth, the classifying stack will also classify $G$-bundles on stacks. (Remark 2.13) (Here $G$ is a Lie group.) My questions are: (1) What is ...
      6
      votes
      1answer
      334 views

      The plus construction for stacks of n-types

      In Jacob Lurie's Higher Topos Theory, Section 6.5.3, he briefly mentions that to stackify a presheaf of $n$-groupoids, one needs to apply the "+"-construction $\left(n+1\right)$ times, and in general, ...
      2
      votes
      1answer
      351 views

      Weak colimits of weak and strict presheaves in groupoids

      Let $C$ be a small category, and for this question, let groupoid mean an (essentially small) groupoid. There are two 2-categories in question: the 2-category of strict presheaves in groupoids and ...
      8
      votes
      1answer
      758 views

      When is a stack (NOT) geometric?

      Following the terminology of $n$-Lab, a geometric stack $\mathcal{X}$ on a site $\mathcal{(C,J)}$ is a stack for which there exists a representable epimorphism $X \to \mathcal{X}$ from an object $X$ ...

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