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      5
      votes
      1answer
      338 views

      Universal property of sheaf category

      Given a site $C$ with a Grothendieck topology and the category of presheaves $P(C)$ (either in the sense of presheaves of sets or in the $\infty$-sense), and the category $S(C)$ of sheaves with ...
      2
      votes
      0answers
      109 views

      Compact generation of quasicoherent sheaves on mapping stack

      Let $k$ be a field of characteristic $0$, and let $\mathcal{C}= \mathbf{Vect}_k^{\leq 0}$ be the $\infty$-category of vector spaces concentrated in degrees $\leq 0$. Consider the category $\mathbf{Pr}(...
      16
      votes
      1answer
      669 views

      A sheaf is a presheaf that preserves small limits

      There is a common misconception that a sheaf is simply a presheaf that preserves limits. This has been discussed here before many times and I believe I understand it well enough. However when reading ...
      63
      votes
      2answers
      5k views

      What is Homology anyway?

      Disclaimer: I don't feel qualified to ask this question and yet it's been troubling me for some time now and I lost my patience and decided to ask to get some kind of answer. If there are any stupid ...
      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
      194 views

      “2-Sheafification” with Values in non $Cat$ categories?

      Let $X$ be a 2-site and consider the category of 2-presheaves over $X$, which will be denoted as $Pshv(X;Cat)$. These are $Cat$-valued 2-functors, where $Cat$ is the 2-category of categories. There is ...
      12
      votes
      2answers
      538 views

      A bit of history of Verdier duality

      I was wondering who originated the presentation of Verdier duality as an equivalence between categories of sheaves and cosheaves ? I learnt it reading Jacob Lurie's Higher Algebra and Justin Curry's ...
      6
      votes
      1answer
      399 views

      Hypercovers of sheaves in classical and quasi-categories

      I am interested in relating the definition of hypercovers in the $\infty$-topos of sheaves on an $\infty$-Grothendieck site to the classical definition of hypercovers of presheaves on a Grothendieck ...
      9
      votes
      2answers
      613 views

      Is the site of (smooth) manifolds hypercomplete?

      By site of manifolds Man, I mean the category of manifolds (maybe submanifolds to obtain a small category) with continuous maps between them. A Grothendieck topology is given by open covers. Actually, ...
      1
      vote
      1answer
      415 views

      Descent of Morphisms of Sheaves

      While reading Brylinski I am trying to understand the descent of morphisms of sheaves. In trying to form a new definition of a presheaf $A$ over a space $X$, we associate to each surjective local ...

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