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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 2nd 2012
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   Author: Urs
   Format: MarkdownItexbrief paragraph at _[[Dolbeault cohomology]]_

   brief paragraph at Dolbeault cohomology

2. • CommentRowNumber2.
   • CommentAuthorColin Tan
   • CommentTimeJun 28th 2016
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   Author: Colin Tan
   Format: MarkdownItexRecall the philosophy which interprets cohomology as the homset in a (oo,1)-topos. Has such an interpretation been found for Dolbeault cohomology? Namely, is there an (oo,1)-topos $\mathcal{E}$ where the compact Kaehler manifolds form a subcategory, such that, for each p, q, there exists a classifying object $D_{p, q}$ such that the Dolbeault cohomology $H^{(p,q)}(X)$ is naturally isomorphic to the homset of the mapping space $Hom_{\mathcal{E}}(X, D_{p, q})$?

   Recall the philosophy which interprets cohomology as the homset in a (oo,1)-topos. Has such an interpretation been found for Dolbeault cohomology?

   Namely, is there an (oo,1)-topos $\mathcal{E}$ where the compact Kaehler manifolds form a subcategory, such that, for each p, q, there exists a classifying object $D_{p, q}$ such that the Dolbeault cohomology $H^{(p,q)}(X)$ is naturally isomorphic to the homset of the mapping space $Hom_{\mathcal{E}}(X, D_{p, q})$?

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJun 28th 2016
   • (edited Jun 28th 2016)
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   Author: Urs
   Format: MarkdownItexBy the _[[Dolbeault theorem]]_ (see there) Dolbeault cohomology is equivalently the [[abelian sheaf cohomology]] $H^q(X;\Omega_X^p)$, of the [[abelian sheaf]] $\Omega_X^p$ which is the [[Dolbeault complex]] of [[holomorphic p-forms]]. And all abelian sheaf cohomology theories are given by hom-spaces in an $\infty$-topos.

   By the Dolbeault theorem (see there) Dolbeault cohomology is equivalently the abelian sheaf cohomology $H^q(X;\Omega_X^p)$, of the abelian sheaf $\Omega_X^p$ which is the Dolbeault complex of holomorphic p-forms.

   And all abelian sheaf cohomology theories are given by hom-spaces in an $\infty$-topos.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJun 28th 2016
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   Author: Urs
   Format: MarkdownItexFor some discussion along these lines see also page 2 of _[[schreiber:Differential cohomology is Cohesive homotopy theory]]_.

   For some discussion along these lines see also page 2 of Differential cohomology is Cohesive homotopy theory (schreiber).

5. • CommentRowNumber5.
   • CommentAuthorColin Tan
   • CommentTimeJun 28th 2016
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   Author: Colin Tan
   Format: TextThanks Urs!

   Thanks Urs!
6. • CommentRowNumber6.
   • CommentAuthorColin Tan
   • CommentTimeJul 1st 2016
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   Author: Colin Tan
   Format: MarkdownItexAs I understand it, the abelian sheaf cohomology $H^q(X;\Omega^p_X)$ is interpreted as $\Pi_0 Hom(X, \mathbf{B}^q \Omega^p_X)$, where the Hom is taken in the oo-topos $Sh_{(\infty,1)}(X)$ of simplicial sheaves over $X$. Thus, as $X$ varies, the oo-topos $Sh_{(\infty,1)}(X)$ in consideration also varies. This unlike ordinary cohomology of topological spaces where there is a single object $\mathbf{B}^q A$ in a single oo-topos $\infty Grpd$ whereby $\pi_o Hom(X, \mathbf{B}^q A) \simeq H^q(X ; A)$ gives the ordinary cohomology for all topological spaces $X$.

   As I understand it, the abelian sheaf cohomology $H^q(X;\Omega^p_X)$ is interpreted as $\Pi_0 Hom(X, \mathbf{B}^q \Omega^p_X)$, where the Hom is taken in the oo-topos $Sh_{(\infty,1)}(X)$ of simplicial sheaves over $X$. Thus, as $X$ varies, the oo-topos $Sh_{(\infty,1)}(X)$ in consideration also varies.

   This unlike ordinary cohomology of topological spaces where there is a single object $\mathbf{B}^q A$ in a single oo-topos $\infty Grpd$ whereby $\pi_o Hom(X, \mathbf{B}^q A) \simeq H^q(X ; A)$ gives the ordinary cohomology for all topological spaces $X$.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeJul 1st 2016
   • (edited Jul 1st 2016)
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   Author: Urs
   Format: MarkdownItexNo, you are to consider the $\infty$-topos over the site of _all_ complex manifolds, see at _[[complex analytic ∞-groupoid]]_ $$ \mathbf{H} \coloneqq \mathbb{C}Analytic \infty Grpd \coloneqq Sh_\infty(CplxMfd) \,. $$ Then given any particular complex manifold $X$, it represents an object in that $\infty$-topos, and its Dolbeault cohomology is $$ \mathcal{H}^{p,q}(X) \simeq H^q(X, \Omega^p) \simeq \pi_0 \mathbf{H}(X, \mathbf{B}^q \Omega^p) \,. $$

   No, you are to consider the $\infty$-topos over the site of all complex manifolds, see at complex analytic ∞-groupoid

   $$\mathbf{H} \coloneqq \mathbb{C}Analytic \infty Grpd \coloneqq Sh_\infty(CplxMfd) \,.$$

   Then given any particular complex manifold $X$, it represents an object in that $\infty$-topos, and its Dolbeault cohomology is

   $$\mathcal{H}^{p,q}(X) \simeq H^q(X, \Omega^p) \simeq \pi_0 \mathbf{H}(X, \mathbf{B}^q \Omega^p) \,.$$
8. • CommentRowNumber8.
   • CommentAuthorColin Tan
   • CommentTimeJul 2nd 2016
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   Author: Colin Tan
   Format: MarkdownItexHere $\mathbf{B}^q\Omega^p$ is not a single object in $Sh_\infty (CplxMfd)$, but really $\mathbf{B}^q\Omega_X^p$ (which depends on $X$) right?

   Here $\mathbf{B}^q\Omega^p$ is not a single object in $Sh_\infty (CplxMfd)$, but really $\mathbf{B}^q\Omega_X^p$ (which depends on $X$) right?

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeJul 2nd 2016
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   Author: Urs
   Format: MarkdownItexNo, it's a single object. $\Omega^p \in Sh_\infty(CpxMfd)$ is the sheaf on the site of all complex manifolds which assigns to any one its additive group of holomophic $p$-forms. The Yoneda lemma says that then $$ \mathbf{H}(X, \Omega ^p) \simeq \Omega^p_X(X) $$ and the claim about Dolbeault cohomology follows similarly.

   No, it’s a single object. $\Omega^p \in Sh_\infty(CpxMfd)$ is the sheaf on the site of all complex manifolds which assigns to any one its additive group of holomophic $p$-forms.

   The Yoneda lemma says that then

   $$\mathbf{H}(X, \Omega ^p) \simeq \Omega^p_X(X)$$

   and the claim about Dolbeault cohomology follows similarly.

10. • CommentRowNumber10.
    • CommentAuthorColin Tan
    • CommentTimeJul 2nd 2016
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    Author: Colin Tan
    Format: TextAh! Thanks for your clarifications, Urs.

    Ah! Thanks for your clarifications, Urs.