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brief paragraph at Dolbeault cohomology
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 $Hom_{\mathcal{E}}(X, D_{p, q})$?
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.
For some discussion along these lines see also page 2 of Differential cohomology is Cohesive homotopy theory (schreiber).
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 $X$.
No, you are to consider the $\infty$-topos over the site of all complex manifolds, see at complex analytic β-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
$\mathcal{H}^{p,q}(X) \simeq H^q(X, \Omega^p) \simeq \pi_0 \mathbf{H}(X, \mathbf{B}^q \Omega^p) \,.$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?
No, itβs 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.
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