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Recently, U.Bunke and others developed in a number of papers (such as this, this or this) an approach to smooth extensions of cohomology theories based on stable homotopy theory. In this approach a differentiable refinement of a cohomology theory is a sheaf from the category of smooth manifolds $\mathbf{Mfd}$ to the category of spectra. Of course proving that in this way one gets a cohomology theory satisfying an analogue of the Eileneberg-Steenrod axioms.

My naive question is the following: can one make a similar construction for the category of complex manifolds (with holomorphic maps) and obtain some kind of analytic refinement of cohomology theories?

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    $\begingroup$ This would be neatly captured by what Urs Schreiber calls cohesive higher topos theory. Indeed his Habilitation is called "Differential cohomology in a cohesive $\infty$-topos" ncatlab.org/schreiber/show/…, but perhaps a more specific pertinent reference is arxiv.org/abs/1311.3188, which, together with the material at ncatlab.org/nlab/show/complex+analytic+%E2%88%9E-groupoid, should allow a differential holomorphic cohomology theory. $\endgroup$
    – David Roberts ♦
    Commented Nov 1, 2016 at 18:16
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    $\begingroup$ Yes. Essentially it suffices to observe that complex manifolds look like a polydisk locally, so homotopy-invariant sheaves are once again equivalent to spaces. The rest of the construction proceeds in the same way. For example, using holomorphic differential forms instead of smooth forms one recovers holomorphic Deligne cohomology. $\endgroup$
    – Dmitri Pavlov
    Commented Nov 1, 2016 at 22:58
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    $\begingroup$ @DmitriPavlov If you would expand your comment to an answer I'll able to vote and accept it. $\endgroup$
    – Andrei Halanay
    Commented Nov 2, 2016 at 17:55

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