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am starting complex analytic infinity-groupoid (in line with “smooth infinity-groupoid” etc.) and higher complex analytic geometry. Currently there is mainly a pointer to Larusson. To be expanded.
added the actual statement of Larusson’s $\infty$-presheaf-formulation of the Oka condition here (or at least the first part of it)
Cohesive over $\infty Grpd$?
No, that’s the thing. Instead we should take this to be the base topos and have cohesion over it, as at smooth E-infinity-groupoid. I would tend to say “smooth analytic $\infty$-groupoid” for objects in
$Sh_\infty(CartSp, Sh_\infty(SteinSp)) \to Sh_\infty(SteinSp)$but maybe that’s too ambiguous and I should instead say “analytically-fibered smooth infinity-groupoid” or something like this. Not sure yet.
I think I have been really confused about this. It is cohesive, isn’t it.
Take the site of complex polydiscs with its canonical coverage. That’s a dense subsite of that of complex manifolds and of that of Stein spaces, so the hypercomplete $\infty$-toposes will all agree. Now since the coverage by polydiscs is completed by the standard Grothendieck topology, a simplicial presheaf satisfies descent with respect to hypercovers in the complex analytic site precisely if it does so already for hypercovers which are degreewise coproducts of polydiscs. (Unless I am mixed up.) The would-be $\Pi$-functor sends such to the simplicial set obtained by replacing each polydisc by a point, hence to the “complex analytic étale homotopy type” of the polydisc. But that is contractible (because forgetting the complex structure it is in particular the homotopy type of the underlying manifold, hypercovered by open balls). So from this point on the argument for cohesion proceeds as that for smooth manifolds.
I have promoted the argument that $\mathbb{C}Analytic \infty Grpd$ is cohesive to a numbered proposition (here) and added a pointer to the Examples section in the entry on cohesive $\infty$-toposes.
One naturally feels a certain urge to use the forgetful functor $CplxMfd \to SmthMfd$ to induce some geoemtric morphism between $\mathbb{C}Analytic \infty Grpd$ and $Smooth\infty Grpd$, but I am not sure if it works…
Now I see that just this statement is also Hopkins-Quick 12, lemma 2.3 + prop. 2.4 + lemma 2.5 + prop. 2.6. I have added corresponding pointers in the entry.
Had many of the consequences of $\mathbb{C}Analytic \infty Grpd$ being cohesive already been observed? If you find so many of the indications of cohesive (infinity,1)-topos – structures in a candidate for cohesion, can this ever tell you it is cohesive?
So I was trying to understand the following, but so far with little success:
the forgetful functor
$ComplexManifolds \longrightarrow SmoothManifolds$induces an adjunction (not a geometric morphism, though) between the toposes
$ComplexAnalytic \infty Grpd \longrightarrow Smooth \infty Grpd$in order to “include” smooth geoemtry into complex geometry, one might look at the factorization of this adjunction through its induced reflection, as discussed at idempotent monad – The associated idempotent monad .
This gives a reflective subcategory
$Something \hookrightarrow ComplexAnalytic \infty Grpd$being the localization of complex analytic $\infty$-groupoids at those maps which become equivalences when forgetting the complex analytic structure and remembering just the smooth structure.
Is there anything useful to say about this $Something$? I am not sure. Also, I doubt that it is a good thing to consider.
One should probably try something else. Not sure yet what.
I don’t see why you wouldn’t get a geometric morphism. The “inclusion” of complex manifolds into smooth manifolds is cover-reflecting (i.e. has the cover lifting property in the sense of Mac Lane and Moerdijk) so you should get a covariantly-induced geometric morphism. This is certainly true for 1-toposes.
Thanks for mentioning this!
(I have now created covering lifting property, which used to be missing on the nLab, and pointed to it from morphism of sites).
So in the case at hand we have an essential geometric morphism, since the functor also preserves covers.
For certain complex analytic spaces, the action of the Galois group of the complex field over the real field acts by conjugation on each plot can be patched up to a conjugation involution on the said complex analytic space. Examples are complex affine spaces, complex projective spaces, some complex tori and most (all?) complex algebraic groups.
In this way, real spaces provide a common context for complex analytic spaces equipped with a conjugation involution and smooth spaces equipped with the trivial involution.
There is something particularly interesting about ${\mathrm{Something}}$. Namely, I wish to look at the complex affine line ${\mathbb{C}}$. The conjugation involution of ${\mathbb{C}}$ is not holomorphic, but is a smooth equivalence (diffeomorphism). More explicitly, for each complex analytic space $X$, there is its conjugate complex analytic space $\bar{X}$, of which it is anti-biholomorphic to. In particular, there are two distinct (i.e. not biholomorphic) complex analytic spaces ${\mathbb{C}}$ and $\bar{{\mathbb{C}}}$. However, these two complex analytic spaces are indistinguishable in ${\mathrm{Something}}$. One reason to “include” smooth spaces is in other to define, say $(1,1)$-forms. Is there already an internal definition of $(1,1)$-forms and/or anti-holomorphic derivatives?
Is there currently a page or terminology to refer to a sheaf of sets over the site of Stein manifolds? Just 1-geometry, not higher geometry.
I don’t think so. Please create it!
@Colin I don’t see why not…
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