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Let $Sh(\mathsf{\mathbb{C}-fAlg}^{op})$ be the topos of zariski sheaves on finitely genertaed $\mathbb{C}$-algebras.

A complex analytic space for our purpose is a locally ringed space locally isomorphic to an analytic subset of $\mathbb{C}^{n}$ (with the sheaf of holomorphic functions). Denote the category of these by $\mathsf{An}$ (for analytic). There's a grothendiek topology on the category given by jointly sujective open immersions (where open immersion are topological embeddings with locally isomorphic structure sheaves). Let $Sh(An)$ be the topos of analytic sheaves on analytic spaces. We have an affine analytification functor:

$$f: \mathsf{\mathbb{C}-fAlg}^{op} \to \mathsf{An}$$

Which associates to an affine $\mathbb{C}$ scheme the corresponding analytic subspace of $\mathbb{C}^n$. This morphism induces a pushforward for presheaves:

$$f_*: Psh(\mathsf{An}) \to Psh(\mathsf{\mathbb{C}-fAlg}^{op})$$

I'm afraid to say anything about sheaves at this point since I'm not so comfortable with toposes yet really. What I'd like to be able say is the following:

(Extremely conjectural!) The pushforward $f_*$ extends to a morphism of topoi:

$$F_*: Sh(\mathsf{An}) \to Sh(\mathsf{\mathbb{C}-fAlg}^{op})$$

This morphism is geometric with left adjoint the analytification functor:

$$F^*: Sh(\mathsf{\mathbb{C}-fAlg}^{op}) \to Sh(\mathsf{An})$$

In partuicular this functor sends schemes in $Sh(\mathsf{\mathbb{C}-fAlg}^{op})$ to analytic spaces.

Is there a way to make this precise?

I'm trying to see the analytification functor come out of general nonsense because it seems like the existence of it should be natural.

I'm obviously not talking about any powerful statements about properties of this functor like GAGA etc. I'm only looking for a natural way to define this functor.

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  • $\begingroup$ The existence of the geometric morphism follow from general non-sense: if your functor $f$ send covering to covering and is flat, then it defines a geometric morphism (see ncatlab.org/nlab/show/morphism+of+sites ). The flatness is not completely obvious as the categories involved don't have all finite limit, but I'm convinced it is not very hard to check. I am not sure how is defined the analytification functor in this framework so I cannot answer the question more, but maybe it has some kind of universal properties equivalent to the fact that it is indeed $f^*$ ? $\endgroup$ – Simon Henry Apr 7 '16 at 8:18
  • $\begingroup$ Oh wait ! I just realized you started with all finitely presented algebras, without any restriction. So your category does have all finite limits. So the flatness condition is equivalent to the fact those limit are preserved by the functor that attach the analytic space, which is easier to check than flatness. $\endgroup$ – Simon Henry Apr 7 '16 at 8:22
  • $\begingroup$ @SimonHenry Well that's super easy. So basically since $f$ sends zariski covers to analytic covers and preserves finite limits it defines the geometric morphism above. In particular now $f^*$ being a left adjoint preserves colimits so schemes are sent to analytic spaces. That's it? seems to good to be true... $\endgroup$ – Saal Hardali Apr 7 '16 at 8:47
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    $\begingroup$ I don't deal with analytic spaces, but this kind of thing is considered in my thesis. Preservation of finite limits is not necessary if you are willing to give up having a geometric morphism. $\endgroup$ – Zhen Lin Apr 7 '16 at 9:13
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I am not very familiar with the analytic side of the pictures or with the analytification functor but here is what I can claim, it seems from your comment that this answer your question:

If you have two subcanonical site $C$ and $D$ (it means that representable presheaves are sheaves, so it is the case with your example).

If $C$ has all finite limits.

If you have a functor $f :C \rightarrow D$ that preserves finite limits and send covering family in $C$ to covering family in $D$ then:

the precompostion with $f : Prsh(D) \rightarrow Prsh(C)$ send sheaves over $D$ to sheaves over $C$ and the restriction of this functor to sheaves defines the $f_*$ part of a geometric morphism:

$f_*: sh(D) \rightarrow sh(C)$

Moreover (easy to check from the universal property) $f^*$ send representable sheaves in $sh(C)$ to representable sheaves in $sh(D)$ in a way that extend $f: C \rightarrow D$.

You have more general claim of this sort on the nLab page morphism of sites

Or in more standard litterature, have a look to MacLane&Moerdijk "sheaves in geometry and logic" chapter VII, more specifically setion 9 and 10.

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