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Algebraic geometry and analytic geometry are closely related (witness GAGA). But the latter still seems much "bigger" than the former. I'd like to be able to get from algebraic geometry to analytic geometry in many small steps rather than what feels to me like one giant leap. Sort of like how we can interpolate between differential and continuous topology by looking at categories of $C^{(k)}$ manifolds.

Question 1: Are there interesting categories of "spaces" interpolating between the world of algebraic geometry and the world of analytic geometry?

Here's an attempt to be a bit more precise. Let $Alg$ be some category of nice schemes over $\mathbb C$ (if there's something interesting to be said about how the nonarchimedean or non-algebraically-closed case looks, I'd also be interested to known about that.), and let $An$ be some category of nice analytic spaces over $\mathbb C$. Then does there exist any kind of filtration $Alg = \mathcal C^0 \to \mathcal C^1 \to \cdots \to \mathcal C^\omega = An$ where the $\mathcal C^i$ are categories of spaces of an intermediate algebraic/analytic nature, and there are "inclusion" functors $\mathcal C^i \to \mathcal C^{i+1}$?

I suspect that the question is highly sensitive to what precisely is meant by a "nice" scheme or a "nice" analytic space. Indeed, with enough "niceness" conditions, GAGA should tell us that we already have $Alg = An$. Unfortunately, I am too ignorant to formulate a more precise version of the question.

Question 2: Alternatively, is there some natural collection of categories where $An$ is "terminal" and $Alg$ is "initial"?

Here I have in mind some vague idea of schemes where the functions are "closed under differentiation" in some sense: $An$ might then be the case where we have as many functions as possible ("generated coinductively", maybe), while $Alg$ is where we have as few functions as possible.

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    $\begingroup$ An interesting adjacent question is how to define "half-analytic, half-topological" (or "half-AG, half-topological") spaces. $\endgroup$
    – Pulcinella
    May 10 at 19:22
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    $\begingroup$ It would be probably interesting to make your question more precise. What do you mean by "bigger"? And by "analytic" geometry? At least in the layman's mind, there is no way the study of schemes over $\mathbb{Z}$ is a subarea of analytic geometry. $\endgroup$
    – Libli
    May 10 at 19:55
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    $\begingroup$ One type of object that has turned out to be very fruitful in recent years is o-minimal geometry. I suppose this is more between algebraic geometry and real analytic geometry (rather than complex analytic). But you throw in more and more functions: the exponential makes the universal cover $\mathbf C \to \mathbf C^\times$ defined in your theory; together with real analytic functions on compact domains this also gives things like the Weierstrass $\wp$-function if I understand correctly; and altogether enough to seriously say new things about period maps in Hodge theory. $\endgroup$
    – R. van Dobben de Bruyn
    May 10 at 19:55
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    $\begingroup$ @M.G. You can require to have fields of fractions being differentially finitely generated over complex numbers. Differential Galois theory is a well-researched area, which studies locally ringed spaces with such properties. $\endgroup$
    – Denis T
    May 10 at 21:41
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    $\begingroup$ @M.G. Another interpolation is given in condensed mathematics: the analytic complex line is an open subset of the algebraic complex line, and there are larger open subsets. Details can be found in Lectures on Condensed Mathematics and Complex Geometry, and roughly speaking, it is putting conditions on the divergence of functions at infinity. $\endgroup$
    – Z. M
    May 11 at 6:19

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