Consider the functor $\pi_1: \text{Closed Surfaces} \rightarrow \textbf{Grp}$. This is homotopy invariant; every closed topological surface has a unique fundamental group. In the reverse direction, by classification, if two closed surfaces have isomorphic fundamental group, then the surfaces themselves are homeomorphic. This fails if we move up to $3$-manifolds, and fails spectacularly for dimension $\geq 4$. So I'll restrict my questions to small dimensions.
Question: is there a functor $\text{Complex Curves} \rightarrow \textbf{Grp}$ that has the same properties as before? (In this case, it should be conformally invariant, but the preimages of a given group should be conformally equivalent.) I doubt it, if only because there are continuum many complex structures on a torus, and it seems unlikely that something which can be continuously varied like this will give me a nice algebraic invariant that can differentiate these structures. Bonus question: if this is provably false, is it provably false for rings as well? (Feel free to let rings have unity or not and morphisms take unity to unity or not.)
Edit: Zhen Lin noted in the comments below that the Yoneda embedding to PreSheaf works fine. I've restricted now to groups and rings. I want there to be no algebraic invariant in the spirit of, say, homotopy groups or the cohomology ring or the group of line bundles that does the job for us.
With a closed Riemann surface $\Sigma$ of genus $g>0$ there is naturally associated a pair $(\Lambda, H)$, where $H$ is a $g$-dimensional complex vector space and $\Lambda\subset H$ is a full rank lattice (an abelian subgroup isomprphic to $\mathbb Z^{2g}$).
For $H$ you can take $H^1_{DR}(\Sigma; \mathbb R)$ (the degree 1 DeRham cohomology of $\Sigma$ with real coefficients). This is a $2g$-dimensional real vector space, if you just consider the differential structure on $\Sigma$. The complex structure on $\Sigma$ gives $H$ the structure of a complex vector space of dimension $g$. For $\Lambda\subset H$ you can take the cohomology classes with integer "periods" (=integrals along oriented closed curves in $\Sigma$).
Another natural definition of $H$ is the space of holomorphic 1-forms on $\Sigma$ (but it is then less obvious why it is a $g$-dimensional complex vector space).
If $f:\Sigma\to\Sigma'$ is a conformal diffeomorphism, then the induced map $f^*:H^1_{DR}(\Sigma';\mathbb R)\to H^1_{DR}(\Sigma;\mathbb R)$ is a complex linear isomorphism such that $f^*(\Lambda_{\Sigma'})= \Lambda_{\Sigma}.$
It follows that the map $[\Sigma]\mapsto [(\Lambda,H)]$ is well-defined (the brackets idicate equivalence classes with respect to the obvious equivalence relations).
The map $[\Sigma]\mapsto [(\Lambda, H)]$ is injective (a complete invariant), but for $g>1$ it is not surjective. The image is an open subset of the set of all equivalence classes of pairs $(\Lambda, H)$, given by the so-called Riemann bilinear relations.
Reference: Griffith-Harris, Principles of Algebraic Geometry.