Varieties given by non-algebraic equations In algebraic geometry one (mostly) studies varieties given by polynomial equations.
Such equations define algebraic varieties and there are many "dictionaries" available.
For example, the category of rings is anti-equivalent to the category of affine schemes, etc.
What if we enlarge our realm of possible equations to equations of the form
$x^y+y^z = z^x$ over the rational numbers.
These also define "varieties" which are no longer algebraic. It doesn't define a scheme, I think, but can we define a locally ringed space or something similar or maybe more general to it?
Sorry for the vagueness. I just have the feeling that sometimes one should enlarge their category in order to get their hands on what's really going on.

It sounds like you're talking about a complex analytic space, sometimes called an analytic variety.  It is locally defined by the vanishing of a holomorphic function in some $\mathbb C^n$.  For instance $\mathbb Z$ is not an algebraic subvariety of $\mathbb C$ but it is an analytic subvariety because it is the zero set of $\sin \pi z$.
