Call a category $C$ *rigid* if every equivalence $C \to C$ is isomorphic to the identity. I don't know if this is standard terminology. Most of the usual algebraic categories is rigid, for example sets, semigropus, (abelian) groups, rings, but also the category of topological spaces. See [M] for a survey and the general strategy for proving rigidity, the case of rings was recently discussed on MO [here][1]. The philosophy is that a category is rigid if every object can be defined in a categorical way, which is a quite interesting property.
 
**Question**: Is the category of schemes rigid?

Here is what I've done so far: The initial scheme is $\emptyset$ and the terminal scheme is $\text{Spec}(\mathbb{Z})$. Spectra of fields are characterized by the property that they are non-initial and and every morphism from a non-unitial object to them is an epimorphism, see Kevin's answer [here][2]. The underlying set $|X|$ of a scheme is the set of equivalence classes of morphisms $Y \to X$, where $Y$ is the spectrum of a field. So this recovers $|X|$ from $X$ in a categorical manner. If $x \in |X|$, then $\text{Spec}(\kappa(x))$ is the terminal spectrum of a field which maps to $X$ and has (set) image $x$.

However, I'm not able to recover the topology from $X$. I don't know how to characterize open or closed immersions. They are exactly the étale resp. proper monomorphisms, see [this][3] MO question, but it seems to be hard to characterize étale and propert categorically. After all, if we were able to characterize affine schemes, then we would be done, since the category of affine schemes is rigid and every scheme is the canonical colimit of the affine schemes mapping into it.
 
In order to characterize affine schemes, it is enough to characterize the ring object $\mathbb{A}^1_\mathbb{Z}$ in the category of schemes, since we can then define the ring of global sections of a scheme categorically and then say that affine schemes $Y$ are characterized by the property that for all schemes $X$ the map $Hom(X,Y) \to Hom(\mathcal{O}(Y),\mathcal{O}(X))$ is bijective.

Any ideas concerning the categorical characterization of other properties / objects are appreciated.
 
[M] E. Makai jun, *Automorphisms and Full Embeddings of Categories in Algebra and Topology*, [online][4]


  [1]: http://mathoverflow.net/questions/55931/invariance-of-zx-under-a-self-equivalence-of-the-category-of-commutative-ring/56183#56183
  [2]: http://mathoverflow.net/questions/56564/what-are-the-epimorphisms-in-the-category-of-schemes
  [3]: http://mathoverflow.net/questions/56591/what-are-the-monomorphisms-in-the-category-of-schemes
  [4]: http://www.heldermann.de/R&E/RAE18/ctw13.pdf