What is an étale covering

Question

I'm reading Buouville's book: Complex Algebraic Surfaces and in particular i'm interested about the part exposing nice properties of surfaces of general type.
During the lecture i'have found the term étale covering of a surface but on Bouville there's no references about that.
On the web there are lot of different defintions and i need something more precise. Any suggestion about references or some hint about this?

Answer 1

My understanding of étale topologies is from a viewpoint of categories, but I believe it will be the same situation for surfaces.

For a topology we need open sets, which is usually a nice natural object. But if you consider something such as a category, how could you go about defining open sets?

Grothendieck answered this for us, he introduced a whole class of topologies (Grothendieck topologies) and étale topology is one of these. A Grothendieck topology is given by a function $t$ which assigns to each object $U$ of a category $C$ a collection $t(U)$ consisting of families of morphisms $\{p_i:U_i \to U\}$ for $i$ in some index set such that it satisfies some properties. Then the families in $t(U)$ are called covering families for $U$.

So we can see it as letting the class of morphisms play the role of open sets in analogy to topological spaces.

Now, we have a generic Grothendieck topology, for the étale one we consider étale maps, i.e. flat and unramified maps. Therefore for the families $\{p_i:U_i \to U\}$ we require $p_i$ to be étale maps!