Someone recently asked if one can talk about a map being etale dense just like one can talk about it being Zariski dense. My main question is: has anyone discussed such a notion?
On a simple approximation, this might seem to mean that a map $X \to Y$ is etale dense if for any etale open $U \to Y$, the scheme $X \times_Y U$ is nonempty. One thing to note is that this is tied specifically to the small (rather than big) site and doesn't even use the concept of covering (just the underlying category of the small etale site).
But I also wonder if there's any kind of notion like this in the category of sites. In topological spaces, a map is dense if it is an epimorphism in the category of Hausdorff topological spaces. Is there an analogue for sites?
Ultimately, the motivation for talking about etale dense instead of Zariski dense is that an algebraic space doesn't have (AFAIK) a small Zariski site, but it does have a small etale site.
