Sheaves and streams II: sheafification, and stratified étale maps

In part I, I explained how one can build the étale space of a presheaf F over a topological space X. I will show how one can retrieve a sheaf from an étale map, leading to a nice adjunction and its associated monad, sheafification. This is all well-known, but then I would like to apply all that to the presheaf of locally monotone functions of a prestream, which we had already started to examine last time. We will obtain a funny structure that I will call stratified étale maps. Read the full post.