Sheaves and homotopy theory
This is an incomplete draft. The goal was to explain the connections between sheaf theory and homotopy theory from a very naive point-of-view, and to show how one can be led naturally to Jardine's model category on simplicial presheaves, as well as Morel and Voevodsky's homotopy theory of schemes. The main point is that the category of sheaves is a solution to a certain universal problem (that of co-completing a category in a sensible way), and that simplicial presheaves is a solution to a similar universal problem. Jardine's category of simplicial presheaves on a site C to some extent represents the `universal homotopy theory' built from C, subject to the `relations' that certain diagrams become homotopy colimits. One purpose of the paper is to make this precise.
As I said, this is an incomplete draft. I stopped writing when I began to realize that the paper had gotten so long that no one was ever going to want to read it. Most of the new ideas have now appeared (with less exposition) in the `Universal homotopy theories' paper, and to some extent in `Hypercovers and simplicial presheaves'.