I'm just learning some basics of model categories, so please forgive me if my question turns out to be trivial. I hope it does at least make sense.

A natural temptation is to relate this machinery to birational geometry; in particular one would like to find a model category structure having the birational morphisms as weak equivalences. More precisely it would be nice to have such a model structure on the category $Sch_k$ of schemes of finite type over a field $k$.

A natural problem arises: a model category is required by definition to have all small limits and colimits, and $Sch_k$ does not satisfy this. For limits the situation is not that bad. I believe the original work of Quillen required only the existence of *finite* limits and colimits. Since $Sch_k$ has finite products and fiber products, it has all finite limits.

On the other hand finite colimits need not exist. A simple way to see this is to realize that categorical quotients by equivalence relations do not always exist in $Sch_k$, and these are just some coequalizers. So my questions are:

Is there a canonical way to enlarge a category to add finite limits?

If this is the case, what do we obtain when applying this to $Sch_k$? The resulting category would have to contain algebraic spaces, as these arise as quotients of schemes by étale equivalence relations. How much bigger would it be?

Assuming one has a decent notion of birational morphism for these objects: is there a model structure on the enlarged category such that birational morphisms are the weak equivalences?

someonehas thought about what happens if you formally invert the class of birational maps in the category schemes, and decided that what you get isn't very interesting. I actually thought that had been asked here before, but I can't find it. $\endgroup$ – Charles Rezk Jun 29 '10 at 18:28