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?