Question

It might be a not very specific problem. I just wanna know how much do we rely on the property of "base change closed". In the definition of Grothendieck pretopology, we require a collection of morphisms satisfied the "invariant under base change"

As far as I know, in noncommutative algebra. The flat morphism does not respect to base change in general. I heard from some guys in category theory telling me that the "topology" which satisfied the "base change closed" is called "square topology"(I am not sure whether I spell it in correct way).

My question is: if we do not have the property of "invariant under base change". How much algebraic geometry can we recover? Or Is there a big difference from classcial algebraic geometry?

In fact, I know some work of Kontsevich-Rosenberg on noncommutative algebraic geometry dealing with the topology which is lack of base change property. And they called "quasi-topology". I just wonder know if there are other people in category theory or in algebraic geometry ever considered this question.

Thanks in advance!

Answer 1

We want invariance under base change because we want to be able to pull back covering sieves. This is pretty much the entire point of a Grothendieck topology. Without the base-change axiom, we only have the local converse, if the pullback by every map of a covering sieve of a sieve S is a covering sieve, then S is a covering sieve. Base-change corresponds to the ability to pull back open covers of topological spaces by preimages of continuous maps.