Why do we say "This functor is left exact, but not right exact" instead of "This functor preserves limits, but not colimits". It seems more natural to base the theory of derived functors on the second statement instead of the first. I've never really understood what it meant to say that the global sections functor is left exact. From looking at the definition on the wikipedia page, it says given an exact sequence $$0 \to A \to B \to C \to 0,$$ the functor is left exact when $$0 \to F(A) \to F(B) \to F(C)$$ is exact. However, if you consider the category of open sets on a space $X$, do exact sequences make sense in this case? $0$ is supposed to be a zero object, which means it is both initial and terminal. I don't think the category of open sets has that.
Also, are the following equivalent?
- $F$ preserves limits/colimits
- $F$ is left exact/right exact
- $F$ is right adjoint/left adjoint
Edit Number One: I think I have a better understanding now. I think has more to do with preserving arbitrary limits/colimits vs finite limits/colimits. If a functor is right adjoint, then it preserves arbitrary limits. From Goldbatt's Topoi book, I see that the definition of left exact is for the functor to preserve finite limits. So, do we lose flavor of the theory of Derived Functors when we say the functor preserves arbitrary limits/colimits instead of the restricted finite limits/colimits? What if we say that the functor is left adjoint/right adjoint?
Edit Number Two: Should the wiki article on Exact Functors be edited? Comparing to the article given over at nlab, I think the wiki article does a poor job. It seems to me that the wiki article gives the definition via short exact sequences, and then derives the "preserving limits/colimits" fact. This approach however assumes that the category has a zero object, and the source of some of my confusion. The nlab approach is better; it gives the definition of left exact functor as preserving finite limits.