Basically my question is:
Suppose I meet an alien mathematician which understands everything through category theory and category theory alone. How would I convince said mathematician that certain cohomology computations are more meaningful than others?
Let $C \to P := Psh(C)$ be a category embedded inside its presheaf topos.
One has the following diagram for abelian group objects in $C$ and $P$ respectively:
$$\mathsf{AbC \to AbP \to Ab}$$
Where the last morphism comes from the morphism to the terminal topos.
The right arrow is the global sections for presheaves and is a functor between abelian categories. A priori we don't know that $\mathsf{AbC}$ is abelian (we expect it not to be in general) so the composition isn't something we can derive to get cohomology.
In order to define cohomology (so that in particular we get cohomology for abelian group objects) it is reasonable to look for reflective subtoposes of $P$ containing $C$ (image of the yoneda).
Several questions arise in this context:
- What is the name for the smallest reflective subtopos of $P$ containing $C$? This should be like the topos of sheaves for the canonical topology - but i'm told that there doesn't always exist a sheafification functor. Is this a pathology which can be rectified by cleverly choosing a "smaller" subcategory of $C$?
- Can we ensure that the embedding $AbC \to T$ (with $T$ our chosen subtopos) preserve finite colimits and limits which already exist in $AbC$? Is this a good thing to ask for if we want to identify good meaningful topologies?
My hope is that given any subcanonical topology on $C$ one can identify the exact sequences in $AbC$ which are preserved by the embedding into the corresponding topos and perhaps treat this subcategory as the one for which the cohomological computations are meaningful.
Examples:
- I've heard it said that the zariski topology is enough for coherent sheaves, does this mean that the embedding from abelian cones ("total spaces" of coherent sheaves) over a scheme to zariski sheaves preserves all exact sequences?
- I've heard it said that etale is enough for smooth group schemes. Does this mean that the embedding from smooth group schemes over a scheme to etale sheaves preserves exact sequences?
I suspect that both points I made above are overly simplistic but I'd very much like to understand this issue.
From an entirely different directions we have for a given topology all the sheafifications of the constant sheaves living in $P$ and these have cohomology groups as well. A priori it seems like all the topologies are on equal footing in this regard.
