There is a notion of relative cohomology in both algebraic topology and algebraic geometry and the flexibility granted by this relative viewpoint is quite powerful in both cases. However, the two notions of relativeness are different:
In algebraic topology, we consider cohomology (or homology or homotopy) of the form $H^k(X,A;R)$ where $A \subset X$ is a pair of spaces and $R$ is a ring. This roughly corresponds to the cohomology of $X/A$ where we identify $A$ with a point.
On the other hand, in algebraic geometry, we consider relative cohomology for a pair of schemes of the form $\pi: X\to A$ and the "relative" cohomology corresponds to gluing together the cohomology of the fibers of $\pi$ into one object.
Why is there this different between the two notions of relative? Are they equivalent (maybe formally or even just in the way they are applied) and if not, are there attempts to introduce the other form of relative into algebraic topology/geometry?
Primarily, I am thinking of introducing the algebraic geometry defn of relative cohomology into algebraic topology because the notion of $X/A$ is not so well defined for schemes $A \subset X$.
More generally, in algebraic topology we seem to be interested in the comma category with an initial object while in algebraic geometry, we are interested in the comma category with a terminal object. Why this difference?
