Today I want to talk about section VII.3 in the manuscript. Here they define and study a functor on solid sheaves which is left adjoint to the usual pullback functor. But why is this called relative homology?
To explain this name, you have to remember that in the usual formalism of etale cohomology, homology is realized as the compactly supported cohomology of the dualizing complex. Symbolically, if is a variety with structure map , then the homology of is given by . Now, you might ask whether this works in families: if is some map of varieties, maybe I can find some complex on whose stalk at realizes the homology of ? If the constant sheaf is -ULA, then formation of commutes with any base change, and exactly the same formula works, but in general there is no naive sheaf with this property.
The punchline now is that does have this property: when the constant sheaf is -ULA (e.g. if is a point) it agrees with by Proposition VII.5.2, and its formation commutes with arbitrary base change, so it really does give a complex on whose stalks realize the homology of the fibers of . The only twist is that is a solid sheaf in general, not a classical etale sheaf.