Let $f : X \to Y$ be a map between compact, orientable, smooth manifolds.
Then the Gysin map is defined by requiring that $\int_X f_! \alpha \wedge \beta = \int_Y \alpha \wedge f^*(\beta)$, for forms of suitable dimension.
This looks very much like an adjunction, if the integration was replaced by Hom. I guess this is not a coincidence (or maybe I am overly excited) but I don't know how to relate these things; is there a categorification of this formula that makes sense?
I guess that somehow there would be morphisms between $k$ and $n-k$ forms, and the space of these morphisms can be naturally given a volume, which would be the volume normally associated to the integral of the wedge product. Less vaguely than that I don't know how to proceed.