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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.

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    $\begingroup$ It's an adjunction in the sense of the adjoint of a map between two inner product spaces, more or less. The inner product is given by taking the cup product and integrating (which gives you zero unless the result is in the right degree). $\endgroup$ – Qiaochu Yuan Sep 12 '16 at 2:10
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    $\begingroup$ First of all, you have $\int_X$ and $\int_Y$ reversed. In the case that $f$ is a fiber bundle, you can think of $f_!$ as integration over the fiber. $\endgroup$ – Ted Shifrin Sep 13 '16 at 16:54

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