Spin-dependent quark beam function at NNLO

We calculate the beam function for longitudinally polarized quarks through next-to-next-to-leading order (NNLO) in QCD perturbation theory. This is the last missing ingredient needed to apply the factorization theorem for the $N$-jettiness event-shape variable in a variety of polarized collisions through the NNLO level. We present all technical details of our derivation. As a by-product of our calculation we provide the first independent check of the previously obtained unpolarized quark beam function. We anticipate that our result will have phenomenological applications in describing data from polarized collisions.