  We construct for each choice of a quiver $Q$, a cohomology theory $A$ and a
poset $P$ a "loop Grassmannian" $\mathcal{G}^P(Q,A)$. This generalizes loop
Grassmannians of semisimple groups and the loop Grassmannians of based
quadratic forms. The addition of a dilation torus
$\mathcal{D}\subset\mathbb{G}_m^2$ gives a quantization
$\mathcal{G}^P_\mathcal{D}(Q,A)$. The construction is motivated by the program
of introducing an inner cohomology theory in algebraic geometry adequate for
the Geometric Langlands program ['Some extensions of the notion of loop
Grassmannians', Rad Hrvat. Akad. Znan. Umjet. Mat. Znan., the Mardesic issue.
No. 532, (2017) 53--74.] and on the construction of affine quantum groups from
generalized cohomologies ArXiv: 1708.01418.
