The loop Grassmannians of reductive groups will be reconsidered as a construction in the setting of “local spaces” over a curve. The notion of a local space is a version of the fundamental structure of a factorization space introduced and developed by Beilinson and Drinfeld. The weakening of the requirements formalizes some well-known examples of “almost factorization spaces.” The change of emphases leads to new constructions. The main example will be generalizations of loop Grassmannians corresponding to quadratic forms Q on based lattices. The quadratic form corresponding to the loop Grassmannian of a simply connected group G is the basic level of G.[-]