Local spaces:finite subschemes::Factorization spaces:finite subsets.
A local space over X is a compatible collection of spaces over the Hilbert schemes of arbitrary numbers of points in X satisfying a factorization property for disjoint union. This is very close to the notion of a factorization space, in which we have a space over the Ran space Ran(X) parametrizing all finite subsets of X. The point is we're allowed (and encouraged!) to keep track of multiplicities. The idea is that in practice objects that don't care about multiplicities arise by allowing all multiplicities at once: vertex/factorization algebras/spaces naturally arise as objects living over formal discs in X, which can be filtered by objects living over increasing infinitesimal neighborhoods of points, which will form local spaces. This gives useful extra structure and allows the construction of interesting factorization spaces by "accumulation of dust" (to quote the creator, Mirkovic).
The amazing application for which local spaces were introduced is the geometric construction of the most interesting factorization space, the affine Grassmannian (and hence of reductive groups themselves) "out of combinatorics" - starting from a torus with an invariant bilinear form and a sequence of natural operations on local spaces Mirkovic builds the affine Grassmannian. This is a beautiful reformulation of the theory of "Zastava spaces" (also a cocreation of Mirkovic), which are models for transverse slices to semiinfinite orbits in the Grassmannian --- in retrospect their construcion fits into the language of local spaces and the idea is one can assemble these slices together with the orbits themselves to reconstruct the Grassmannian. Mirkovic has many exciting related ideas, including applications in higher dimensions. A recent talk on the subject is available
here.
