Ran space

In mathematics, the Ran space of a topological space X is a topological space whose underlying set is the set of all nonempty finite subsets of X: for a metric space X the topology is induced by the Hausdorff distance. The notion is named after Ziv Ran.

Definition

In general, the topology of the Ran space is generated by sets
for any disjoint open subsets.
There is an analog of a Ran space for a scheme: the Ran prestack of a quasi-projective scheme X over a field k, denoted by, is the category whose objects are triples consisting of a finitely generated k-algebra R, a nonempty set S and a map of sets, and whose morphisms consist of a k-algebra homomorphism and a surjective map that commutes with and. Roughly, an R-point of is a nonempty finite set of R-rational points of X "with labels" given by. A theorem of Beilinson and Drinfeld continues to hold: is acyclic if X is connected.

Properties

A theorem of Beilinson and Drinfeld states that the Ran space of a connected manifold is weakly contractible.

Topological chiral homology

If F is a cosheaf on the Ran space, then its space of global sections is called the topological chiral homology of M with coefficients in F. If A is, roughly, a family of commutative algebras parametrized by points in M, then there is a factorizable sheaf associated to A. Via this construction, one also obtains the topological chiral homology with coefficients in A. The construction is a generalization of Hochschild homology.