Ran space

In mathematics, the Ran space (or Ran's space) of a topological space "X" is a topological space formula_1 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 formula_3.
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 formula_1, is the category whose objects are triples formula_5 consisting of a finitely generated "k"-algebra "R", a nonempty set "S" and a map of sets formula_6, and whose morphisms formula_7 consist of a "k"-algebra homomorphism formula_8 and a surjective map formula_9 that commutes with formula_10 and formula_11. Roughly, an "R"-point of formula_1 is a nonempty finite set of "R"-rational points of "X" "with labels" given by formula_10. A theorem of Beilinson and Drinfeld continues to hold: formula_1 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 formula_15, 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.