Ran space

In mathematics, the Ran space (or Ran's 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.  It seems the notion was first introduced and popularized by Alexander Beilinson and Vladimir Drinfeld in the context of Chiral algebras.

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 where the objects are triples  consisting of a finitely generated k-algebra R, a nonempty set S and a map of sets  and where a morphism   consists of a k-algebra homomorphism , 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.

See also 
Chiral homology

Notes

References 
Dennis Gaitsgory, Contractibility of the space of rational maps, 2012
http://www.math.harvard.edu/~lurie/282ynotes/LectureVIII-Poincare.pdf
Jacob Lurie, Higher Algebra, last updated March 2016.
http://pantodon.shinshu-u.ac.jp/topology/literature/ja/exponential_space.html

Category:Topological spaces