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In mathematics, the Ran space (or Ran's space) of a topological space X is a topological space $$\operatorname{Ran}(X)$$ whose underlying set is the set of all nonempty finite subsets of X: for a metric space X the topology is given by Hausdorff distance. The notion is named after Ziv Ran. It seems the notion was first introduced and popularized by A. Beilinson and V. Drinfeld, Chiral algebras.

In general, the topology of the Ran space is generated by sets

$$\{ S \in \operatorname{Ran}(U_1 \cup \dots \cup U_m) \mid S \cap U_1 \ne \emptyset, \dots, S \cap U_m \ne \emptyset \}$$

for any disjoint open subsets $$U_i \subset X, 1 \le i \le m$$.

A theorem of Beilinson and Drinfeld states that the Ran space of a connected manifold is weakly contractible.[1]

There is an analog of a Ran space for a scheme:[2] the Ran prestack of a quasi-projective scheme X over a field k, denoted by $$\operatorname{Ran}(X)$$, is the category where the objects are triples $$(R, S, \mu)$$ consisting of a finitely generated k-algebra R, a nonempty set S and a map of sets \mu: S \to X(R) and where a morphism $$(R, S, \mu) \to (R', S', \mu')$$ consists of a k-algebra homomorphism $$R \to R'$$, a surjective map S \to S' that commutes with $$\mu$$ and $$\mu'$$. Roughly, an R-point of $$\operatorname{Ran}(X)$$ is a nonempty finite set of R-rational points of X "with labels" given by $$\mu$$. A theorem of Beilinson and Drinfeld continues to hold: $$\operatorname{Ran}(X)$$ is acyclic if X is connected.

Topological chiral homology

If F is a cosheaf on the Ran space $$\operatorname{Ran}(M)$$, 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.[3]

Chiral homology
Exponential space

Notes

Lurie 2012, Theorem 5.3.1.6.
http://www.math.harvard.edu/~lurie/282ynotes/LectureVII-Stacks.pdf

Lurie 2012, Theorem 5.3.3.11

References

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

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