Ran space

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 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


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

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

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 $$\operatorname{Ran}(X)$$, is the category whose 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 whose morphisms $$(R, S, \mu) \to (R', S', \mu')$$ consist of a k-algebra homomorphism $$R \to R'$$ and 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.

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 $$\operatorname{Ran}(M)$$, 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.