Ran space

In mathematics, the Ran space (or Ran's space) of a topological space X is a topological space ${\displaystyle \operatorname {Ran} (X)}$ 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.

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

${\displaystyle \{S\in \operatorname {Ran} (U_{1}\cup \dots \cup U_{m})\mid S\cap U_{1}\neq \emptyset ,\dots ,S\cap U_{m}\neq \emptyset \}}$

for any disjoint open subsets ${\displaystyle U_{i}\subset X,i=1,...,m}$.

There is an analog of a Ran space for a scheme:¹ the Ran prestack of a quasi-projective scheme X over a field k, denoted by ${\displaystyle \operatorname {Ran} (X)}$, is the category whose objects are triples ${\displaystyle (R,S,\mu )}$ consisting of a finitely generated k-algebra R, a nonempty set S and a map of sets ${\displaystyle \mu :S\to X(R)}$, and whose morphisms ${\displaystyle (R,S,\mu )\to (R',S',\mu ')}$ consist of a k-algebra homomorphism ${\displaystyle R\to R'}$ and a surjective map ${\displaystyle S\to S'}$ that commutes with ${\displaystyle \mu }$ and ${\displaystyle \mu '}$. Roughly, an R-point of ${\displaystyle \operatorname {Ran} (X)}$ is a nonempty finite set of R-rational points of X "with labels" given by ${\displaystyle \mu }$. A theorem of Beilinson and Drinfeld continues to hold: ${\displaystyle \operatorname {Ran} (X)}$ is acyclic if X is connected.

A theorem of Beilinson and Drinfeld states that the Ran space of a connected manifold is weakly contractible.²

If F is a cosheaf on the Ran space ${\displaystyle \operatorname {Ran} (M)}$, 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