In mathematics, the Ran space (or Ran's space) of a topological space X is a topological space Ran ( X ) {\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.
Definition In general, the topology of the Ran space is generated by sets
{ S ∈ Ran ( U 1 ∪ ⋯ ∪ U m ) ∣ S ∩ U 1 ≠ ∅ , … , S ∩ U m ≠ ∅ } {\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 U i ⊂ X , i = 1 , . . . , m {\displaystyle 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 Ran ( X ) {\displaystyle \operatorname {Ran} (X)} , is the category whose objects are triples ( R , S , μ ) {\displaystyle (R,S,\mu )} consisting of a finitely generated k-algebra R, a nonempty set S and a map of sets μ : S → X ( R ) {\displaystyle \mu :S\to X(R)} , and whose morphisms ( R , S , μ ) → ( R ′ , S ′ , μ ′ ) {\displaystyle (R,S,\mu )\to (R',S',\mu ')} consist of a k-algebra homomorphism R → R ′ {\displaystyle R\to R'} and a surjective map S → S ′ {\displaystyle S\to S'} that commutes with μ {\displaystyle \mu } and μ ′ {\displaystyle \mu '} . Roughly, an R-point of Ran ( X ) {\displaystyle \operatorname {Ran} (X)} is a nonempty finite set of R-rational points of X "with labels" given by μ {\displaystyle \mu } . A theorem of Beilinson and Drinfeld continues to hold: Ran ( X ) {\displaystyle \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 Ran ( M ) {\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.
See also Chiral homology
Notes
References Gaitsgory, Dennis (2012). "Contractibility of the space of rational maps". arXiv:1108.1741 [math.AG]. Lurie, Jacob (19 February 2014). "Homology and Cohomology of Stacks (Lecture 7)" (PDF). Tamagawa Numbers via Nonabelian Poincare Duality (282y). Lurie, Jacob (18 September 2017). "Higher Algebra" (PDF). "Exponential space と Ran space". Algebraic Topology: A Guide to Literature. 2018.
