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. 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

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

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

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 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.

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.

See also

• Chiral homology

References

• Gaitsgory, Dennis (2012). "Contractibility of the space of rational maps". arXiv:1108.1741. {{cite arXiv}}: Cite has empty unknown parameter: |publisher= (help)
• 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.