Ran space

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

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

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 $\mathrm{Ran}(X)$, 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 $(R,S,\mu )\to (R^{\prime },S^{\prime },{\mu }^{\prime })$ consist of a k-algebra homomorphism $R\to R^{\prime }$ and a surjective map $S\to S^{\prime }$ that commutes with $\mu$ and ${\mu }^{\prime }$. Roughly, an R-point of $\mathrm{Ran}(X)$ 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: $\mathrm{Ran}(X)$ 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 $\mathrm{Ran}(M)$, 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 [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.