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 given by Hausdorff distance. The notion is named after Ziv Ran. It seems the notion was first introduced and popularized by A. Beilinson and V. Drinfeld, Chiral algebras.

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 ${\displaystyle U_{i}\subset X,1\leq i\leq m}$.

A theorem of Beilinson and Drinfeld states that the Ran space of a connected manifold is weakly contractible.^[1]

There is an analog of a Ran space for a scheme:^[2] the Ran prestack of a quasi-projective scheme X over a field k, denoted by ${\displaystyle \operatorname {Ran} (X)}$, is the category where the 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 where a morphism ${\displaystyle (R,S,\mu )\to (R',S',\mu ')}$ 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 ${\displaystyle \operatorname {Ran} (X)}$ 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: ${\displaystyle \operatorname {Ran} (X)}$ is acyclic if X is connected.

Topological chiral homology

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]

See also

• Chiral homology
• Exponential space

Notes

1. Lurie 2016, Theorem 5.5.1.6.
2. http://www.math.harvard.edu/~lurie/282ynotes/LectureVII-Stacks.pdf
3. Lurie 2016, Theorem 5.5.3.11

References

• D. Gaitsgory, Contractibility of the space of rational maps, 2012
• http://www.math.harvard.edu/~lurie/282ynotes/LectureVIII-Poincare.pdf
• J. Lurie, Higher Algebra, last updated March 2016.
• http://pantodon.shinshu-u.ac.jp/topology/literature/ja/exponential_space.html