Ran space
In mathematics, the Ran space (or Ran's space) of a topological space X is a topological space [math]\displaystyle{ \operatorname{Ran}(X) }[/math] 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
- [math]\displaystyle{ \{ S \in \operatorname{Ran}(U_1 \cup \dots \cup U_m) \mid S \cap U_1 \ne \emptyset, \dots, S \cap U_m \ne \emptyset \} }[/math]
for any disjoint open subsets [math]\displaystyle{ U_i \subset X, i = 1, ..., m }[/math].
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 [math]\displaystyle{ \operatorname{Ran}(X) }[/math], is the category whose objects are triples [math]\displaystyle{ (R, S, \mu) }[/math] consisting of a finitely generated k-algebra R, a nonempty set S and a map of sets [math]\displaystyle{ \mu: S \to X(R) }[/math], and whose morphisms [math]\displaystyle{ (R, S, \mu) \to (R', S', \mu') }[/math] consist of a k-algebra homomorphism [math]\displaystyle{ R \to R' }[/math] and a surjective map [math]\displaystyle{ S \to S' }[/math] that commutes with [math]\displaystyle{ \mu }[/math] and [math]\displaystyle{ \mu' }[/math]. Roughly, an R-point of [math]\displaystyle{ \operatorname{Ran}(X) }[/math] is a nonempty finite set of R-rational points of X "with labels" given by [math]\displaystyle{ \mu }[/math]. A theorem of Beilinson and Drinfeld continues to hold: [math]\displaystyle{ \operatorname{Ran}(X) }[/math] 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.[2]
Topological chiral homology
If F is a cosheaf on the Ran space [math]\displaystyle{ \operatorname{Ran}(M) }[/math], 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
Notes
- ↑ Lurie 2014
- ↑ Beilinson, Alexander; Drinfeld, Vladimir (2004). Chiral algebras. American Mathematical Society. p. 173. ISBN 0-8218-3528-9. https://archive.org/details/chiralalgebras00abei.
- ↑ Lurie 2017, Theorem 5.5.3.11
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)". Tamagawa Numbers via Nonabelian Poincare Duality (282y). http://www.math.harvard.edu/~lurie/282ynotes/LectureVIII-Poincare.pdf.
- Lurie, Jacob (18 September 2017). "Higher Algebra". http://www.math.harvard.edu/~lurie/papers/HA.pdf.
- "Exponential space と Ran space". Algebraic Topology: A Guide to Literature. 2018. http://pantodon.shinshu-u.ac.jp/topology/literature/ja/exponential_space.html.
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