Ran space
In mathematics, the Ran space (or Ran's space) of a topological space X is a topological space 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
[edit]In general, the topology of the Ran space is generated by sets
for any disjoint open subsets .
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 , is the category whose objects are triples consisting of a finitely generated k-algebra R, a nonempty set S and a map of sets , and whose morphisms consist of a k-algebra homomorphism and a surjective map that commutes with and . Roughly, an R-point of is a nonempty finite set of R-rational points of X "with labels" given by . A theorem of Beilinson and Drinfeld continues to hold: is acyclic if X is connected.
Properties
[edit]A theorem of Beilinson and Drinfeld states that the Ran space of a connected manifold is weakly contractible.[2]
Topological chiral homology
[edit]If F is a cosheaf on the Ran space , 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
[edit]Notes
[edit]- ^ Lurie 2014
- ^ Beilinson, Alexander; Drinfeld, Vladimir (2004). Chiral algebras. American Mathematical Society. p. 173. ISBN 0-8218-3528-9.
- ^ Lurie 2017, Theorem 5.5.3.11
References
[edit]- 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.