Ran space explained
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
In general, the topology of the Ran space is generated by sets
\{S\in\operatorname{Ran}(U1\cup...\cupUm)\midS\capU1\ne\emptyset,...,S\capUm\ne\emptyset\}
for any disjoint open subsets
.
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
, 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
A theorem of Beilinson and Drinfeld states that the Ran space of a connected manifold is weakly contractible.[1]
Topological chiral homology
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.
See also
References
Notes and References
- Book: Beilinson. Alexander. Alexander Beilinson. Drinfeld. Vladimir. Vladimir Drinfeld . Chiral algebras. limited. 2004. American Mathematical Society. 0-8218-3528-9. 173.
