Ran space

In general, the topology of the Ran space is generated by sets

: $\backslash \{\; S\; \backslash in\; \backslash operatorname\{Ran\}(U\_1\; \backslash cup\; \backslash dots\; \backslash cup\; U\_m)\; \backslash mid\; S\; \backslash cap\; U\_1\; \backslash ne\; \backslash emptyset,\; \backslash dots,\; S\; \backslash cap\; U\_m\; \backslash ne\; \backslash emptyset\; \backslash \}$

for any disjoint open subsets $U\_i\; \backslash subset\; X,\; i\; =\; 1,\; ...,\; m$.

There is an analog of a Ran space for a scheme:{{harvnb|Lurie|2014}} the Ran prestack of a quasi-projective scheme X over a field k, denoted by $\backslash operatorname\{Ran\}(X)$, is the category whose objects are triples $(R,\; S,\; \backslash mu)$ consisting of a finitely generated k-algebra R, a nonempty set S and a map of sets $\backslash mu:\; S\; \backslash to\; X(R)$, and whose morphisms $(R,\; S,\; \backslash mu)\; \backslash to\; (R\text{'},\; S\text{'},\; \backslash mu\text{'})$ consist of a k-algebra homomorphism $R\; \backslash to\; R\text{'}$ and a surjective map $S\; \backslash to\; S\text{'}$ that commutes with $\backslash mu$ and $\backslash mu\text{'}$. Roughly, an R-point of $\backslash operatorname\{Ran\}(X)$ is a nonempty finite set of R-rational points of X "with labels" given by $\backslash mu$. A theorem of Beilinson and Drinfeld continues to hold: $\backslash operatorname\{Ran\}(X)$ is acyclic if X is connected.

A theorem of Beilinson and Drinfeld states that the Ran space of a connected manifold is weakly contractible.{{cite book|last1=Beilinson|first1=Alexander|author-link1=Alexander Beilinson|last2=Drinfeld|first2=Vladimir|author-link2=Vladimir Drinfeld | title=Chiral algebras|url=https://archive.org/details/chiralalgebras00abei|url-access=limited|date=2004|publisher=American Mathematical Society|isbn=0-8218-3528-9|page=[https://archive.org/details/chiralalgebras00abei/page/n174 173]}}

If F is a cosheaf on the Ran space $\backslash operatorname\{Ran\}(M)$, 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.{{harvnb|Lurie|2017|loc=Theorem 5.5.3.11}}

• Chiral homology

{{reflist}}

• {{cite arXiv |author-link=Dennis Gaitsgory |first=Dennis |last=Gaitsgory |title=Contractibility of the space of rational maps |date=2012 |class=math.AG |eprint=1108.1741}}
• {{cite web |first=Jacob |last=Lurie |title=Homology and Cohomology of Stacks (Lecture 7) |date=19 February 2014 |work=Tamagawa Numbers via Nonabelian Poincare Duality (282y) |url=http://www.math.harvard.edu/~lurie/282ynotes/LectureVIII-Poincare.pdf}}
• {{cite web |first=Jacob |last=Lurie |title=Higher Algebra |date=18 September 2017 |url=http://www.math.harvard.edu/~lurie/papers/HA.pdf}}
• {{cite web |title=Exponential space と Ran space |date=2018 |work=Algebraic Topology: A Guide to Literature |url=http://pantodon.shinshu-u.ac.jp/topology/literature/ja/exponential_space.html}}

Category:Topological spaces