Ran space




The Ran space RanXRan X of a space, XX, is the space of all finite subsets of XX, equipped with geometric structure that knows about several coincident points merging to a single point.

The Ran space is defined, in topological and algebro-geometric contexts, for instance in (Lurie 14, lectures 7-9). It is naturally a stratified space (Ayala-Francis-Tanaka 14, section 3.7)

For cardinality ii

Ran iX=colim(Fin surj,i opStratStTop) Ran_{\leq i} X = \text{colim} ( Fin_{surj,\leq i}^{op} \to Strat \to StTop )

where Fin surj,iFin_{surj,\leq i} has objects that are finite sets of cardinality at most ii. When ii is infinite the subscript is left off as above.

It is named after Ziv Ran.


The Ran space of a connected manifold is weakly contractible (BeilDrin04, p. 173).


Ran 2k+1(S 1)S 2k+1 Ran_{\leq 2k+1}(S^1) \;\simeq\; S^{2k+1}

(Bott 52, Tuffley 02)


Apparently the concept of the Ran space was first introduced in

Review includes

Textbook accounts include

Ran spaces of the circle are discussed in

  • Raoul Bott, On the third symmetric potency of S 1S^1, Fund. Math. 39 (1952), 264–268 (1953)

  • Christopher Tuffley, Finite subsets of S 1S^1, Algebraic & Geometric Topology, Volume 2 (2002) 1119-1145 (arXiv:math/0209077)

Rational functions as functions on the Ran space in the sense of functions on the complement of finitely many points as these range over all points, is discussed in

Acyclicity of (some version of) the Ran space of a connected, quasi-projective scheme over an algebraically closed field is shown in lecture 10 of

  • Jacob Lurie, Tamagawa Numbers via Nonabelian Poincare Duality (282y), lecture notes, 2014, (website)

Discussion in the context of conformal field theory includes

See also

Last revised on July 19, 2019 at 17:16:42. See the history of this page for a list of all contributions to it.