nLab Ran space




The Ran space RanXRan X of a space, XX, is the space of all nonempty 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).


Algebraic case: Ran Prestack

Let X:CAlg kSpacesX: CAlg_k \to Spaces be a derived prestack. Consider the diagram in prestacks

X ():N(fSet)PrSt k,IX I X^{(-)}: N(fSet) \to PrSt_k, I \mapsto X^I

where fSetfSet is the category of non-empty finite sets, with morphisms being surjective maps. The functor sends morphisms to the corresponding diagonal maps. The Ran prestack by definition is the (homotopy) colimit of this diagram:

Ran X:=colim IX IRan_X := colim_I X^I


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

See also

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)

Any flat quasicoherent sheaf on the Ran space of a smooth algebraic variety over kk canonically acquires a D-module structure as shown in

  • James Tao, nn-Excisive Functors, Canonical Connections, and Line Bundles on the Ran Space (arXiv:1906.07976)

Discussion in the context of conformal field theory includes

Relation to configuration spaces of points

Expressing the rational cohomology of ordered configuration spaces of points via factorization homology and Ran spaces:

  • Quoc P. Ho, Higher representation stability for ordered configuration spaces and twisted commutative factorization algebras (arXiv:2004.00252)

Last revised on July 27, 2020 at 15:25:18. See the history of this page for a list of all contributions to it.