nLab Ran space

Idea

The Ran space $Ran X$ of a space, $X$, is the space of all nonempty finite subsets of $X$, 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 $i$

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

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

It is named after Ziv Ran.

Properties

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

Constructions

Algebraic case: Ran Prestack

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

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

where $fSet$ 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:

$$Ran_X := colim_I X^I$$

Examples

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

(Bott 52, Tuffley 02)

Related concepts

• nonabelian Poincaré duality

• factorization algebra

References

General

Apparently the concept of the Ran space was first introduced in

• Alexander Beilinson, Vladimir Drinfeld , Chiral algebras, American Mathematical Society Colloquium Publications 51, (2004).

Review includes

• David Ayala, John Francis, Hiro Lee Tanaka, section 3.7 of Local structures on stratified spaces, Adv. Math. 307, 903-1028 (2017) (arXiv:1409.0501)

Textbook accounts include

• Jacob Lurie, section 5.5.1 of Higher Algebra

See also

• Wikipedia

Ran spaces of the circle are discussed in

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

• Christopher Tuffley, Finite subsets of $S^1$, 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

• Dennis Gaitsgory, Contractibility of the space of rational maps (arXiv:1108.1741)

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 $k$ canonically acquires a D-module structure as shown in

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

Discussion in the context of conformal field theory includes

• Pierre Deligne, Pavel Etingof, Dan Freed, L. Jeffrey, David Kazhdan, John Morgan, David Morrison and Edward Witten (eds.) volume II, part III of Quantum Fields and Strings, A course for mathematicians, 2 vols. Amer. Math. Soc. Providence 1999. (web version)

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)