Contents

# Contents

## Idea

The Ran space $Ran X$ of a space, $X$, is the space of all 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).

## Examples

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

### General

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 $S^1$, 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

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

### 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)