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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 $i$
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.
The Ran space of a connected manifold is weakly contractible (BeilDrin04, p. 173).
Let $X: CAlg_k \to Spaces$ be a derived prestack. Consider the diagram in prestacks
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:
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 $S^1$, 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
Any flat quasicoherent sheaf on the Ran space of a smooth algebraic variety over $k$ canonically acquires a D-module structure as shown in
Discussion in the context of conformal field theory includes
Expressing the rational cohomology of ordered configuration spaces of points via factorization homology and Ran spaces:
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