higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
function algebras on ∞-stacks?
derived smooth geometry
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 $i$
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).
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
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.