Ran Grassmannian and Factorization Geometry

• The Ran Grassmannian is a moduli prestack over the Ran space that parametrizes G-bundles with trivializations off finite subsets of points on a smooth curve.
• It assembles Beilinson–Drinfeld Grassmannians via a colimit construction and exhibits rich factorization and unital structures essential for bridging local modifications with global geometry.
• It underpins advanced sheaf-theoretic and semi-infinite category techniques in geometric Langlands, leveraging stratification and arc-group actions for profound representation-theoretic applications.

The Ran Grassmannian is the Ran-space version of the affine Grassmannian, usually denoted $\mathrm{Gr}_{G,\mathrm{Ran}(X)}$ or $Gr_{G,Ran}$, for a smooth algebraic curve $X$ and a reductive group $G$. It is a moduli object that parametrizes $G$-bundles on $X$ together with trivializations away from a finite nonempty subset of points of the curve, and it is constructed by assembling the Beilinson–Drinfeld Grassmannians over all finite configurations of points. In geometric representation theory it functions as a basic factorization space, a receptacle for semi-infinite and Hecke-theoretic categories, and a bridge between local and global constructions in the geometric Langlands program (Gaitsgory, 2017, Nocera, 2020, Tao, 2020, Nocera et al., 7 Sep 2025).

1. Definition and moduli interpretation

Let $X$ be a connected smooth complex algebraic curve. The Ran space $Ran(X)$ is the prestack encoding finite nonempty subsets of $X$; concretely,

$$Ran(X)(R)=\left\{S\subset X(R)\ \text{finite, nonempty}\right\},$$

and equivalently

$Gr_{G,Ran}$0

with transition maps given by diagonals (Nocera, 2020, Nocera et al., 7 Sep 2025).

For a finite set $Gr_{G,Ran}$1, the Beilinson–Drinfeld Grassmannian $Gr_{G,Ran}$2 parametrizes triples

$Gr_{G,Ran}$3

where $Gr_{G,Ran}$4, $Gr_{G,Ran}$5, and $Gr_{G,Ran}$6 is a trivialization of $Gr_{G,Ran}$7 away from the union of the graphs $Gr_{G,Ran}$8 (Nocera, 2020, Nocera et al., 7 Sep 2025). Passing from ordered tuples to unordered finite subsets gives the Ran Grassmannian: $Gr_{G,Ran}$9 Its $X$0-points may be described as pairs consisting of a finite nonempty subset $X$1 and a $X$2-bundle on $X$3 trivialized away from $X$4 (Nocera, 2020, Nocera et al., 7 Sep 2025).

A basic compatibility statement is that pulling back $X$5 along a singleton $X$6 recovers the usual affine Grassmannian $X$7 (Nocera, 2020). Thus the Ran Grassmannian is not a different local object but a globalized multi-point version of the affine Grassmannian.

2. Colimit, factorization, and unital structure

The Ran Grassmannian is defined as a presheaf colimit, but its structure is richer than a mere union over finite sets. For a surjection $X$8, the Beilinson–Drinfeld Grassmannians satisfy a factorization isomorphism on the incidence locus $X$9: $G$0 This expresses the principle that modifications at disjoint points factor into independent pieces (Nocera, 2020).

In the formulation of Dhillon and Lysenko, $G$1 carries both a factorization structure and a unital structure coming from the action of the semigroup $G$2 (Dhillon et al., 3 Aug 2025). The factorization structure encodes the behavior under disjoint union of point configurations, while unitality governs invariance under adding extra Ran-points without changing the underlying $G$3-bundle. These two structures are essential in the construction of well-behaved factorization, perverse, and Hecke-theoretic sheaf categories (Dhillon et al., 3 Aug 2025).

Using Beauville–Laszlo, the fiber of $G$4 over a finite family of points can be interpreted in terms of $G$5-torsors on the formal neighborhood of the corresponding divisor, trivialized away from the support (Dhillon et al., 3 Aug 2025). This is one of the standard geometric mechanisms by which local loop-group data and global curve-theoretic data are identified.

3. Stratifications, arc-group actions, and semi-infinite categories

The Ran Grassmannian is naturally stratified. On $G$6 one has the Schubert stratification indexed by dominant coweights, and on $G$7 one has the incidence stratification recording collisions among points. These combine to give a stratification on $G$8, and hence a colimit stratification on $G$9 (Nocera et al., 7 Sep 2025).

There is also a compatible action of the Beilinson–Drinfeld arc group. For finite $G$0, one has an action of $G$1 on $G$2, and these pass to an action of $G$3 on $G$4 (Nocera et al., 7 Sep 2025). This equivariance is structural rather than decorative: it is built into the sheaf-theoretic categories attached to the Ran Grassmannian.

Gaitsgory defines the semi-infinite category

$G$5

where $G$6 is the unipotent radical of a fixed Borel $G$7, and $G$8 is the corresponding semi-infinite group ind-scheme over the Ran space (Gaitsgory, 2017). The resulting geometry is stratified by locally closed substacks $G$9, indexed by negative coweights and described in terms of defect divisors on $X$0 (Gaitsgory, 2017).

The parabolic analogue replaces $X$1 by the parabolic datum $X$2, where $X$3 is parabolic, $X$4 is its Levi, and $X$5 its unipotent radical. The corresponding category is

$X$6

and the closure of the basic orbit admits a stratification

$X$7

This furnishes the parabolic semi-infinite local models used in the global-local comparison theorems (Dhillon et al., 3 Aug 2025).

4. Semi-infinite IC sheaves and local–global comparison

A central sheaf-theoretic object on the Ran Grassmannian is the semi-infinite intersection cohomology sheaf. In the Ran setting it is defined as a genuine middle extension in the relevant $X$8-structure: $X$9 where $X$0 is the basic stratum and $X$1 is the dualizing sheaf (Gaitsgory, 2017).

This object admits several explicit descriptions. One is a colimit presentation: $X$2 which ties the object directly to geometric Satake and translation/convolution operations (Gaitsgory, 2017). Its stalk and costalk behavior on strata are also computed explicitly: $X$3 and

$X$4

These formulas make the Langlands dual group visible in the local geometry of the Ran Grassmannian (Gaitsgory, 2017).

A decisive local–global statement identifies the Ran semi-infinite IC sheaf with a pullback of the IC sheaf on Drinfeld’s compactification: $X$5 where $X$6 (Gaitsgory, 2017). This is one of the cleanest instances in which the Ran formalism converts local factorization data into a canonically global object.

The parabolic variant has the same pattern. Dhillon and Lysenko define

$X$7

prove a colimit description via Drinfeld–Plücker formalism, compute its $X$8-restrictions to strata, and compare it canonically with the global intersection cohomology sheaf on the Drinfeld compactification of $X$9 (Dhillon et al., 3 Aug 2025). The Ran Grassmannian thereby serves as the local factorization model for both the Borel and parabolic semi-infinite IC theories.

5. Topological and higher-algebraic structures

The Ran Grassmannian also has a substantial topological and operadic life. Mair proves that the Beilinson–Drinfeld Grassmannian over the Ran space can be used to upgrade the convolution product on $Ran(X)$0-equivariant constructible sheaves on the affine Grassmannian to a left $Ran(X)$1-exact $Ran(X)$2-monoidal structure in $Ran(X)$3-categories (Nocera, 2020). The geometric inputs are the Beilinson–Drinfeld Grassmannian, the topological Ran space, factorization, stratified homotopy theory, and the formalism of correspondences. The conclusion is expressed as the existence of

$Ran(X)$4

whose underlying category is the constructible sheaf category on the affine Grassmannian (Nocera, 2020).

A later topological refinement establishes isotopy invariance directly for the Ran Grassmannian. For open metric disks $Ran(X)$5 and any finite set $Ran(X)$6, the inclusion

$Ran(X)$7

is a stratified homotopy equivalence, and the homotopies can be chosen to be stratified isotopies (Nocera et al., 7 Sep 2025). The same holds at the Ran level: $Ran(X)$8 Moreover, these equivalences are equivariant for the Beilinson–Drinfeld arc-group actions (Nocera et al., 7 Sep 2025).

As a consequence, for any metric disk $Ran(X)$9, the stratified topological space $X$0 carries the structure of a non-unital $X$1-algebra in the localization $X$2, canonically independent of the disk (Nocera et al., 7 Sep 2025). Together with the sheaf-theoretic $X$3-construction, this places the Ran Grassmannian at the intersection of factorization spaces, loop-group geometry, and higher monoidal structures.

6. Reducedness, scope, and common distinctions

In characteristic zero, Tao proves that the Ran Grassmannian is the presheaf colimit of the reduced ind-schemes $X$4: $X$5 He also shows that every map from an affine $X$6-scheme to $X$7 factors through a reduced quasi-projective $X$8-scheme (Tao, 2020). The proof uses a generalized notion of reduction for presheaves together with colimit arguments over indexing categories satisfying the amalgamation property (Tao, 2020). This result is foundational for sheaf-theoretic work on the Ran Grassmannian because it reduces many questions to reduced finite-type geometry.

Within geometric representation theory, the Ran Grassmannian is therefore best understood as a factorizable multi-point enhancement of the affine Grassmannian, rather than as an infinite-dimensional analogue in a purely formal sense. A plausible implication is that its enduring importance comes from the simultaneous presence of three compatible structures: moduli of bundle modifications, factorization over configurations of points, and sheaf categories with Hecke and semi-infinite symmetries.

It is also important to distinguish the Ran Grassmannian from several other objects that share the word “Grassmannian.” The finite-dimensional Grassmann manifold $X$9 is the set of all $$Ran(X)(R)=\left\{S\subset X(R)\ \text{finite, nonempty}\right\},$$0-dimensional linear subspaces of $$Ran(X)(R)=\left\{S\subset X(R)\ \text{finite, nonempty}\right\},$$1 and is the standard object in Riemannian geometry and optimization (Bendokat et al., 2020). The totally nonnegative Grassmannian $$Ran(X)(R)=\left\{S\subset X(R)\ \text{finite, nonempty}\right\},$$2, defined by nonnegative Plücker coordinates, belongs to total positivity and topological combinatorics (Galashin et al., 2017). By contrast, the Ran Grassmannian is a moduli prestack or ind-scheme over the Ran space of a curve, and its natural habitat is factorization geometry, affine Grassmannians, and the geometric Langlands program (Gaitsgory, 2017, Nocera, 2020, Tao, 2020).