2.1 Lax prestacks

A lax prestack is a functor $\mathrm{AffSch}^{op} \to \mathrm{Cat}$ . The category $\mathrm{Fun}(\mathrm{AffSch}^{op}, \mathrm{Cat})$ of lax prestacks will be denoted $\mathrm{LaxPrStk}$ .

Let $\mathrm{Spc}$ denote the $\infty$ -category of $\infty$ -groupoids. Then there is a natural inclusion $\mathrm{Spc} \to \mathrm{Cat}$ and so every prestack $\mathrm{AffSch}^{op} \to \mathrm{Spc}$ may be thought of as a particular kind of lax prestack. We will denote the left adjoint to the inclusion $\mathrm{PrStk} \to \mathrm{LaxPrStk}$ by $\mathrm{str}: \mathrm{LaxPrStk} \to \mathrm{PrStk}$ . Roughly, for an affine scheme $S$ , the space $\mathcal{Y}_{\mathrm{str}}(S) := \mathrm{str}(\mathcal{Y})(S)$ is the universal groupoid admitting a map from $\mathcal{Y}(S)$ .

Here is an equivalent formulation of $\mathcal{Y}_{\mathrm{str}}$ . There is a fully faithful functor $\mathrm{Cat} \to \mathrm{Fun}(\Delta ^{op}, \mathrm{Spc})$ , whose essential image consists of the complete Segal spaces. Hence, with a lax prestack $\mathcal{Y}$ we may associate a simplicial prestack $s\mathcal{Y}: \mathrm{AffSch}^{op} \to \mathrm{Fun}(\Delta ^{op},\mathrm{Spc})$ satisfying the Segal conditions. The following lemma gives a more concrete construction of $\mathcal{Y}_{\mathrm{str}}$ .

Lemma 2.2. Let $\mathcal{Y}$ be a lax prestack and let $s\mathcal{Y}$ be the associated simplicial prestack. Then there is a canonical equivalence of prestacks

\begin{equation*}\mathcal{Y}_{\mathrm{str}} \longrightarrow |s\mathcal{Y}|,\end{equation*}

where $|s\mathcal{Y}|$ is the prestack obtained from $s\mathcal{Y}$ by composing with the geometric realization functor $\mathrm{Fun}(\Delta ^{op}, \mathrm{Spc}) \to \mathrm{Spc}$ .

Proof. This follows from the fact that the straightening functor $\mathrm{Cat} \to \mathrm{Spc}$ is equivalent to the composition

\begin{equation*}\mathrm{Cat} \longrightarrow \mathrm{Fun}(\Delta ^{op},\mathrm{Spc}) \longrightarrow \mathrm{Spc},\end{equation*}

where the first arrow is the functor taking a category to its associated complete Segal space, and the second arrow is geometric realization.

Recall that a category $\mathscr{C}$ is weakly contractible if $\mathscr{C}_{\mathrm{str}}$ is equivalent to a point in $\mathrm{Spc}$ . We will say that a lax prestack $\mathcal{Y}$ is weakly contractible if for all affine schemes $S$ , the category $\mathcal{Y}(S)$ is weakly contractible.

Consider the following example of a lax prestack, which will be important throughout the text. Let $\mathcal{M}$ be a presheaf of commutative monoids on $\mathrm{AffSch}_k$ . That is, for every affine scheme $S$ the groupoid $\mathcal{M}(S)$ is a set, and $\mathcal{M}$ has a factorization through the forgetful functor from commutative monoids to sets.

The functor $\mathcal{M}$ can be upgraded to a lax prestack as follows. Define $\mathcal{M}^{\to }(S)$ to be the category whose objects are given by the set $\mathcal{M}(S)$ and such that there is a unique arrow $x \to y$ for every $m$ such that $mx=y$ . For every morphism $S \to T$ of affine schemes, the morphism $\mathcal{M}(T) \to \mathcal{M}(S)$ of monoids given by restriction can be upgraded to a functor

\begin{equation*}\mathcal{M}^{\to }(T) \longrightarrow \mathcal{M}^{\to }(S)\end{equation*}

in the obvious way. It is easy to see that the associated simplicial prestack is given by the simplicial object

induced by multiplication in $\mathcal{M}$ .

2.2 Sheaves on lax prestacks

We will now briefly review sheaves on lax prestacks [Reference RaskinRas19]. Let us assume we are given a sheaf theory $\mathrm{Shv}: \mathrm{AffSch}^{op} \to \mathrm{Cat}$ . For all intents and purposes $\mathrm{Shv}(S)$ will be the category of quasicoherent sheaves, ind-coherent sheaves, or $D$ -modules (see [Reference GaitsgoryGai12] for a survey of ind-coherent sheaves and [Reference Gaitsgory and RozenblyumGR14] for a survey of $D$ -modules). For a map $f: S \to T$ of affine schemes, denote the resulting functor between categories of sheaves by $f^!$ . We can then define the category of sheaves on an arbitrary lax prestack.

Unwinding the definition of $\mathrm{Shv}(\mathcal{Y})$ , we see that a sheaf $c$ on $\mathcal{Y}$ is the following data:

1. (i) for every $S$ -point $f:S \to \mathcal{Y}$ of $\mathcal{Y}$ we have a sheaf $c_{S,f} \in \mathrm{Shv}(S)$ ;

2. (ii) for every map $\varphi : (S,f) \to (T,g)$ of affine schemes over $\mathcal{Y}$ , we have an isomorphisms $\varphi ^!(c_{T,g}) \simeq c_{S,f}$ with homotopy coherent compatibilities;

3. (iii) for every morphism $f \to g$ of $S$ -points, we have morphisms $c_{S,f} \to c_{S,g}$ with homotopy coherent compatibilities.

By definition of $\mathcal{Y}_{\mathrm{str}}$ , it follows that the category $\mathrm{Shv}(\mathcal{Y}_{\mathrm{str}})$ is equivalent to the full subcategory $\mathrm{Shv}_{\mathrm{str}}(\mathcal{Y})$ of sheaves on $\mathcal{Y}$ where all the morphisms $c_{S,f} \to c_{S,g}$ in (iii) are isomorphisms. Denote the resulting full subcategory of $\mathrm{Shv}(\mathcal{Y})$ by $\mathrm{Shv}_{\mathrm{str}}(\mathcal{Y})$ . There is the following equivalent simplicial perspective on $\mathrm{Shv}_{\mathrm{str}}(\mathcal{Y})$ .

Lemma 2.6. Let $\mathcal{Y}$ be a lax prestack, and let $s\mathcal{Y}$ be the associated simplicial prestack. Then there is an equivalence of categories

\begin{equation*}\mathrm{Shv}_{\mathrm{str}}(\mathcal{Y}) \longrightarrow \mathrm{Tot}(\mathrm{Shv}(s\mathcal{Y})), \end{equation*}

where $\mathrm{Tot}(\mathrm{Shv}(s\mathcal{Y}))$ denotes the totalization of the cosimplicial category $\mathrm{Shv}(s\mathcal{Y})$ .

2.3 Left fibrations of lax prestacks

Recall that a functor $F: \mathscr{C} \to \mathscr{D}$ of $\infty$ -categories is a left fibration if it satisfies the right lifting property with respect to all horn inclusions, except possibly the right outer ones. That is, whenever the outer square of

commutes, there exists a dotted arrow as above for all $0 \leq k \lt n$ making the diagram commute. In higher category theory, the left fibrations play a role similar to that of categories cofibered in groupoids in ordinary category theory. In fact, any left fibration is also a coCartesian fibration, and if the base is a Kan complex then it is also a Kan fibration.

We will define here an analog of left fibrations for lax prestacks. Consider the lax prestack $\mathrm{Spc}_{/-}$ which sends an affine scheme to the slice category $\mathrm{PrStk}_{/S}$ . For a lax prestack $\mathcal{Y}$ , define $\mathrm{Spc}_{/\mathcal{Y}}$ to be the category of natural transformations

\begin{equation*} \Phi : \mathcal{Y} \longrightarrow \mathrm{Spc}_{/-}.\end{equation*}

One should think of $\Phi$ as assigning to an $S$ -point $f: S \to \mathcal{Y}$ the fiber $\Phi _f$ of the associated left fibration. For any morphism $f \to g$ of $S$ -points, the structure of natural transformation $\mathcal{Y} \to \mathrm{Spc}_{/-}$ gives a morphism

\begin{equation*}\Phi _f \longrightarrow \Phi _g\end{equation*}

as prestacks over $S$ . By passing to global sections and using the Grothendieck construction, for each $\Phi$ there is an associated lax prestack $\Phi _{\mathrm{Spc}} \to \mathcal{Y}$ lying over $\mathcal{Y}$ . For an affine scheme $S$ , the functor $\Phi _{\mathrm{Spc}}(S) \to \mathcal{Y}(S)$ is a left fibration of categories.

We will refer to a natural transformation $\Phi : \mathcal{Y} \to \mathrm{Spc}/-$ as a left fibration of lax prestacks or simply as a left fibration. Since left fibrations are examples of coCartesian fibrations, we will also sometimes call such functors $\Phi$ coCartesian fibrations of lax prestacks. Note that we have the lax prestack $\mathcal{Y}^{\operatorname {op}}$ taking an affine scheme $S$ to $\mathcal{Y}(S)^{\operatorname {op}}$ . Hence we also have the notion of right fibration of lax prestacks or Cartesian fibrations of lax prestacks. Additionally, one can define a pointwise (co)Cartesian fibration of lax prestacks which is a map $\mathcal{X} \to \mathcal{Y}$ of lax prestacks such that for all affine schemes $S$ the induced functor

\begin{equation*} \mathcal{X}(S) \longrightarrow \mathcal{Y}(S)\end{equation*}

is a (co)Cartesian fibration and for any map $T \to S$ the induced map $\mathcal{X}(S) \to \mathcal{X}(T)$ sends (co)Cartesian arrows to (co)Cartesian arrows. Although for $\Phi$ as above the map $\Phi _{\mathrm{Spc}} \to \mathcal{Y}$ is a pointwise coCartesian fibration, it is not clear that the converse holds.

An advantage of this perspective is that one can immediately see how to assign to a lax prestack $\mathcal{Y}$ a sheaf of categories, given the existence of such a $\Phi$ . Recall that a sheaf of categories is defined to be a lax natural transformation

\begin{equation*}\mathcal{Y} \longrightarrow \operatorname {ShvCat}_/,\end{equation*}

where $\operatorname {ShvCat}$ is the lax prestack which assigns to an affine scheme $S$ the category of modules for the symmetric monoidal category $\mathrm{QCoh}(S)$ . Note there is a natural transformation

\begin{equation*}\mathfrak{D}^{\mathrm{Spc} \to \mathrm{Cat}}: \mathrm{Spc}_{/-} \longrightarrow \operatorname {ShvCat}_{/-},\end{equation*}

which takes a prestack $\mathcal{X} \to S$ over an affine scheme to $\mathfrak{D}(\mathcal{X})$ . Hence to any left fibration $\Phi$ over $\mathcal{Y}$ we may assign a sheaf of categories given as the composition $\mathfrak{D}^{\mathrm{Spc} \to \mathrm{Cat}}(\Phi ) := \mathfrak{D}^{\mathrm{Spc} \to \mathrm{Cat}} \circ \Phi$ .

Note that, whenever there is a map $f:\mathcal{X} \to \mathcal{Y}$ of lax prestacks, we can define a pullback functor

\begin{equation*}f^{\mathrm{Spc},*}: \mathrm{Spc}_{/\mathcal{Y}} \to \mathrm{Spc}_{/\mathcal{X}}\end{equation*}

simply by composing a lax natural transformation $\mathcal{Y} \to \mathrm{Spc}_/$ with $f$ . We therefore obtain a functor

\begin{equation*}\mathrm{LaxPrStk}^{\mathrm{op}} \longrightarrow \mathrm{Cat,}\end{equation*}

which takes a lax prestack $\mathcal{Y}$ to $\mathrm{Spc}_{/\mathcal{Y}}$ .

2.4 Correspondences

In this section we will discuss categories of correspondences. These will be important in defining notions of factorization spaces and their variants (e.g. unital, effective). Of course, the definition of factorization spaces is well known, and we are not adding anything new to the theory here. However, since we are considering factorization over several different bases, it is convenient to have a unified language to discuss all of them.

Let $\mathscr{C}$ be a category with all finite limits, and let $\mathrm{Corr}(\mathscr{C})$ be the 2-category of correspondences Footnote ⁵ in $\mathscr{C}$ . Objects in $\mathrm{Corr}(\mathscr{C})$ are just objects of $\mathscr{C}$ , but morphisms $\mathcal{Y} \to \mathcal{G}$ are given by correspondences

Composition of morphisms $\mathcal{Y} \to \mathcal{G} \to \mathcal{H}$ is given by the outer legs of the diagram

Note that $\mathrm{Corr}(\mathscr{C})$ is naturally symmetric monoidal under the Cartesian monoidal structure. Hence we may consider commutative algebras (also known as commutative monoids) in $\mathrm{Corr}(\mathscr{C})$ . Roughly, a commutative algebra in $\mathrm{Corr}(\mathscr{C})$ is a pair of correspondences

(2.1)

satisfying natural associativity and unitality conditions up to coherent homotopy.

We will apply the above picture to both $\mathrm{PrStk}$ and $\mathrm{LaxPrStk}$ to obtain the categories $\mathrm{Corr}(\mathrm{PrStk})$ and $\mathrm{Corr}(\mathrm{LaxPrStk})$ . Note that $\mathrm{Spc}_{/(-)}$ is naturally a commutative algebra in lax prestacks, where multiplication is induced by the functors

\begin{equation*}\mathrm{Spc}_{/S} \times \mathrm{Spc}_{/S} \longrightarrow \mathrm{Spc}_{/S}\end{equation*}

taking a pair of prestacks $(\mathcal{Y},\mathcal{X})$ to the fiber product $\mathcal{Y} \times _S \mathcal{X}$ . The unit is provided by the inclusion of the identity $S \to S$ as an object of $\mathrm{Spc}_{/S}$ .

2.5 Multiplicative spaces

In this section, we will review the category of multiplicative spaces over a commutative algebra in correspondences. A general survey of the notion of multiplicative spaces can be found in [Reference ButsonBut20]. As we will later see, when the commutative algebra is the Ran space or the configuration space, multiplicative spaces will coincide with the familiar notion of factorization spaces. Our definition of multiplicative space proceeds by adapting Raskin’s construction of multiplicative sheaves of categories to a geometric context.

Denote by

\begin{equation*}\mathrm{Spc}_{/(-)}: \mathrm{LaxPrStk}^{\mathrm{op}} \longrightarrow \mathrm{Cat,}\end{equation*}

the functor taking a lax prestack $\mathcal{Y}$ to its category $\mathrm{Spc}_{/\mathcal{Y}}$ of left fibrations over $\mathcal{Y}$ . As in Chiral categories, we can construct the category $\operatorname {Groth}(\mathrm{Spc}_{/(-)})$ whose objects are pairs $(\mathcal{Y},\Phi )$ , where $\mathcal{Y}$ is a lax prestack and $\Phi : \mathcal{Y} \to \mathrm{Spc}_{/(-)}$ is a left fibration over $\mathcal{Y}$ . Morphisms $(\mathcal{Y},\Phi ) \to (\mathcal{X}, \Psi )$ are given by a correspondence

together with a map

\begin{equation*}f^{\mathrm{Spc},*}(\Phi ) \to g^{\mathrm{Spc},*}(\Psi )\end{equation*}

satisfying various homotopy coherent compatibilities.

Since $\mathrm{Spc}_{/(-)}$ is lax symmetric monoidal with respect to the Cartesian monoidal structure on both sides, the category $\operatorname {Groth}(\mathrm{Spc}_{/(-)})$ inherits a symmetric monoidal structure $\otimes$ , given pointwise on objects by

\begin{equation*}(\mathcal{Y},\Phi ) \otimes (\mathcal{X},\Psi ) =(\mathcal{Y} \times \mathcal{X}, \Phi \times \Psi ).\end{equation*}

There is a natural forgetful functor

\begin{equation*}\mathrm{oblv}_{\mathrm{Spc}}: \operatorname {Groth}(\mathrm{Spc}_{/(-)}) \longrightarrow \mathrm{Corr}(\mathrm{LaxPrStk}),\end{equation*}

which has the structure of a symmetric monoidal functor.

Now let $\mathcal{Y}$ be a commutative algebra in $\mathrm{Corr}(\mathrm{LaxPrStk})$ . Define a weakly multiplicative space over $\mathcal{Y}$ to be a commutative algebra object of $\operatorname {Groth}(\mathrm{Spc}_{/(-)})$ mapping to $\mathcal{Y}$ under $\mathrm{oblv}_{\mathrm{Spc}}$ as a commutative algebra. Note that, among other features, a weakly multiplicative space on $\mathcal{Y}$ is a left fibration $\Phi$ over $\mathcal{Y}$ together with a morphism

(2.2) \begin{equation} (\iota _{\mathcal{Y}})^*(\Phi \times \Phi ) \longrightarrow (\mu _{\mathcal{Y}})^*(\Phi ), \end{equation}

where $\iota _{\mathcal{Y}}$ and $\mu _{\mathcal{Y}}$ are the left and right legs, respectively, of the multiplication correspondence in (2.1). We also have analogous maps for all $n$ -ary operations of $\mathcal{Y}$ .

Let us unwind this definition a bit more. By the Grothendieck construction, a left fibration $\Phi : \mathcal{Y} \to \mathrm{Spc}_/$ corresponds to a particular kind of lax prestack $\Phi _{\tau }$ with a map

\begin{equation*}\Phi ^{\mathrm{Groth}}: \Phi _{\tau } \to \mathcal{Y},\end{equation*}

where here the notation $\tau$ stands for total space, i.e. the total space of the fibration $\Phi$ . Then if $\Phi$ has the structure of a multiplicative space over $\mathcal{Y}$ we have isomorphisms

\begin{equation*}(\Phi _{\tau } \times \Phi _{\tau }) \times _{\mathcal{Y} \times \mathcal{Y}} \operatorname {mult}_{\mathcal{Y}} \overset {\sim }{\longrightarrow } \Phi _{\tau } \times _{\mathcal{Y}} \operatorname {mult}_{\mathcal{Y}}\end{equation*}

satisfying various commutativity and associativity compatibilities.

2.6 Compatibility of straightening with algebras

In this section, we will discuss when it is possible to push a multiplicative space forward along the canonical map $\mathcal{Y} \to \mathcal{Y}_{\mathrm{str}}$ for a commutative algebra $\mathcal{Y}$ in $\mathrm{Corr}(\mathrm{LaxPrStk})$ .

Note that the straightening functor

\begin{equation*}\mathrm{str}: \mathrm{LaxPrStk} \longrightarrow \mathrm{PrStk}\end{equation*}

preserves products but not fiber products in general. As such, it is not clear how to upgrade $\mathrm{str}$ to a functor between categories of correspondences. To remedy this, we will restrict the allowable morphisms in $\mathrm{LaxPrStk}$ .

Recall that a functor $\mathscr{C} \to \mathscr{D}$ between categories $\mathscr{C}$ and $\mathscr{D}$ is called a realization fibration if for all functors $\mathscr{E} \to \mathscr{D}$ the diagram

is Cartesian. We will call a map $\mathcal{X} \to \mathcal{Y}$ of lax prestacks a realization fibration if for all affine schemes $S$ the functor

\begin{equation*}\mathcal{X}(S) \longrightarrow \mathcal{Y}(S),\end{equation*}

is a realization fibration of categories. Denote by $\mathrm{LaxPrStk}_{\operatorname {rf}}$ the subcategory of $\mathrm{LaxPrStk}$ consisting of lax prestacks and realization fibrations as morphisms. Note that, by an application of the pasting lemma, realization fibrations are stable under base change and hence it makes sense to consider the category $\mathrm{Corr}(\mathrm{LaxPrStk}_{\operatorname {rf}})$ .

Since the functor $\mathrm{str}$ preserves fiber products when restricted to $\mathrm{LaxPrStk}_{\operatorname {rf}}$ , we may upgrade $(-)_{\mathrm{str}}$ to a functor

\begin{equation*}\operatorname {\mathrm{str}}^{\operatorname {enh}}: \operatorname {CAlg}(\mathrm{Corr}(\mathrm{LaxPrStk}_{\operatorname {rf}})) \longrightarrow \operatorname {CAlg}(\mathrm{Corr}(\mathrm{PrStk})),\end{equation*}

where for a category $\mathscr{C}$ with all finite limits we define $\operatorname {CAlg}(\mathscr{C})$ to be the category of commutative algebras in $\mathscr{C}$ . In other words, given a commutative algebra $\mathcal{Y}$ in correspondences of lax prestacks where each leg of the multiplication and unit correspondences is a realization fibration, the straightening $\mathcal{Y}_{\mathrm{str}}$ has a natural structure of a commutative algebra in correspondences of prestacks.

Proposition 2.8. Let $\mathcal{Y}$ be a commutative algebra in correspondences such that all maps in the diagrams (2.1) are realization fibrations. Then there is a natural functor

\begin{equation*}\operatorname {LKE_{\mathcal{Y} \to \mathcal{Y}_{\mathrm{str}}}}: \operatorname {MultSpc}(\mathcal{Y}) \longrightarrow \operatorname {MultSpc}(\mathrm{str}^{\operatorname {enh}}(\mathcal{Y})),\end{equation*}

taking multiplicative spaces over $\mathcal{Y}$ to multiplicative spaces over the commutative algebra $\mathrm{str}^{\operatorname {enh}}(\mathcal{Y})$ .

Proof. Let $\Phi : \mathcal{Y} \to \mathrm{Spc}_{/(-)}$ be a multiplicative space over $\mathcal{Y}$ , and denote also by $\Phi _{\tau }$ the total space of the associated left fibration over $\mathcal{Y}$ . Define $\operatorname {LKE}_{\mathcal{Y} \to \mathcal{Y}_{\mathrm{str}}}(\Phi )$ by left Kan extension as follows:

where by left Kan extension we mean the functor $\mathcal{Y}_{\mathrm{str}} \to \mathrm{Spc}_/$ which on an affine scheme $S$ is the left Kan extension of $\mathcal{Y}(S) \to \mathrm{Spc}_{/S}$ along $\mathcal{Y}(S) \to \mathcal{Y}_{\mathrm{str}}(S)$ . Note that this gives a well-defined morphism of lax prestacks because $\mathrm{PrStk}$ is locally Cartesian closed.

We need to show that the resulting diagram

(2.3)

and all the analogous ones for higher operations commute (and similarly for the unit diagram, which we omit). By the universal property of Kan extensions, we know that (2.3) commutes up to a possibly non-invertible 2-morphism. We wish to show that this 2-morphism is an equivalence.

Let $(\Phi _{\tau })_{\mathrm{str}} \to \mathcal{Y}_{\mathrm{str}}$ denote the left fibration associated with $\operatorname {LKE}_{\mathcal{Y} \to \mathcal{Y}_{\mathrm{str}}}$ via the Grothendieck construction. Equivalently, $(\Phi _{\tau })_{\mathrm{str}}$ is the straightening of the lax prestack $\Phi _{\tau }$ . To show that (2.3) commutes, it suffices to show that the corresponding map

\begin{equation*}((\Phi _{\tau })_{\mathrm{str}} \times (\Phi _{\tau })_{\mathrm{str}}) \times _{\mathcal{Y}_{\mathrm{str}} \times \mathcal{Y}_{\mathrm{str}}} \operatorname {mult}_{\mathcal{Y}_{\mathrm{str}}} \longrightarrow (\Phi _{\tau })_{\mathrm{str}} \times _{\mathcal{Y}_{\mathrm{str}}} \operatorname {mult}_{\mathcal{Y}_{\mathrm{str}}},\end{equation*}

is an equivalence, and similarly for the higher operations. But this follows from the definitions together with our assumption on $\mathcal{Y}$ .

3.1 The Ran space

Define a prestack $\mathrm{Ran}$ as follows (see [Reference RaskinRas19] for a more detailed overview). For an affine scheme $S$ , a map $S \to \mathrm{Ran}$ is a nonempty finite subset of $\mathrm{Map}(S,X)$ . Equivalently, we can write

\begin{equation*}\mathrm{Ran} \overset {\sim }{\longrightarrow } \mathrm{colim}_{\operatorname {FinSet}^{\mathrm{op}}_{\operatorname {surj}}} X^I, \end{equation*}

where the colimit is over the category opposite to the one whose objects are (possibly empty) finite sets and whose morphisms are surjective functions. Here $X^I$ denotes the scheme of maps $x_I : I \to X$ from a finite set $I$ to $X$ .

For a point $x$ of $X$ , define $\Gamma _x$ to be its graph. Accordingly, for a point $x_I=(x_i)_{i \in I}$ of $\mathrm{Ran}$ , write $\Gamma _{x_I}$ for the union $\cup _{i \in I} \Gamma _{x_i}$ of graphs. Inside the product $\mathrm{Ran} \times \mathrm{Ran}$ we have the subprestack $[\mathrm{Ran} \times \mathrm{Ran}]_{\mathrm{disj}}$ given by pairs $(x_I, x_J)$ such that $\Gamma _{x_I} \cap \Gamma _{x_J} = \emptyset$ . Note that $\mathrm{Ran}$ is equipped with the structure of a monoid

\begin{equation*}\mathrm{add}_{\mathrm{Ran}} : \mathrm{Ran} \times \mathrm{Ran} \to \mathrm{Ran,}\end{equation*}

given by union of finite subsets. By abuse of notation, we will denote by $\mathrm{add}_{\mathrm{Ran}}$ the restriction of the same named map to $[\mathrm{Ran} \times \mathrm{Ran}]_{\mathrm{disj}}$ .

An important feature of $\mathrm{Ran}$ is that it has a structure of a commutative algebra in correspondences. Namely, there is a correspondence

(3.1)

whose second leg is given $\mathrm{add}_{\mathrm{Ran}}$ . The unit diagram for $\mathrm{Ran}$ is just given by the inclusion

\begin{equation*}\{\emptyset \} \hookrightarrow {\quad } \mathrm{Ran}\end{equation*}

of the empty subset of $X$ .

Define a factorization space over $\mathrm{Ran}$ to be a multiplicative space for the correspondence (3.1). A bit more explicitly, a factorization space is a prestack $\mathcal{Y} \to \mathrm{Ran}$ together with unital and associative isomorphisms

\begin{equation*}\mathrm{fact}_{\mathcal{Y}}: (\mathcal{Y} \times \mathcal{Y}) \times _{\mathrm{Ran} \times \mathrm{Ran}} [\mathrm{Ran} \times \mathrm{Ran}]_{\mathrm{disj}} \overset {\sim }{\longrightarrow } \mathcal{Y} \times _{\mathrm{Ran}} [\mathrm{Ran} \times \mathrm{Ran}]_{\mathrm{disj}}.\end{equation*}

We will often denote the left-hand side by $[\mathcal{Y} \times \mathcal{Y}]_{\mathrm{disj}}$ and the right-hand side by $\mathcal{Y}_{\mathrm{disj}}$ .

3.2 The unital Ran space

We will also need an enhancement of $\mathrm{Ran}$ as a lax prestack. Define

\begin{equation*}\mathrm{Ran}^{\mathrm{un}}: \mathrm{AffSch}_k^{\mathrm{op}} \longrightarrow \mathrm{Cat}\end{equation*}

to be the functor sending $S$ to the following category. Objects in $\mathrm{Ran}^{\mathrm{un}}(S)$ are given by finite subsets $x_I \subseteq \mathrm{Map}(S,X)$ and for any such pair $x_I$ and $x_J$ there is a single morphism $x_I \to x_J$ precisely if $x_I \subseteq x_J$ . Note we are not requiring that $x_I$ be nonempty. We call the lax prestack $\mathrm{Ran}^{\mathrm{un}}$ the unital Ran space.

As in the case of the usual Ran space, there is a correspondence

which makes $\mathrm{Ran}^{un}$ into a commutative algebra in correspondences of lax prestacks. Define a unital factorization space to be a multiplicative space over $\mathrm{Ran}^{\mathrm{un}}$ .

More explicitly, we may think of a unital factorization space as a prestack $\mathcal{Y} \to \mathrm{Ran}$ together with associative and unital maps $\mathcal{Y}_{x_I} \to \mathcal{Y}_{x_J}$ between fibers whenever we have an inclusion $x_I \subseteq x_J$ . By abuse of notation, we will often call $\mathcal{Y}$ a unital factorization space when the associated structure of a left fibration is clear from context.

Let $\mathcal{Y}^{\mathrm{un}} \to \mathrm{Ran}^{\mathrm{un}}$ be a unital factorization space, and let $\mathcal{Y}$ be the underlying ordinary factorization space. Since $\mathcal{Y}^{\mathrm{un}}$ is a lax prestack, we may consider the category $\mathfrak{D}_{\mathrm{str}}(\mathcal{Y}^{\mathrm{un}})$ of unital D-modules on $\mathcal{Y}$ . There is a natural forgetful functor $\mathrm{oblv}_{\mathcal{Y}}: \mathfrak{D}_{\mathrm{str}}(\mathcal{Y}^{\mathrm{un}}) \to \mathfrak{D}(\mathcal{Y})$ obtained by $!$ -pull back along the composition

\begin{equation*}\mathcal{Y} \to \mathcal{Y}^{\mathrm{un}} \to \mathcal{Y}_{\mathrm{str}},\end{equation*}

where the first map is the projection $\mathcal{Y}= \mathcal{Y}^{\mathrm{un}} \times _{\mathrm{Ran}^{\mathrm{un}}} \mathrm{Ran} \to \mathcal{Y}^{\mathrm{un}}$ . The following proposition is standard.

Proposition 3.1. The forgetful functor

\begin{equation*} \mathrm{oblv}_{\mathcal{Y}}: \mathfrak{D}_{\mathrm{str}}(\mathcal{Y}^{\mathrm{un}}) \longrightarrow \mathfrak{D}(\mathcal{Y})\end{equation*}

is fully faithful.

Hence we may call $\mathfrak{D}_{\mathrm{str}}(\mathcal{Y}^{\mathrm{un}})$ the unital subcategory of $\mathfrak{D}(\mathcal{Y})$ , and often denote it by $\mathfrak{D}_{\mathrm{untl}}(\mathcal{Y})$ .

3.3 An alternative description of the unital category

A unital factorization space $\mathcal{Y}^{\mathrm{un}}$ is determined by an ordinary factorization space $\mathcal{Y} \to \mathrm{Ran}$ together with an action

\begin{equation*} \mathrm{ act}_{\mathcal{Y}}: \mathrm{Ran} \times \mathcal{Y} \to \mathcal{Y,} \end{equation*}

which extends the natural map over the disjoint locus provided by factorization. In particular, we obtain a simplicial prestack

whose associated geometric realization $|\mathcal{Y}^{\bullet }|$ is canonically isomorphic to $\mathcal{Y}^{\mathrm{un}}_{\mathrm{str}}$ . It follows that there is a canonical equivalence of categories

\begin{equation*}\mathrm{Tot}(\mathfrak{D}(\mathcal{Y}^{\bullet })) \overset {\sim }{\longrightarrow } \mathfrak{D}_{\mathrm{str}}(\mathcal{Y}^{\mathrm{un}}),\end{equation*}

which is compatible with factorization. One should think of $\mathrm{Tot}(\mathfrak{D}(\mathcal{Y}^{\bullet }))$ as a category of $\mathrm{Ran}$ -equivariant sheaves on $\mathcal{Y}$ .

Note that by homological contractibility of Ran, the category $\mathrm{Tot}(\mathfrak{D}(\mathcal{Y}^{\bullet }))$ may be explicitly described as follows. It is the full subcategory of $\mathfrak{D}(\mathcal{Y})$ of sheaves $c$ such that there exists a (necessarily unique) isomorphism

\begin{equation*}\operatorname {pr}_{\mathcal{Y}}^!(c) \overset {\sim }{\longrightarrow } \mathrm{ act}_{\mathcal{Y}}^!(c),\end{equation*}

i.e. equivariance is a property rather than an extra structure.

3.4 The configuration space

We will now discuss a ‘graded’ analog of $\mathrm{Ran}$ (see [Reference RaskinRas21]). Let $\mathrm{Div}^{-\mathrm{eff}}$ denote the moduli of anti-effective divisors on $X$ and let $\check {\Lambda }^+$ be the monoid of dominant weights of $G$ . Define the configuration space $\mathrm{Conf}$ as the space of homomorphisms $\check {\Lambda }^+ \to \mathrm{Div}^{-\mathrm{eff}}$ . Equivalently, $\mathrm{Conf}$ is the moduli of sums $\sum _j \lambda _j x_j$ where $\lambda _j \in \Lambda ^-$ is a negative coweight for each $j$ and the $x_j$ values are pairwise distinct. We will refer to the points of $\mathrm{Conf}$ as colored divisors on $X$ .

The prestack $\mathrm{Conf}$ is actually a scheme with $\pi _0(\mathrm{Conf})= \Lambda ^-$ where each connected component $\mathrm{Conf}^{\lambda }$ consists of colored divisors with total degree $\lambda$ . Explicitly, a choice of simple coroots determines an isomorphism

\begin{equation*} \mathrm{Conf}^{\lambda } \overset {\sim }{\longrightarrow } \prod _{\alpha _i \in \mathcal{I}_G} X^{(n_i)},\end{equation*}

where $X^{(n_i)}= X^{n_i}//S_{n_i}$ is the GIT quotient of $X^{n_i}$ by the symmetric group. We will often denote $\mathrm{Conf}^{\lambda }$ by $X^{\lambda }$ and call it a partially symmetrized power of $X$ .

Consider the incidence divisor in the product $X \times \mathrm{Conf}$ . Given an $S$ -point $D: S \to \mathrm{Conf}$ , define $\Gamma _D$ to be the relative effective divisor in $X \times S$ , equal to the pullback along

\begin{equation*}\operatorname {pr_X} \times D: X \times S \to X \times \mathrm{Conf}\end{equation*}

of the incidence divisor. Here, $\operatorname {pr}_X$ is the projection onto $X$ . As in the case of the Ran space, there is a subscheme $[\mathrm{Conf} \times \mathrm{Conf}]_{\mathrm{disj}}$ of $\mathrm{Conf} \times \mathrm{Conf}$ consisting of pairs $(D,E)$ such that $\Gamma _D \cap \Gamma _E = \emptyset$ . As before, there is a correspondence

which makes $\mathrm{Conf}$ into a commutative algebra in correspondences. Here, the second leg of the correspondence is the restriction of the map $\mathrm{add}_{\mathrm{Conf}}: \mathrm{Conf} \times \mathrm{Conf} \to \mathrm{Conf}$ given by addition of divisors. Hence we may define a factorization space over $\mathrm{Conf}$ to be a multiplicative space for $\mathrm{Conf}$ .

Parallel to what happens for the Ran space, the configuration space may be upgraded to a lax prestack

\begin{equation*} \mathrm{Conf}^{\to }: \mathrm{AffSch}^{\mathrm{op}}_k \longrightarrow \mathrm{Cat,}\end{equation*}

where there is a unique arrow $D \to E$ if and only if $\Gamma _D \subseteq \Gamma _E$ . In typical fashion, there is a natural structure of a commutative algebra in correspondences on $\mathrm{Conf}^{\to }$ . As before, we may consider left fibrations over $\mathrm{Conf}^{\to }$ endowed with a multiplicative structure. To distinguish them from the Ran situation, we will call such objects effective factorization spaces.

Given an effective factorization space $\mathcal{Y}^{\to } \to \mathrm{Conf}^{\to }$ with underlying factorization space $\mathcal{Y} \to \mathrm{Conf}$ , we may consider the category $\mathfrak{D}_{\mathrm{eff}}(\mathcal{Y}) := \mathfrak{D}(\mathcal{Y}^{\to }_{\mathrm{str}})$ of effective sheaves on $\mathcal{Y}$ . There is also a description of this category in terms of equivariant sheaves with respect to a corresponding defect action $\mathrm{Conf} \times \mathcal{Y} \to \mathcal{Y}$ . Note, however, that since $\mathrm{Conf}$ is not contractible, the forgetful functor $\mathfrak{D}_{\mathrm{eff}}(\mathcal{Y}) \to \mathfrak{D}(\mathcal{Y})$ is no longer fully faithful.

4.1 The configuration space semi-infinite IC sheaf

We will also need a version of $\mathrm{IC}^{\infty /2}_{\mathrm{Ran}}$ which lies over the configuration space. To this end, define the configuration space affine Grassmannian $\mathrm{Gr}_{G,\mathrm{Conf}}$ to be the prestack whose $S$ -points are given by triples $(D, \mathscr{P}_G, \alpha )$ , where:

1. (i) $D$ is an $S$ -family of colored divisors;

2. (ii) $\mathscr{P}_G \to X \times S$ is a $G$ -bundle;

3. (iii) $\alpha$ is a trivialization of $\mathscr{P}_G$ on $(X \times S) \setminus \Gamma _D$ .

As before, we define a closed substack ${\overline {S}}\vphantom {S}^0_{\mathrm{Conf}}$ inside $\mathrm{Gr}_{G, \mathrm{Conf}}$ by a positivity condition. More precisely, the trivialization $\alpha$ induces meromorphic maps

\begin{equation*}\mathcal{O}_{X \times S} \longrightarrow V^{\check {\lambda }}_{\mathscr{P}_G} \end{equation*}

for every dominant weight $\check {\lambda }$ and we require that these extend to $X \times S$ . The space ${\overline {S}}\vphantom {S}^0_{\mathrm{Conf}}$ is also equipped with a canonical map ${\overline {\mathfrak{p}}}\vphantom {\mathfrak{p}}_{\mathrm{Conf}}$ to ${\overline {\mathrm{Bun}}}\vphantom {\mathrm{Bun}}_N$ .

We may endow $\mathrm{Gr}_{G,\mathrm{Conf}}$ with the structure of an effective factorization space as follows. Let $D$ be a colored divisor and let $D'$ be another colored divisor such that $\Gamma _D \subseteq \Gamma _{D'}$ . Suppose we have a point $(D,\mathscr{P}_G,\alpha )$ of $\mathrm{Gr}_{G,\mathrm{Conf}}$ lying over $D$ . The effective structure is given bythe map

\begin{equation*} (D,\mathscr{P}_G, \alpha ) \mapsto (D',\mathscr{P}_G, \alpha '),\end{equation*}

where $\alpha '$ is the restriction of $\alpha$ to $X \times S \setminus \Gamma _{D'}$ . The factorization structure is again given by gluing $G$ -bundles. It is evident that ${\overline {S}}\vphantom {S}^0_{\mathrm{Conf}}$ is closed under both structures.

Define the correspondence configuration space $\mathrm{Conf}_{\mathrm{corr}}$ as the subspace of $\mathrm{Conf} \times \mathrm{Ran}$ given by the condition that the divisor is set-theoretically supported on the point of $\mathrm{Ran}$ .

We have a correspondence

(4.4)

defined in the obvious way. For a space $\mathcal{Y} \to \mathrm{Conf}$ lying over the configuration space, define

\begin{equation*}\mathcal{Y}_{\mathrm{corr}} := \mathcal{Y} \times _{\mathrm{Conf}} \mathrm{Conf}_{\mathrm{corr}}\end{equation*}

as a space over $\mathrm{Conf}_{\mathrm{corr}}$ . Similarly, define

\begin{equation*}\mathcal{Y}_{\mathrm{corr},R} := \mathcal{Y} \times _{\mathrm{Ran}} \mathrm{Conf}_{\mathrm{corr}} \end{equation*}

for a space $\mathcal{Y} \to \mathrm{Ran}$ .

We also have a correspondence analogous to (4.4) whose left leg is $\mathrm{Conf}^{\to }$ and whose total space is denoted $\mathrm{Conf}^{\to }_{\mathrm{corr}}$ . For a space $\mathcal{Y} \to \mathrm{Conf}^{\to }$ , denote by $\mathcal{Y}^{\to }_{\mathrm{corr}}$ its pullback to $\mathrm{Conf}^{\to }_{\mathrm{corr}}$ . Similarly, for a space $\mathcal{Y} \to \mathrm{Ran}$ we have the space $\mathcal{Y}^{\to }_{\mathrm{corr},R}$ over $\mathrm{Conf}^{\to }_{\mathrm{corr}}$ .

Let us show that $\mathrm{Conf}^{\to }_{\mathrm{corr}}$ is a commutative algebra in $\mathrm{Corr}(\mathrm{LaxPrStk})$ . Define

\begin{equation*}[\mathrm{Conf}^{\to }_{\mathrm{corr}} \times \mathrm{Conf}^{\to }_{\mathrm{corr}}]_{\mathrm{disj}} := (\mathrm{Conf}^{\to }_{\mathrm{corr}} \times \mathrm{Conf}^{\to }_{\mathrm{corr}}) \times _{\mathrm{Ran} \times \mathrm{Ran}} [\mathrm{Ran} \times \mathrm{Ran}]_{\mathrm{disj}}, \end{equation*}

and note that, in addition to the obvious map

\begin{equation*}\operatorname {inc}_{\mathrm{corr}}: [\mathrm{Conf}^{\to }_{\mathrm{corr}} \times \mathrm{Conf}^{\to }_{\mathrm{corr}}]_{\mathrm{disj}} \to \mathrm{Conf}^{\to }_{\mathrm{corr}} \times \mathrm{Conf}^{\to }_{\mathrm{corr}},\end{equation*}

we have another map

\begin{equation*}\mathrm{add}_{\mathrm{corr}}: [\mathrm{Conf}^{\to }_{\mathrm{corr}} \times \mathrm{Conf}^{\to }_{\mathrm{corr}}]_{\mathrm{disj}} \longrightarrow \mathrm{Conf}^{\to }_{\mathrm{corr}},\end{equation*}

which sends a pair $( (D,x_I), (E,x_J) )$ to $(D+E,x_I \cup x_J)$ .

The inclusion of the pair

\begin{equation*}(0,\emptyset ) \hookrightarrow {\quad } \mathrm{Conf}^{\to }_{\mathrm{corr}}\end{equation*}

provides the necessary unit for $\mathrm{Conf}^{\to }_{\mathrm{corr}}$ , and it is easy to see that the above maps endow $\mathrm{Conf}^{\to }_{\mathrm{corr}}$ with the structure of a commutative algebra in $\mathrm{Corr}(\mathrm{LaxPrStk})$ . Note that the inclusion of $(\emptyset ,\emptyset )$ into $\mathrm{Conf}^{\to }_{\mathrm{corr}}$ is an inclusion of a connected component.

Now the map $\mathfrak{c}_L$ upgrades to a lax morphism of commutative algebras, while the second projection $\mathfrak{c}_R$ is a strict morphism of commutative algebras. It follows that for an effective factorization space $\mathcal{Y}^{\to } \to \mathrm{Conf}^{\to }$ , the pullback $\mathcal{Y}^{\to }_{\mathrm{corr}}$ to $\mathrm{Conf}^{\to }_{\mathrm{corr}}$ obtains a structure of a multiplicative space over $\mathrm{Conf}^{\to }_{\mathrm{corr}}$ . Similarly, if $\mathcal{Y}$ is a factorization space over $\mathrm{Ran}$ then $\mathcal{Y}_{\mathrm{corr},R}$ obtains a structure of a multiplicative space over $\mathrm{Conf}^{\to }_{\mathrm{corr}}$ .

In addition to the usual structure on $\mathrm{Conf}^{\to }_{\mathrm{corr}}$ of a commutative algebra in correspondences, there is a richer structure ${'}{\mathrm{Conf}_{\mathrm{corr}}^{\to }}$ . The underlying prestack of ${'}{\mathrm{Conf}_{\mathrm{corr}}^{\to }}$ is just $\mathrm{Conf}_{\mathrm{corr}}$ , whereas morphisms are given as follows. For an affine scheme $S$ , there is precisely one morphism $(D,x_I) \to (E,x_J)$ of $S$ -points whenever there is a morphism $D \to E$ in $\mathrm{Conf}^{\to }(S)$ and $x_I \to x_J$ in $\mathrm{Ran}^{\mathrm{un}}(S)$ . Note the lax prestack ${'}{\mathrm{Conf}_{\mathrm{corr}}^{\to }}$ also acquires a structure of a commutative algebra in correspondences such that the inclusion

\begin{equation*}\mathrm{Conf}_{\mathrm{corr}}^{\to } \longrightarrow {'}{\mathrm{Conf}_{\mathrm{corr}}^{\to }} \end{equation*}

is a map of commutative algebras.

The maps $\mathfrak{c}_L$ and $\mathfrak{c}_R$ are compatible with the structures of the previous paragraph, and moreover $\mathfrak{c}_L$ is a pointwise Cartesian fibration. For a multiplicative space $\mathcal{Y}$ (respectively $\mathcal{X}$ ) over $\mathrm{Conf}^{\to }_{\mathrm{corr}}$ (respectively over $\mathrm{Ran}^{\mathrm{un}}$ ), the space $\mathfrak{c}_L^{\mathrm{Spc},*}(\mathcal{Y})=\mathcal{Y}_{\mathrm{corr}}$ (respectively $\mathfrak{c}^{\mathrm{Spc},*}_R(c)=\mathcal{Y}_{\mathrm{corr},R}$ ) over ${'}{\mathrm{Conf}^{\to }_{\mathrm{corr}}}$ has a canonical structure of a multiplicative space. Note also the projection

(4.5) \begin{equation} \mathcal{Y}_{\mathrm{corr}} \longrightarrow \mathcal{Y} \end{equation}

is a pointwise Cartesian fibration by base change.

Let $\mathfrak{D}_{\mathrm{untl}}(\mathcal{Y}_{\mathrm{corr}})$ denote the full subcategory of $\mathfrak{D}(\mathcal{Y}_{\mathrm{corr}})$ consisting of sheaves $c$ such that $c_{S,f} \to c_{S,g}$ is an isomorphism for any arrow $f \to g$ of $S$ -points in $\mathcal{Y}_{\mathrm{corr}}$ whose image in $\mathcal{Y}$ is the identity. For any $\mathcal{Y}$ such that $\mathcal{Y}(S)$ is discrete for all $S$ , by [Reference CisinskiCis19, Proposition 7.1.12] the map (4.5) is a localization and there is a canonical equivalence

(4.6) \begin{equation} \mathfrak{D}(\mathcal{Y}) \overset {\sim }{\longrightarrow } \mathfrak{D}_{\mathrm{untl}}(\mathcal{Y}_{\mathrm{corr}}). \end{equation}

Indeed, a sheaf on $\mathcal{Y}_{\mathrm{corr}}$ is in particular the data of a functor

\begin{equation*}\mathcal{Y}_{\mathrm{corr}}(S) \longrightarrow \mathfrak{D}(S),\end{equation*}

for every affine scheme $S$ , and by our assumptions any such which sends the appropriate morphisms to inverses admits a unique factorization through (4.5). Note an analogous discussion holds over $\mathrm{Ran}$ and we obtain a category $\mathfrak{D}_{\mathrm{untl}}(\mathcal{Y}_{\mathrm{corr},R})$ for any multiplicative space $\mathcal{Y} \to \mathrm{Ran}^{\mathrm{un}}$ .

We claim there is a map $\iota _{\mathrm{corr}}: {\overline {S}}\vphantom {S}^0_{\mathrm{Conf},\mathrm{corr}} \to {\overline {S}}\vphantom {S}^0_{\mathrm{Ran},\mathrm{corr},R}$ which makes the triangle

commute. Indeed, $\iota _{\mathrm{corr}}\big ((x_J,D,\mathscr{P}_G,\alpha )\big )$ is obtained by restricting $\alpha$ to the complement of $\Gamma _{x_J}$ . In the following proposition, denote by $t_{\mathrm{Conf}}$ (respectively $t_{\mathrm{Ran}}$ ) the projection ${\overline {S}}\vphantom {S}^0_{\mathrm{Conf},\mathrm{corr}} \to {\overline {S}}\vphantom {S}^0_{\mathrm{Conf}}$ (respectively ${\overline {S}}\vphantom {S}^0_{\mathrm{Ran},\mathrm{corr}} \to {\overline {S}}\vphantom {S}^0_{\mathrm{Ran}}$ ).

Proposition 4.4. There exists an effective factorization algebra $\mathrm{IC}^{\infty /2}_{\mathrm{Conf}}$ over ${\overline {S}}\vphantom {S}^0_{\mathrm{Conf}}$ together with an isomorphism

\begin{equation*}t_{\mathrm{Conf}}^!(\mathrm{IC}^{\infty /2}_{\mathrm{Conf}}) \overset {\sim }{\longrightarrow } (t_{\mathrm{Ran}} \circ \iota _{\mathrm{corr}})^!(\mathrm{IC}^{\infty /2}_{\mathrm{Ran}}),\end{equation*}

which is compatible with factorization.

Proof. Let us apply the discussion in 4.1 to our present situation. Consider the multiplicative spaces ${\overline {S}}\vphantom {S}^0_{\mathrm{Conf},\mathrm{corr}}$ and ${\overline {S}}\vphantom {S}^0_{\mathrm{Ran},\mathrm{corr},R}$ over ${'}{\mathrm{Conf}^{\to }_{\mathrm{corr}}}$ . It is easy to see that the map $\iota _{\mathrm{corr}}: {\overline {S}}\vphantom {S}^0_{\mathrm{Conf},\mathrm{corr}} \to {\overline {S}}\vphantom {S}^0_{\mathrm{Ran},\mathrm{corr},R}$ is compatible with multiplicative structures on both sides. It follows that the shriek pullback $\iota _{\mathrm{corr}}^!$ restricts to a functor

\begin{equation*}\iota _{\mathrm{corr}}^!: \mathfrak{D}_{\mathrm{untl}}({\overline {S}}\vphantom {S}^0_{\mathrm{Ran},\mathrm{corr},R}) \longrightarrow \mathfrak{D}_{\mathrm{untl}}({\overline {S}}\vphantom {S}^0_{\mathrm{Conf},\mathrm{corr}}).\end{equation*}

Hence, we will obtain the object $\mathrm{IC}^{\infty /2}_{\mathrm{Conf}} \in \mathfrak{D}({\overline {S}}\vphantom {S}^0_{\mathrm{Conf}})$ corresponding uniquely under the equivalence (4.6) to $(\iota _{\mathrm{Ran}} \circ \iota _{\mathrm{corr}})^!(\mathrm{IC}^{\infty /2}_{\mathrm{Ran}})$ provided $\mathrm{IC}^{\infty /2}_{\mathrm{Ran}}$ belongs to the unital subcategory of $\mathfrak{D}({\overline {S}}\vphantom {S}^0_{\mathrm{Ran}})$ . The latter is the content of [Reference GaitsgoryGai21b, § 4]. Note that $\mathrm{IC}^{\infty /2}_{\mathrm{Conf}}$ immediately inherits the structure of an effective factorization algebra from $\mathrm{IC}^{\infty /2}_{\mathrm{Ran}}$ .

5.1 Drinfeld’s compactification

Let $\mathrm{Bun}_B$ denote the moduli of $B$ -bundles on $X$ , and consider the induction morphism $\operatorname {pr}_{B,G}:\mathrm{Bun}_B \longrightarrow \mathrm{Bun}_G.$ Although the fibers of this map are not proper in general, there is a proper morphism

\begin{equation*}\overline {\operatorname {pr}}_{B,G}: {\overline {\mathrm{Bun}}}\vphantom {\mathrm{Bun}}_B \longrightarrow \mathrm{Bun}_G,\end{equation*}

together with an open embedding $j: \mathrm{Bun}_B \to {\overline {\mathrm{Bun}}}\vphantom {\mathrm{Bun}}_B$ with dense image which commutes with the projection to $\mathrm{Bun}_G$ . The stack ${\overline {\mathrm{Bun}}}\vphantom {\mathrm{Bun}}_B$ is known as Drinfeld’s compactification, and is explicitly defined as follows. Let $S$ be an affine scheme. A map $S \to {\overline {\mathrm{Bun}}}\vphantom {\mathrm{Bun}}_B$ is the data of a $G$ -bundle $\mathscr{P}_G$ and a $T$ -bundle $\mathscr{P}_T$ on $X \times S$ , together with embeddings of coherent sheaves

\begin{equation*}\check {\lambda }(\mathscr{P}_T) \longrightarrow V^{\check {\lambda }}_{\mathscr{P}_G}\end{equation*}

for every $\check {\lambda } \in \check {\Lambda }$ which satisfy the Plücker relations. There is an evident map ${\overline {\mathrm{Bun}}}\vphantom {\mathrm{Bun}}_B \to \mathrm{Bun}_T$ whose preimage over $\mathrm{Bun}_T^{\lambda }$ will be denoted ${\overline {\mathrm{Bun}}}\vphantom {\mathrm{Bun}}_B^{\lambda }$ . Here, $\mathrm{Bun}_T^{\lambda }$ is the connected component of $\mathrm{Bun}_T$ corresponding to $\lambda \in \Lambda$ consisting of $T$ -bundles of total degree $\lambda$ .

There is a stratification of ${\overline {\mathrm{Bun}}}\vphantom {\mathrm{Bun}}_B$ by smooth substacks indexed by positive coweights $\Lambda ^+$ . Each stratum ${_{=\lambda }}{{\overline {\mathrm{Bun}}}\vphantom {\mathrm{Bun}}_B}$ is canonically isomorphicFootnote ⁶ to $X^{\lambda } \times \mathrm{Bun}_B$ , and we will write $\iota _{\lambda }$ for the inclusion ${_{=\lambda }}{{\overline {\mathrm{Bun}}}\vphantom {\mathrm{Bun}}_B} \to {\overline {\mathrm{Bun}}}\vphantom {\mathrm{Bun}}_B$ . The union of strata indexed by coweights which are less than or equal to some fixed coweight $\mu$ is an open substack of ${\overline {\mathrm{Bun}}}\vphantom {\mathrm{Bun}}_B$ and will be denoted by ${_{\leq \mu }}{{\overline {\mathrm{Bun}}}\vphantom {\mathrm{Bun}}_B}$ .

The maps $\iota _{\lambda }$ extend to finite morphisms

(5.1) \begin{equation} \overline {\iota }_{\lambda }: X^{\lambda } \times {\overline {\mathrm{Bun}}}\vphantom {\mathrm{Bun}}_B \longrightarrow {\overline {\mathrm{Bun}}}\vphantom {\mathrm{Bun}}_B, \end{equation}

which assemble into an action of the configuration space on ${\overline {\mathrm{Bun}}}\vphantom {\mathrm{Bun}}_B$ . Moreover, the diagram

(5.2)

commutes, where $a_{X^{\lambda }}: X^{\lambda } \to \mathrm{pt}$ is the unique map. In practice, we will more often use the corresponding structure for $B^-$ .

We will also need a variant ${\overline {\mathrm{Bun}}}\vphantom {\mathrm{Bun}}_N$ of Drinfeld’s compactification for the unipotent radical $N$ of $B$ . By definition, ${\overline {\mathrm{Bun}}}\vphantom {\mathrm{Bun}}_N$ sits in a pullback square

where $\mathrm{Spec}(k) \to \mathrm{Bun}_T$ is the $k$ -point of $\mathrm{Bun}_T$ corresponding to the trivial $T$ -bundle. Since $T$ acts on $N$ by the adjoint action, it also naturally acts on the stack ${\overline {\mathrm{Bun}}}\vphantom {\mathrm{Bun}}_N$ .

We will denote by ${\overline {\mathscr{F}}}\vphantom {\mathscr{F}}_{N,B^-}$ the fiber product

\begin{equation*}{\overline {\mathrm{Bun}}}\vphantom {\mathrm{Bun}}_N \times _{\mathrm{Bun}_G} {\overline {\mathrm{Bun}}}\vphantom {\mathrm{Bun}}_{B^-}. \end{equation*}

Define the compactified Zastava space ${\overline {\mathcal{Z}}}\vphantom {\mathcal{Z}}$ to be the open locus of ${\overline {\mathscr{F}}}\vphantom {\mathscr{F}}_{N,B^-}$ where the two generalized reductions are generically transverse. Explicitly, an $S$ -point of ${\overline {\mathcal{Z}}}\vphantom {\mathcal{Z}}$ consists of a $G$ -bundle $\mathscr{P}_G$ and a $T$ -bundle $\mathscr{P}_T$ on $X \times S$ together with diagrams

(5.3) \begin{equation} \mathcal{O}_{X \times S} \overset {\kappa ^{\check {\lambda }}}{\longrightarrow } V^{\check {\lambda }}_{\mathscr{P}_G} \overset {(\kappa ^-)^{\check {\lambda }}}{\longrightarrow } \check {\lambda }(\mathscr{P}_T), \end{equation}

for every dominant weight $\check {\lambda }$ , where the collections $\{\kappa ^{\check {\lambda }}\}$ and $\{(\kappa ^-)^{\check {\lambda }}\}$ satisfy the Plücker equations and each composition $(\kappa ^-)^{\check {\lambda }} \circ \kappa ^{\check {\lambda }}$ is nonzero.

Denote by $\overline {\mathfrak{p}}$ (respectively $\overline {\mathfrak{q}}$ ) the projection ${\overline {\mathcal{Z}}}\vphantom {\mathcal{Z}} \to {\overline {\mathrm{Bun}}}\vphantom {\mathrm{Bun}}_N$ (respectively ${\overline {\mathcal{Z}}}\vphantom {\mathcal{Z}} \to {\overline {\mathrm{Bun}}}\vphantom {\mathrm{Bun}}_{B^-}$ ). Note that ${\overline {\mathcal{Z}}}\vphantom {\mathcal{Z}}$ splits into connected components ${\overline {\mathcal{Z}}}\vphantom {\mathcal{Z}}^{\lambda }$ indexed by negative coweights $\lambda$ , where ${\overline {\mathcal{Z}}}\vphantom {\mathcal{Z}}^{\lambda }$ is given as the preimage of ${\overline {\mathrm{Bun}}}\vphantom {\mathrm{Bun}}_{B^-}^{\lambda }$ under $\overline {\mathfrak{q}}$ . By taking the zeros of the composition of (5.3) we obtain a map

\begin{equation*}\overline {\pi }^{\lambda }: {\overline {\mathcal{Z}}}\vphantom {\mathcal{Z}}^{\lambda } \longrightarrow X^{\lambda }.\end{equation*}

It is well known that each ${\overline {\mathcal{Z}}}\vphantom {\mathcal{Z}}^{\lambda }$ is a scheme of finite type, and that $\overline {\pi }^{\lambda }$ is proper. Moreover, there is a natural factorization structure on ${\overline {\mathcal{Z}}}\vphantom {\mathcal{Z}}$ with respect to $\overline {\pi }$ .

By commutativity of the diagram (5.2) for $B$ replaced by $B^-$ , the maps $\overline {\iota }_{\lambda }$ give rise to an action of $\mathrm{Conf}$ on the fiber product ${\overline {\mathscr{F}}}\vphantom {\mathscr{F}}_{N,B^-}$ . Moreover, since the condition defining ${\overline {\mathcal{Z}}}\vphantom {\mathcal{Z}}$ is generic, the compactified Zastava is preserved by this action. This equips ${\overline {\mathcal{Z}}}\vphantom {\mathcal{Z}}$ with the structure of an effective factorization space.

By requiring the maps $(\kappa ^-)^{\check {\lambda }}$ in (5.3) to be surjective, we obtain the affine Zastava $\mathcal{Z}$ . The space $\mathcal{Z}$ admits a factorization structure and the dense open embedding

\begin{equation*}j_Z : \mathcal{Z} \hookrightarrow {\overline {\mathcal{Z}}}\vphantom {\mathcal{Z}}\end{equation*}

is compatible with factorization on both sides. Note, however, that the factorization structure on $\mathcal{Z}$ is not effective. Similarly, by requiring that the maps $\kappa ^{\check {\lambda }}$ are regular embeddings of vector bundles, we obtain the factorizable open subscheme

\begin{equation*}j_Z^-: \mathcal{Z}^- \hookrightarrow {\overline {\mathcal{Z}}}\vphantom {\mathcal{Z}},\end{equation*}

which we call the opposite affine Zastava. The intersection $\mathring {\mathcal{Z}} := \mathcal{Z} \cap \mathcal{Z}^-$ is called the open Zastava and is known to be smooth. Note that each variation of the Zastava space carries a $T$ -action.

5.2 Zastava spaces via the affine Grassmannian

Suppose we have a point of the compactified Zastava space, consisting of two generically transverse generalized $N$ and $B^-$ reductions of a $G$ -bundle $\mathscr{P}_G$ . Over the locus of transversality both reductions are genuine, and intersect in a single point over each fiber of the projection $\mathscr{P}_G \to X \times S$ . It follows that we may equip $\mathscr{P}_G$ with a section away from the support of a colored divisor $D$ , and hence we obtain a point of $\mathrm{Gr}_{G,\mathrm{Conf}}$ . The result is a closed embedding

\begin{equation*}{\overline {\mathcal{Z}}}\vphantom {\mathcal{Z}} \hookrightarrow {\quad } \mathrm{Gr}_{G,\mathrm{Conf}},\end{equation*}

which we will now describe more explicitly.

Define ${\overline {S}}\vphantom {S}^{-}_{\mathrm{Conf}}$ to be the subspace of $\mathrm{Gr}_{G,\mathrm{Conf}}$ whose $S$ -points consist of triples $(D, \mathscr{P}_G, \alpha )$ such that for each dominant weight $\check {\lambda } \in \check {\Lambda }^+$ the composition

\begin{equation*}V^{\check {\lambda }}_{\mathscr{P}_G} \overset {\alpha }{\longrightarrow } V^{\check {\lambda }}_{\mathscr{P}_G^0} \longrightarrow \mathcal{O}_{X \times S}\end{equation*}

of meromorphic maps extends to a regular map

\begin{equation*}V^{\check {\lambda }}_{\mathscr{P}_G^0} \longrightarrow \mathcal{O}_{X \times S}(-\check {\lambda }(D)),\end{equation*}

where the map $V^{\check {\lambda }}_{\mathscr{P}_G^0} \to \mathcal{O}_{X \times S}$ corresponds to the unique $N^-$ -invariant functional on $V^{\check {\lambda }}$ . Note that ${\overline {S}}\vphantom {S}^-_{\mathrm{Conf}}$ inherits an effective structure from the one on $\mathrm{Gr}_{G,\mathrm{Conf}}$ .

The image of the compactified Zastava space in $\mathrm{Gr}_{G,\mathrm{Conf}}$ may be described as the intersection

\begin{equation*}{\overline {S}}\vphantom {S}^0_{\mathrm{Conf}} \times _{\mathrm{Gr}_{G,\mathrm{Conf}}} {\overline {S}}\vphantom {S}^-_{\mathrm{Conf}} \hookrightarrow {\quad } \mathrm{Gr}_{G,\mathrm{Conf}}.\end{equation*}

As a result of the definitions, we see that the embedding ${\overline {\mathcal{Z}}}\vphantom {\mathcal{Z}} \hookrightarrow \mathrm{Gr}_{G,\mathrm{Conf}}$ is compatible with the effective factorization structures on both sides. We are now ready to define our central object of study.

In the remaining sections we will give a more explicit description of $\mathrm{IC}_{\mathrm{Z}}^{\infty /2}$ and show that it is, in a precise sense, equivalent to $\mathrm{IC}^{\infty /2}_{\mathrm{Ran}}$ .

6.1 Factorization over the configuration space to Ran

In this section we will show that the assignment $\mathcal{Y} \mapsto \mathcal{Y}_{\mathrm{Ran}}$ sends effective factorization spaces to unital factorization spaces.

Proof. By Proposition 2.8, it suffices to show that the maps $\operatorname {inc}_{\mathrm{corr}}$ and $\mathrm{add}_{\mathrm{corr}}$ (as well as the corresponding maps for the unital diagram) are realization fibrations. By [Reference SteinebrunnerSte22, Theorem 2.1] it is enough to show that $\operatorname {inc}_{\mathrm{corr}}$ and $\mathrm{add}_{\mathrm{corr}}$ are both pointwise Cartesian and coCartesian fibrations. Note that, while the right leg of the unital diagram is not a pointwise (co)Cartesian fibration, it is still a realization fibration since it is the inclusion of a disjoint base point.

We will show that $\mathrm{add}_{\mathrm{corr}}$ is a pointwise coCartesian fibration, and the other cases will be left to the reader. Let

\begin{equation*}f: (D+E,x_I \cup x_J) \longrightarrow (D',x_I \cup x_J)\end{equation*}

be a morphism in $\mathrm{Conf}^{\to }_{\mathrm{corr}}$ with $( (D,x_I), (E,x_J) )$ in $[\mathrm{Conf}^{\to }_{\mathrm{corr}} \times \mathrm{Conf}^{\to }_{\mathrm{corr}}]_{\mathrm{disj}}$ . Since $x_I$ and $x_J$ are disjoint, we can write

\begin{equation*}D'=D'(x_I)+D'(x_J),\end{equation*}

where $D'(x_I)$ is set-theoretically supported on $x_I$ and $D'(x_J)$ is set-theoretically supported on $x_J$ . Then it is easy to see that there exists

\begin{equation*}f':((D,x_I),(E,x_J) ) \longrightarrow ((D'(x_I),x_I),(D'(x_J),x_J) ),\end{equation*}

which is a coCartesian lift of $f$ .

To see that $\mathcal{Y}_{\mathrm{Ran}}$ is a unital factorization space, first recall the action of $\mathrm{Ran}$ on $\mathcal{Y}^{\to }_{\mathrm{corr}}$ constructed in § 4. Clearly, this action is compatible with the morphisms in $\mathcal{Y}^{\to }_{\mathrm{corr}}$ and hence descends to an action on $\mathcal{Y}_{\mathrm{Ran}}$ . Lastly, it is easy to see that this action is compatible with the factorization structure constructed above.

Corollary 6.3.1. There is a natural equivalence of categories

\begin{equation*}\mathfrak{D}_{\mathrm{str}}(\mathcal{Y}^{\to }_{\mathrm{corr}}) \overset {\sim }{\longrightarrow } \mathfrak{D}(\mathcal{Y}_{\mathrm{Ran}}),\end{equation*}

which sends factorization algebras to factorization algebras.

The next proposition relates the effective category of $\mathcal{Y}^{\to }$ with the unital subcategory of $\mathfrak{D}(\mathcal{Y}_{\mathrm{Ran}})$ . Note that the natural projection $\operatorname {pr}_{\mathcal{Y}}: \mathcal{Y}^{\to }_{\mathrm{corr}} \to \mathcal{Y}^{\to }$ is a morphism of lax prestacks, and hence we obtain a functor

\begin{equation*}\operatorname {pr}^!_{\mathcal{Y}_{\mathrm{str}}}: \mathfrak{D}_{\mathrm{eff}}(\mathcal{Y}) \longrightarrow \mathfrak{D}(\mathcal{Y}_{\mathrm{Ran}}),\end{equation*}

from the category of effective sheaves on $\mathcal{Y}$ and the category of D-modules on $\mathcal{Y}_{\mathrm{Ran}}$ given by !-pullback along the induced map of straightenings.

Proposition 6.4. Let $\varphi ^{\to }:\mathcal{Y}^{\to } \to \mathrm{Conf}$ be an effective factorization space and assume the underlying prestack $\mathcal{Y}$ takes values in sets. Then the functor

\begin{equation*}\operatorname {pr}_{\mathcal{Y}_{\mathrm{str}}}^!: \mathfrak{D}_{\mathrm{eff}}(\mathcal{Y}) \longrightarrow \mathfrak{D}(\mathcal{Y}_{\mathrm{Ran}}) \end{equation*}

is fully faithful and its essential image is the unital subcategory of $\mathfrak{D}(\mathcal{Y}_{\mathrm{Ran}})$ .

Proof. By § 4 we may endow $\mathcal{Y}^{\to }_{\mathrm{corr}}$ with an action of $\mathrm{Ran}$ by functors. This gives us a modified lax prestack ${'}{\mathcal{Y}^{\to }_{\mathrm{corr}}}$ whose $S$ -objects are given by $\mathcal{Y}^{\to }_{\mathrm{corr}}(S)$ with a morphism

\begin{equation*}(y,x_I) \longrightarrow (y',x_J)\end{equation*}

for every morphism $y \to y'$ provided $x_I$ is contained in $x_J$ . By definition of the unital structure on $\mathcal{Y}_{\mathrm{Ran}}$ , the straightening of ${'}{\mathcal{Y}^{\to }_{\mathrm{corr}}}$ is tautologically isomorphic to $(\mathcal{Y}_{\mathrm{Ran}})_{\mathrm{str}}$ .

We claim that the obvious map

\begin{equation*}{'}{\mathcal{Y}^{\to }_{\mathrm{corr}}} \longrightarrow \mathcal{Y}^{\to },\end{equation*}

induces an isomorphism on the level of straightenings after sheafification in the fppf topology. The reason sheafification is needed is because $\mathrm{Conf}_{\mathrm{corr}} \to \mathrm{Conf}$ is only surjective locally in the fppf topology by results of [Reference BarlevBar14]. Now let $y_0$ be an $S$ -object of $\mathcal{Y}^{\to }$ . By Lemma 6.2, it suffices to show that the category

\begin{equation*}\mathcal{Y}^{\to }_{\mathrm{corr},y_0} := \mathcal{Y}^{\to }_{\mathrm{corr}}(S) \times _{\mathcal{Y}^{\to }(S)} \mathcal{Y}^{\to }_{y_0/}(S)\end{equation*}

is weakly contractible.

By our assumption on $\mathcal{Y}$ , the category $\mathcal{Y}^{\to }_{\mathrm{corr},y_0}$ is equivalent to the category whose objects are pairs $(y,x_J)$ as in the definition of $\mathcal{Y}^{\to }_{\mathrm{corr}}$ and $y_0$ admits a (necessarily unique) map to $y$ . Let $(y,x_I)$ and $(y',x_J)$ be two such objects. Since $\mathcal{Y}^{\to } \to \mathrm{Conf}^{\to }$ is a pointwise coCartesian fibration, we may lift the morphism

\begin{equation*}\varphi ^{\to }(y) \longrightarrow \varphi ^{\to }(y)+ \varphi ^{\to }(y') \end{equation*}

to a coCartesian morphism $y \to y_1$ . We obtain an analogous morphism $y' \to y_1'$ .

Let $\lambda$ be the degree of $\varphi ^{\to }(y)+ \varphi ^{\to }(y')$ . Then, as points of $\mathcal{Y}$ , $y_1$ and $y_1'$ lie in $\mathcal{Y}^{\lambda }$ over $\mathrm{Conf}^{\lambda }$ , and since $y_1$ and $y_1'$ admit a map from $y_0$ , we conclude that there is a (necessarily unique) isomorphism $y_1 \overset {\sim }{\to } y_1'$ . Hence we obtain maps

and by our hypothesis on $\mathcal{Y}$ , we conclude that $\mathcal{Y}^{\to }_{\mathrm{corr},/y_0}$ is a directed set and is therefore weakly contractible.

To conclude the proof, notice we have constructed a commutative diagram

where the bottom arrow is an isomorphism. It follows that $\operatorname {pr}^!_{\mathcal{Y}_{\mathrm{str}}}$ is fully faithful and has essential image given by $\mathfrak{D}_{\mathrm{untl}}(\mathcal{Y}_{\mathrm{Ran}})$ .

We will now apply the machinery above to the affine Grassmannian. An application of Lemma 6.2 shows that $(\mathrm{Gr}_{G,\mathrm{Conf}})_{\mathrm{Ran}}$ is canonically isomorphic to $\mathrm{Gr}_{G,\mathrm{Ran}}$ . Moreover, the unital structure on $\mathrm{Gr}_{G,\mathrm{Ran}}$ is inherited from the one on $\mathrm{Gr}^{\to }_{G,\mathrm{Conf},\mathrm{corr}}$ via Proposition 6.3. Likewise, we have a canonical isomorphism

\begin{equation*}({\overline {S}}\vphantom {S}^0_{\mathrm{Conf}})_{\mathrm{Ran}} \overset {\sim }{\longrightarrow } {\overline {S}}\vphantom {S}^0_{\mathrm{Ran}}\end{equation*}

inducing the unital structure on the right-hand side.

The next proposition shows that we can actually obtain all of $\mathrm{Gr}_{G,\mathrm{Ran}}$ from ${\overline {S}}\vphantom {S}^{-,\to }_{\mathrm{Conf}}$ , where we recall that ${\overline {S}}\vphantom {S}^{-,\to }_{\mathrm{Conf}}$ is the effective factorization space obtained from the space ${\overline {S}}\vphantom {S}^-_{\mathrm{Conf}}$ defined in § 5.2. Denote by $\iota ^-_{\mathrm{corr}}$ the inclusion ${\overline {S}}\vphantom {S}^{-,\to }_{\mathrm{Conf},\mathrm{corr}} \hookrightarrow \mathrm{Gr}^{\to }_{G,\mathrm{Conf},\mathrm{corr}}$ .

Lemma 6.5. The inclusion

\begin{equation*}\iota ^-_{\mathrm{corr}}: {\overline {S}}\vphantom {S}^{-,\to }_{\mathrm{Conf},\mathrm{corr}} \hookrightarrow \mathrm{Gr}^{\to }_{G,\mathrm{Conf},\mathrm{corr}}\end{equation*}

induces an equivalence

\begin{equation*}({\overline {S}}\vphantom {S}^{-,\to }_{\mathrm{Conf}})_{\mathrm{Ran}} \overset {\sim }{\longrightarrow } (\mathrm{Gr}_{G,\mathrm{Conf}})_{\mathrm{Ran}} \end{equation*}

and hence $({\overline {S}}\vphantom {S}^{-,\to }_{\mathrm{Conf}})_{\mathrm{Ran}}$ is canonically equivalent to $\mathrm{Gr}_{G,\mathrm{Ran}}$ .

Proof. Let $(D,x_I,\mathscr{P}_G,\alpha )$ be an $S$ -point of $\mathrm{Gr}^{\to }_{G,\mathrm{Conf},\mathrm{corr}}$ . We will construct a point $(D',x_I, \mathscr{P}_G,\alpha ')$ of ${\overline {S}}\vphantom {S}^{-,\to }_{\mathrm{Conf}}$ together with a morphism $(D,x_I,\mathscr{P}_G,\alpha ) \to (D',x_I,\mathscr{P}_G,\alpha ')$ . To this end, let $\omega _i$ be a fundamental weight of $G$ , and consider the meromorphic map

\begin{equation*}\beta _i: V^{\omega _i}_{\mathscr{P}_G} \longrightarrow \mathcal{O}_{X \times S}\end{equation*}

induced by $\alpha$ .

Since $\beta _i$ is defined away from the divisor $\Gamma _D$ , there exists an integer $n_i$ large enough such that the composition

\begin{equation*}V^{\omega _i}_{\mathscr{P}_G} \longrightarrow \mathcal{O}_{X \times S} \longrightarrow \mathcal{O}_{X \times S}(n_i \Gamma _D)\end{equation*}

is regular. Define the colored divisor

\begin{equation*}D' := D + \sum _{i \in \mathcal{I}_G} (-n_i)\alpha _i \Gamma _D\end{equation*}

and note that there is an obvious map $D \to D'$ . Define $\alpha '$ to be the restriction of the map $\alpha$ to the complement of $\Gamma _{D'}$ . Tautologically the tuple $(D',x_I,\mathscr{P}_G,\alpha ')$ admits a map from $(D,x_I, \mathscr{P}_G, \alpha )$ and $(D',x_I,\mathscr{P}_G,\alpha ')$ is a point of ${\overline {S}}\vphantom {S}^{-,\to }_{\mathrm{Conf}}$ .

Now consider the composition of

\begin{equation*}{\overline {S}}\vphantom {S}^{-,\to }_{\mathrm{Conf},\mathrm{corr}} \longrightarrow \mathrm{Gr}^{\to }_{G,\mathrm{Conf},\mathrm{corr}} \longrightarrow (\mathrm{Gr}_{G,\mathrm{Conf}})_{\mathrm{Ran}}, \end{equation*}

where the first arrow is the inclusion and the second arrow is the natural map. By the discussion above, this map is essentially surjective for every affine scheme $S$ , and the fact that it induces an equivalence

\begin{equation*}({\overline {S}}\vphantom {S}^{-}_{\mathrm{Conf}})_{\mathrm{Ran}} \overset {\sim }{\longrightarrow } (\mathrm{Gr}_{G,\mathrm{Conf}})_{\mathrm{Ran}}\end{equation*}

is a direct application of Lemma 6.2. Indeed, the fibers of the functor ${\overline {S}}\vphantom {S}^{-,\to }_{\mathrm{Conf},\mathrm{corr}}(S) \to \mathrm{Gr}_{G,\mathrm{Ran}}(S)$ are easily seen to be directed sets for every affine scheme $S$ .

We are now ready to prove the theorem.

Proof of Theorem 6.1. Recall we have a natural embedding ${\overline {\mathcal{Z}}}\vphantom {\mathcal{Z}}^{\to } \hookrightarrow \mathrm{Gr}_{G,\mathrm{Conf}}^{\to }$ whose essential image is the intersection ${\overline {S}}\vphantom {S}^{0,\to }_{\mathrm{Conf}} \cap {\overline {S}}\vphantom {S}^{-,\to }_{\mathrm{Conf}}$ . It is easy to see that ${\overline {S}}\vphantom {S}^{-,\to }_{\mathrm{Conf},\mathrm{corr}} \to \mathrm{Gr}_{G,\mathrm{Ran}}$ is both a Cartesian and coCartesian fibration, and hence a realization fibration. By Lemma 6.5 the inclusion

\begin{equation*}{\overline {\mathcal{Z}}}\vphantom {\mathcal{Z}} \hookrightarrow {\overline {S}}\vphantom {S}^0_{\mathrm{Conf}}\end{equation*}

induces a canonical isomorphism

\begin{equation*}{\overline {\mathcal{Z}}}\vphantom {\mathcal{Z}}_{\mathrm{Ran}} \overset {\sim }{\longrightarrow } ({\overline {S}}\vphantom {S}^0_{\mathrm{Conf}})_{\mathrm{Ran}} \simeq {\overline {S}}\vphantom {S}^0_{\mathrm{Ran}}\end{equation*}

on the level of fppf sheafifications in such a way that $\mathcal{Z}^-_{\mathrm{Ran}}$ is identified with $S^0_{\mathrm{Ran}}$ . We conclude the existence of a natural equivalence $\mathscr{F}:\mathfrak{D}_{\mathrm{eff}}({\overline {\mathcal{Z}}}\vphantom {\mathcal{Z}}) \overset {\sim }{\longrightarrow } \mathfrak{D}_{\mathrm{untl}}({\overline {S}}\vphantom {S}^0_{\mathrm{Ran}})$ by Proposition 6.4.

By construction, the equivalence $\mathscr{F}$ is the composition of two equivalences

\begin{equation*}\mathfrak{D}_{\mathrm{eff}}({\overline {\mathcal{Z}}}\vphantom {\mathcal{Z}}) \overset {\mathscr{F}_0}{\longrightarrow } \mathfrak{D}_{\mathrm{eff}}({\overline {S}}\vphantom {S}^0_{\mathrm{Conf}}) \overset {\mathscr{F}_1}{\longrightarrow } \mathfrak{D}_{\mathrm{untl}}({\overline {S}}\vphantom {S}^0_{\mathrm{Ran}})\end{equation*}

and tracing through the proof of Proposition 4.4 we see that $\mathscr{F}_1(\textrm {IC}^{\infty /2}_{\mathrm{Conf}})$ is canonically equivalent to $\mathrm{IC}^{\infty /2}_{\mathrm{Ran}}$ as a factorization algebra. Moreover, by definition of $\mathrm{IC}_{\mathrm{Z}}^{\infty /2}$ and the fact that the inclusion ${\overline {\mathcal{Z}}}\vphantom {\mathcal{Z}}^{\to } \hookrightarrow {\overline {S}}\vphantom {S}^{0,\to }_{\mathrm{Conf}}$ induces an equivalence on the level of straightenings, we have that $\mathscr{F}_0(\mathrm{IC}_{\mathrm{Z}}^{\infty /2})$ is canonically equivalent to $\mathrm{IC}^{\infty /2}_{\mathrm{Conf}}$ as an effective factorization algebra, whence the theorem.

6.2 The generic Zastava

The following sections on the generic Zastava will not be used in the rest of the text. Nevertheless, the results contained therein may be of independent interest as they connect the theory of Zastava spaces and the semi-infinite intersection cohomology sheaf to moduli spaces of rationally defined maps and principal bundles (see [Reference BarlevBar14] as well as [Reference GaitsgoryGai21b, Appendix A]).

Denote by $\mathrm{Gr}^{\mathrm{gen}}_G$ the straightening of the lax prestack $\mathrm{Gr}_{G,\mathrm{Ran}}^{\to }$ associated with the canonical unital structure. We wish to give a more explicit description of $\mathrm{Gr}^{\mathrm{gen}}_G$ . Let $S$ be an affine scheme, and recall that an open subscheme $U \subseteq X \times S$ is called a domain if it is universally dense with respect to the projection

\begin{equation*}p_S: X \times S \longrightarrow S.\end{equation*}

This means that for any closed point $s \in S$ , the intersection of the fiber $X_s$ of the projection $p_S$ over $s$ with $U$ is nonempty.

It is well known that $\mathrm{Gr}^{\mathrm{gen}}_G$ is the moduli space whose $S$ -points are triples $(\mathscr{P}_G, U, \alpha )$ where $\mathscr{P}_G \to X \times S$ is a $G$ -bundle, $U \subseteq X \times S$ is a domain, and $\alpha$ is a trivialization of $\mathscr{P}_G$ defined over $U$ . An isomorphism of such triples is an isomorphism of $G$ -bundles which commutes with the trivializations over some subdomain of the intersection of their domains of definition.

Let us generalize the above construction. For $H \subseteq G$ a subgroup, let $\mathrm{Bun}_{G,H}^{\mathrm{gen}}$ denote the following moduli space (see [Reference GaitsgoryGai21b, Appendix A] for more information). For an affine scheme $S$ , a morphism

\begin{equation*}S \to \mathrm{Bun}_{G,H}^{\mathrm{gen}}\end{equation*}

is a triple $(\mathscr{P}_G, U, \kappa )$ , where $\mathscr{P}_G \to X \times S$ is a $G$ -bundle, $U \subseteq X \times S$ is a domain, and $\kappa$ is a reduction of $\mathscr{P}_G$ to $H$ over $U$ . An isomorphism

\begin{equation*}(\mathscr{P}_G,U,\kappa ) \overset {\sim }{\longrightarrow } (\mathscr{P}_G',U',\kappa '),\end{equation*}

is an isomorphism $\mathscr{P}_G \overset {\sim }{\to } \mathscr{P}_G'$ over $X \times S$ which commutes with $\kappa$ and $\kappa '$ over a domain contained in $U \cap U'$ . In this notation, note that we have a tautological isomorphism

\begin{equation*}\mathrm{Gr}_G^{\mathrm{gen}} \overset {\sim }{\longrightarrow } \mathrm{Bun}_{G,1}^{\mathrm{gen}},\end{equation*}

where $1 \subseteq G$ denotes the trivial subgroup.

For any subgroup $H$ of $G$ , there is a morphism

\begin{equation*}\mathrm{Gr}_G^{\mathrm{gen}} \longrightarrow \mathrm{Bun}_{G,H}^{\mathrm{gen}},\end{equation*}

which takes a triple $(\mathscr{P}_G,U,\alpha )$ to $(\mathscr{P}_G,U,\alpha _H)$ , where $\alpha _H$ is the isomorphism given by the composition

\begin{equation*}\mathscr{P}_G \overset {\sim }{\longrightarrow } \mathscr{P}_G^0 \simeq \mathrm{ind}_1^G(\mathscr{P}_1^0) \overset {\sim }{\longrightarrow } \mathrm{ind}_H^G(\mathscr{P}_H^0) \end{equation*}

defined over $U$ .

When $H=N$ , there is also a morphism ${\overline {\mathrm{Bun}}}\vphantom {\mathrm{Bun}}_N \to \mathrm{Bun}_{G,N}^{\mathrm{gen}}$ obtained by restricting a generalized reduction $\kappa$ of a $G$ -bundle $\mathscr{P}_G$ to its non-degeneracy domain, i.e. the maximal domain over which the Plücker maps are injective maps of vector bundles. Now define a prestack ${\overline {S}}\vphantom {S}^{\mathrm{gen}}$ by the condition that it sits in a Cartesian square

Explicitly, an $S$ -point of ${\overline {S}}\vphantom {S}^{\mathrm{gen}}$ is given by a $G$ -bundle $\mathscr{P}_G \to X \times S$ , a domain $U \subseteq X \times S$ , and a trivialization $\alpha$ of $\mathscr{P}_G$ defined over $U$ such that the following condition holds. For all dominant weights $\lambda \in \check {\Lambda }^+$ , the meromorphic maps

\begin{equation*} \mathcal{O}_{X \times S} \longrightarrow V^{\check {\lambda }}_{\mathscr{P}_G^0} \overset {\alpha }{\longrightarrow } V^{\check {\lambda }}_{\mathscr{P}_G}\end{equation*}

induced by $\alpha$ extend to regular maps defined over all of $X \times S$ . As usual, the map $\mathcal{O}_{X \times S} \to V^{\check {\lambda }}_{\mathscr{P}_G^0}$ is the inclusion of the highest weight line.

In what follows, denote by $\underline {\mathrm{Map}}^{\mathrm{gen}}(X,N)$ the group prestack of generically defined maps $X \to N$ (see [Reference BarlevBar14] for a careful definition). In other words, an $S$ -point of $\underline {\mathrm{Map}}^{\mathrm{gen}}(X,N)$ is a map $U \to N$ , where $U \subseteq X \times S$ is a domain. Two such maps $U \to N$ and $U' \to N$ are equivalent if they agree on a subdomain of $U \cap U'$ .

By Theorem 1.8.2 in [Reference GaitsgoryGai13], the space $\underline {\mathrm{Map}}^{\mathrm{gen}}(X,N)$ is homologically contractible.Footnote ⁷ As the next proposition shows, the projection $\mathfrak{p}^{\mathrm{gen}}: {\overline {S}}\vphantom {S}^{\mathrm{gen}} \to {\overline {\mathrm{Bun}}}\vphantom {\mathrm{Bun}}_N$ is very close to being an equivalence. Note the following result is closely related to arguments from [Reference GaitsgoryGai21b, Appendix A].

Proof. For a subgroup $H$ of $G$ , define a prestack $\mathrm{Bun}_H^{\mathrm{gen}}$ by declaring a morphism $S \to \mathrm{Bun}_H^{\mathrm{gen}}$ to be a domain $U \subseteq X \times S$ as well as an $H$ -bundle

\begin{equation*} \mathscr{P}_H \to U \subseteq X \times S.\end{equation*}

An isomorphism $(U, \mathscr{P}_H) \to (U',\mathscr{P}'_H)$ of pairs as above is an isomorphism $\mathscr{P}_H \to \mathscr{P}'_H$ defined over a domain contained in $U \cap U'$ . There is an obvious morphism $\mathrm{Bun}_H^{\mathrm{gen}} \to \mathrm{Bun}_G^{\mathrm{gen}}$ . Moreover, it is easy to see that there is a canonical isomorphism

\begin{equation*} \psi _H:\mathrm{Bun}_{G,H}^{\mathrm{gen}} \overset {\sim }{\longrightarrow } \mathrm{Bun}_H^{\mathrm{gen}} \times _{\mathrm{Bun}_G^{\mathrm{gen}}} \mathrm{Bun}_G.\end{equation*}

Now since $N$ is unipotent, for any affine scheme $S$ the étale cohomology group $H^1_{\acute {e}t}(S,N)$ vanishes, and hence there is an isomorphism

\begin{equation*}\beta : \mathrm{Bun}_N^{\mathrm{gen}} \overset {\sim }{\longrightarrow } B(\underline {\mathrm{Map}}^{\mathrm{gen}}(X,N)), \end{equation*}

which commutes with the obvious map from $\mathrm{Spec}(k)$ to both sides. Therefore under the equivalences $\psi _1$ , $\psi _N$ , and $\beta$ , the map $\mathrm{Gr}_G^{\mathrm{gen}} \to \mathrm{Bun}_{G,N}^{\mathrm{gen}}$ corresponds to the map

\begin{equation*}(u \times \mathrm{id}_{\mathrm{Bun}_G}): \mathrm{Spec}(k) \times _{\mathrm{Bun}_G^{\mathrm{gen}}} \mathrm{Bun}_G \longrightarrow B(\underline {\mathrm{Map}}^{\mathrm{gen}}(X,N)) \times _{\mathrm{Bun}_G^{\mathrm{gen}}} \mathrm{Bun}_G\end{equation*}

obtained by base change from

\begin{equation*}u: \mathrm{Spec}(k) \to B(\underline {\mathrm{Map}}^{\mathrm{gen}}(X,N)).\end{equation*}

Since $\mathfrak{p}^{\mathrm{gen}}$ is a further base change of $(u \times \mathrm{id}_{\mathrm{Bun}_G})$ , we conclude that it is a torsor for $\underline {\mathrm{Map}}^{\mathrm{gen}}(X,N)$ .

The claim that $\mathfrak{p}^{\mathrm{gen}}$ is universally homologically contractible follows from the following general observation. Let $\mathcal{Y}$ be a prestack and let $\pi : \mathcal{X} \to \mathcal{Y}$ be a torsor for a homologically contractible group prestack $\mathcal{H}$ . Then $\pi$ is universally homologically contractible (see [Reference BarlevBar14, Lemma 6.2.10]).

6.3 Equivalence between the generic Zastava and a moduli space of rational data

Define the generic Zastava $\mathcal{Z}^{\mathrm{gen}}$ to be the straightening ${\overline {\mathcal{Z}}}\vphantom {\mathcal{Z}}^{\to }_{\mathrm{str}}$ . Denote by

\begin{equation*}\mathfrak{t}^{\mathrm{gen}}_Z: {\overline {\mathcal{Z}}}\vphantom {\mathcal{Z}} \longrightarrow \mathcal{Z}^{\mathrm{gen}},\end{equation*}

the quotient map.

In this section we will identify $\mathcal{Z}^{\mathrm{gen}}$ with ${\overline {S}}\vphantom {S}^{\mathrm{gen}}$ . First note that there is a canonical equivalence

\begin{equation*}(\mathrm{Gr}_{G,\mathrm{Conf}}^{\to })_{\mathrm{str}} \longrightarrow \mathrm{Gr}_G^{\mathrm{gen}}.\end{equation*}

Indeed, this follows from the fact that for every domain $U \subseteq X \times S$ there exists a Zariski cover $S' \to S$ such that $U \times _{X \times S} (X \times S')$ is a divisor complement, as well as an application of Lemma 6.2.

6.4 The generic Zastava in detail

In this section we will give an alternative description of $\mathcal{Z}^{\mathrm{gen}}$ using the moduli space of rational reductions to the opposite Borel $B^-$ .

Recall the stack $\mathrm{Bun}_{G,B^-}^{\mathrm{gen}}$ with its map to $\mathrm{Bun}_G$ . Then we can form the fiber product

\begin{equation*}{\overline {\mathrm{Bun}}}\vphantom {\mathrm{Bun}}_N \times _{\mathrm{Bun}_G} \mathrm{Bun}_{G,B^-}^{\mathrm{gen}}\end{equation*}

together with its open locus $\mathcal{Z}^{\mathrm{gen}}_0$ consisting of a generalized $N$ -reduction with a generically transverse generically defined $B^-$ -reduction. Note this is a well-defined condition. We claim there is an isomorphism

(6.1) \begin{equation} \mathcal{Z}^{\mathrm{gen}} \overset {\sim }{\longrightarrow } \mathcal{Z}^{\mathrm{gen}}_0. \end{equation}

Indeed, $\mathcal{Z}^{\mathrm{gen}}$ is the quotient of ${\overline {\mathcal{Z}}}\vphantom {\mathcal{Z}}$ by the action of the configuration space. But $\mathrm{Conf}$ inherits its action from a similar action on ${\overline {\mathrm{Bun}}}\vphantom {\mathrm{Bun}}_{B^-}$ , and it is easy to see that the quotient of the latter is equivalent to $\mathrm{Bun}_{G,B^-}^{\mathrm{gen}}$ .

From this perspective, we get another understanding of Proposition 6.6. The fiber of the map

\begin{equation*}\mathrm{Bun}_{G,B^-}^{\mathrm{gen}} \longrightarrow \mathrm{Bun}_G,\end{equation*}

over a $G$ -bundle $\mathscr{P}_G \to X \times S$ is given by the prestack

\begin{equation*}\operatorname {GSect}_{X \times S}((G/B^-)_{\mathscr{P}_G}) \overset {\sim }{\longrightarrow } \mathrm{Bun}_{G,B^-}^{\mathrm{gen}} \times _{\mathrm{Bun}_G} S,\end{equation*}

of generically defined sections of the bundle $(G/B^-)_{\mathscr{P}_G}$ associated with $\mathscr{P}_G$ with fiber $G/B^-$ (see [Reference BarlevBar14] for a definition).

In particular, the fiber over the trivial bundle is tautologically equivalent to the prestack $\underline {\mathrm{Map}}^{\mathrm{gen}}(X \times S, G/B^-)$ . Inside the latter we have the open subprestack consisting of generically defined maps $X \times S \to G/B^-$ which factor through the open cell $N \hookrightarrow G/B^-$ . This is precisely the fiber of the map