The Atiyah-Bott formula and connectivity in chiral Koszul duality

The $\otimes^\star$-monoidal structure on the category of sheaves on the $\mathrm{Ran}$ space is not pro-nilpotent in the sense of Francis-Gaitsgory. However, under some connectivity assumptions, we prove that Koszul duality induces an equivalence of categories and that this equivalence behaves nicely with respect to Verdier duality on the $\mathrm{Ran}$ space and integrating along the $\mathrm{Ran}$ space, i.e. taking factorization homology. Based on ideas sketched by Gaitsgory, we show that these results also offer a simpler alternative to one of the two main steps in the proof of the Atiyah-Bott formula given in Gaitsgory-Lurie and Gaitsgory.

Let X be a smooth and complete algebraic curve, and G a simply-connected semi-simple algebraic group over an algebraically closed field k. 1 Then we know that for some finite dimensional vector space V over Λ, where Λ is ℓ when k = p (ℓ = p), and Λ is any field of characteristic 0 when k has characteristic 0. Let Bun G denote the moduli stack of principal G-bundles over X . In the differential geometric setting, i.e. when k = , the cohomology ring of Bun G was computed by Atiyah and Bott in [AB83] using Morsetheoretic methods.
Theorem 1.1.1 (Atiyah-Bott). We have the following equivalence where ω X is the dualizing sheaf of X .
In the recent work [GL14], Gaitsgory and Lurie gave a purely algebro-geometric proof of the theorem above in the framework of étale cohomology (see also [Gai15] for an alternative perspective). In the case where X and G come from objects over k = q , the isomorphism in Theorem 1.1.1 was proved to be compatible with the Frobenius actions on both sides. The Grothendieck-Lefschetz trace formula for Bun G then gives an expression for the number of k-points on Bun G and hence, confirms the conjecture of Weil that the Tamagawa number of G is 1.
Following ideas suggested in [Gai12], this paper aims to provide an alternative (and simpler) proof of one of the two main steps in the original proofs, as given in [GL14] and [Gai15]. This is possible due to a family of new results regarding connectivity in the theory of chiral Koszul duality proved in this paper which are of independent interest.

Prerequisites and guides to the literature.
For the reader's convenience, we include a quick review of the necessary background as well as pointers to the existing literature in §2. The readers who are unfamiliar with the language used in the introduction are encouraged to take a quick look at §2 before returning to the current section. 1 This corresponds to the case of constant group G × X over X . For simplicity's sake, we will restrict ourselves to this case in the introduction.

A sketch of Gaitsgory and Lurie's method.
We will now provide a sketch of the strategy employed by [GL14] and [Gai15]. In both cases, the proofs utilize the theory of factorization algebras. Broadly speaking, there are two main steps: non-abelian Poincaré duality and Verdier duality on the Ran space.
The readers who are only interested in Koszul duality in the setting of factorization algebras in its own rights can safely skip to §1.4.
1.3.1. Non-abelian Poincaré duality. The first step involves a factorizable sheaf A on Ran X from f ! ω Gr Ran X where f is the natural map f : Gr Ran X → Ran X , and Gr Ran X is the Beilinson-Drinfeld factorizable affine Grassmannian. The crucial observation is that the natural map Gr Ran X → Bun G has homologically contractible fibers, and hence, we get an equivalence (1.3.2) C * c (Bun G , ω Bun G ) ≃ C * c (Ran X , A). 1.3.3. Verdier duality. The right hand side of (1.3.2) is, however, not directly computable. If one thinks of factorizable sheaves on Ran X as E 2 -algebras, then one reason that makes it hard to compute the factorization homology of A is the fact that it's not necessarily commutative (i.e. not E ∞ ). A, however, also has a commutative co-algebra structure, via the diagonal map 2 Gr → Gr × Gr .
Thus, its Verdier dual D Ran X A naturally has the structure of a commutative algebra. In fact, it is proved that D Ran X A is a commutative factorization algebra.
1.3.4. Computing the Verdier dual. One can prove something even better: D Ran X A is isomorphic to the commutative factorization algebra B coming from C * (BG). Namely, the co-stalk of B at any closed point ι x : x → X is ι ! x B ≃ C * (BG) and in fact B| X ≃ C * (BG) ⊗ ω X .
A natural map from one to the other is given by a certain pairing between A and B. Since these are factorizable, showing that this map is an equivalence amounts to showing that its restriction to X is also an equivalence. This is now a purely local problem, and hence, for example, one can reduce it to the case of 1 to prove it. Remark 1.3.5. Note that in the above, co-stalk, rather than stalk, appears. This is because in [GL14,Gai15], sheaves on (pre-)stacks are set up using the !-functors rather than * -functors.
1.3.6. Conclusion. Note from the above that is a free commutative algebra, where V is some explicit chain complex that we can compute. But factorization homology with coefficients in a free commutative factorization algebra is easy to compute. Hence, we conclude Unlike the case of Lie ⋆ and ComCoAlg ⋆ , for a co-Lie algebra g ∈ coLie ⋆ (X ), coChev(g) ∈ ComAlg ⋆ (Ran X ) doesn't necessarily live inside Fact ⋆ (X ). However, we have the following Theorem 1.5.5 (Theorem 4.1.3). Restricted to the full subcategory coLie ⋆ (X ) ≥1 , where we are using the perverse t-structure on X , the functor coChev factors through Fact ⋆ , i.e.
1.5.6. Interaction between coChev and factorization homology. In [FG11], it is proved that the functor of taking factorization homology C * c : Shv(Ran X ) → Vect commutes with Chev. This is because Chev is computed as a colimit, and moreover, C * c has the following two useful properties (i) C * c is symmetric monoidal with respect to the ⊗ ⋆ -monoidal structure on Shv(Ran X ) and the usual monoidal structure on Vect, and (ii) C * c is continuous. The functor coChev, however, is constructed as a limit, so we need some extra conditions to make it behave nicely with C * c .
1.6.1. Factorizability of D Ran X Chev a. The initial observation is that the sheaf A mentioned above lies in the essential image of Chev, i.e.
This is a direct result of Theorem 1.5.3 and the fact that A satisfies this connectivity constraint on the ComCoAlg ⋆ side. As mentioned above, we have a pairing which induces a map B → D Ran X A, compatible with the commutative algebra structures on both sides. Thus, we get a map which we want to be an equivalence. By construction, the LHS is factorizable. Corollary 1.5.11 can be used to show that the RHS is also factorizable. Thus it suffices to show that they are isomorphic over X , which is now a local problem, and the same proof as in [Gai15] applies.
1.6.2. Verdier duality vs. linear dual. The results proved in this paper could also be used to give an alternative proof of the equivalence ≃ coChev(C * c (X , D X a)) (Theorem 1.5.7) ≃ coChev(C * c (X , a) ∨ ) ≃ Chev(C * c (X , a)) ∨ (Theorem 1.5.9 for X = pt)

PRELIMINARIES
In this section, we will set up the language and conventions used throughout the paper. Since the material covered here are used in various places, the readers should feel free to skip it and backtrack when necessary.
The mathematical content in this section has already been treated elsewhere. Hence, results are stated without any proof, and we will do our best to provide the necessary references. It is important to note that it is not our aim to be exhaustive. Rather, we try to familiarize the readers with the various concepts and results used in the text, as well as to give pointers to the necessary references for the background materials.

Notation and conventions.
2.1.1. Category theory. We will use DGCat to denote the (∞, 1)-category of stable infinity categories, DGCat pres to denote the full subcategory of DGCat consisting of presentable categories, and DGCat pres,cont the (non-full) subcategory of DGCat pres where we restrict to continuous functors, i.e. those commuting with colimits. Spc will be used to denote the category of spaces, or more precisely, ∞-groupoids.
The main references for this subject are [Lur17a] and [Lur17b]. For a slightly different point of view, see also [GR17].
2.1.2. Algebraic geometry. Throughout this paper, k will be an algebraically closed ground field. We will denote by Sch the ∞-category obtained from the ordinary category of separated schemes of finite type over k. All our schemes will be objects of Sch. A scheme X ∈ Sch is said to be smooth if it is smooth over k.
In most cases, we will use the calligraphic font to denote prestacks, for eg. X, Y etc., and the usual font to denote schemes, for eg. X , Y etc.
2.1.3. t-structures. Let C be a stable infinity category, equipped with a t-structure. Then we have the following diagram of adjoint functors We use τ ≤0 and τ ≥1 to denote respectively. Shifts of these functors, for e.g. τ ≥n and τ ≤n , are defined in the obvious ways.
2.2. Prestacks. The theory of sheaves on prestacks has been developed in [GL14] and [Gai15]. In this subsection and the next, we will give a brief review of this theory, including the definition of the category of sheaves as well as various pull and push functors. We will state them as facts, without any proof, which (unless otherwise specified), could all be found in [Gai15]. 2.2.6. We also have relative versions of the definitions above in an obvious manner. Namely, we can speak of a morphism f : Y → S, where Y is a prestack and S is a scheme, being pseudo-schematic (resp. pseudo-proper, finitary).
2.2.7. More generally, a morphism f : Y 1 → Y 2 is said to be pseudo-schematic (resp. pseudo-proper, finitary) if for any scheme S, equipped with a morphism S → Y 2 , the morphism f S in the following pull-back diagram is pseudo-schematic (resp. pseudo-proper, finitary).
2.3. Sheaves on prestacks. As we mentioned above, proofs of all the results in mentioned in this section, unless otherwise specified, could be found in [Gai15].
2.3.1. Sheaves on schemes. We will adopt the same conventions as in [Gai15], except that for simplicity, we will restrict ourselves to the "constructible setting." Namely, for a scheme S, (i) when the ground field is , and Λ is an arbitrary field of characteristic 0, we take Shv(S) to be the ind-completion of the category of constructible sheaves on S with Λ-coefficients. (ii) for any ground field k in general, and Λ = ℓ , ℓ with ℓ = char k, we take Shv(S) to be the indcompletion of the category of constructible ℓ-adic sheaves on S with Λ-coefficients. See also [GL14,§4], [LZ12], and [LZ14]. The theory of sheaves on schemes is equipped with the various pairs of adjoint functors for any morphism f : S 1 → S 2 between schemes. Moreover, we have box-product ⊠ as well as ⊗ and ! ⊗.

2.3.2.
Throughout the text, we will use the perverse t-structure on Shv(S), when S is a scheme.
2.3.3. We will also use Vect to denote the category of sheaves on a point, i.e. Vect denotes the (infinity derived) category of chain complexes in vector spaces over Λ.
2.3.4. Sheaves on prestacks. For a prestack Y, the category Shv(Y) is defined by where the transition functor we use is the !-pullback. Informally speaking, an object F ∈ Shv(Y) is the same as the following data (i) a sheaf F S, y ∈ Shv(S) for each S ∈ Sch and y : S → Y (i.e. y ∈ Y(S)), and (ii) an equivalence of sheaves F S ′ , f ( y) → f ! F S, y for each morphism of schemes f : S ′ → S. Moreover, we require that this assignment satisfies a homotopy-coherent system of compatibilities.
be a morphism between prestacks. Then by restriction, we get a functor which commutes with both limits and colimits. In particular, f ! admits a left adjoint f ! . 6 The functor f ! is generally not computable. However, there are a couple of cases where it is.
2.3.10. The first instance is when the target of f is a scheme Then, by (2.3.8), we have is just a morphism between schemes. 6 It also admits a right adjoint. However, we do not make use of it in this paper. 2.3.11. The second case is where f is pseudo-proper, then f ! satisfies the base change theorem with respect to the (−) ! -pullback. Namely, for any pull-back diagram of prestacks and any sheaf F ∈ Shv(Y), we have a natural equivalence Thus, in particular, if we have a pull-back diagram F and as discussed above, f S! could be computed as an explicit colimit.
2.3.12. Let F ∈ Shv(Y). Then we denote by 2.3.13. In case where F ≃ ω Y is the dualizing sheaf on Y (characterized by the fact that its (−) !pullback to any scheme is the dualizing sheaf on that scheme), then we write 3.14. f * ⊣ f * . When f : Y 1 → Y 2 is a schematic morphism between prestacks, one can also define a pair of adjoint functors (see [Gai15] where the functor f * is defined, and [Ho17] where the adjunction is constructed) The behavior of f * is easy to describe, due to the fact that f * satisfies the base change theorem with respect to the (−) ! -pullback functor. Namely, suppose F ∈ Shv(Y 1 ) and we have a pullback square where S 2 (and hence, S 1 ) is a scheme Then, the pullback could be described in classical terms, since where f S is just a morphism between schemes.
2.3.21. Monoidal structure. The theory of sheaves on prestacks discussed so far naturally inherits the box-tensor structure from the theory of sheaves on schemes. Namely, let F i ∈ Shv(Y i ) where Y i 's are prestacks, for i = 1, 2. Then, for any pair of schemes S 1 , S 2 equipped with maps for any prestack Y, we get the ! ⊗-symmetric monoidal structure on Y in the usual way. More explicitly, for F 1 , F 2 ∈ Shv(Y), we define 2.4. The Ran space/prestack. The Ran space (or more precisely, prestack) of a scheme plays a central role in this paper. The Ran space, along with various objects on it, was first studied in the seminal book [BD04] in the case of curves, and was generalized to higher dimensions in [FG11]. In what follows, we will quickly review the main definitions and results. For proofs, unless otherwise specified, we refer the reader to [Gai15] and [FG11]. The topologically inclined reader could also find an intuitive introduction in [Ho17, §1].
2.4.1. For a scheme X ∈ Sch, we will use Ran X to denote the following prestack: for each scheme S ∈ Sch, (Ran X )(S) = {non-empty finite subsets of X (S)} Alternatively, one has Ran X ≃ colim where fSet surj denotes the category of non-empty finite sets, where morphisms are surjections. Using the fact that X is separated, one sees easily that Ran X is a pseudo-scheme. Moreover, when X is proper, Ran X is pseudo-proper.
2.4.2. The ⊗ ⋆ monoidal structure. There is a special monoidal structure on Ran X which we will use throughout the text: the ⊗ ⋆ -monoidal structure.
Consider the following map union : Ran X × Ran X → Ran X given by the union of non-empty finite subsets of X . One can check that union is finitary pseudo-proper. Given two sheaves F, G ∈ Shv(Ran X ), we define This defines the ⊗ ⋆ -monoidal structure on Shv(Ran X ).

2.4.3.
Since union is pseudo-proper, ⊗ ⋆ has an easy presentation. Namely, for and any non-empty finite set I , we have the following where • X I denotes the open subscheme of X I given by the condition that no two "coordinates" are equal, and where 2.4.5. Factorizable sheaves. Using the ⊗ ⋆ -monoidal structure on Shv(Ran X ), one can talk about various types of algebras/coalgebras in Shv(Ran X ). The ones that are of importance to us in this papers are As the name suggests, these are used, respectively, to denote the categories of commutative algebras, Lie algebras, co-commutative co-algebras and co-Lie co-algebras in Shv(Ran X ) with respect to the ⊗ ⋆monoidal structure defined above.
2.4.6. We use Lie ⋆ (X ) and coLie ⋆ (X ) to denote the full subcategories of Lie ⋆ (Ran X ) and coLie ⋆ (Ran X ) respectively, consisting of objects whose supports are inside the diagonal disj is the open sub-prestack of (Ran X ) n defined by the following condition: for each scheme S, (Ran X ) n (S) consists of n non-empty subsets of X (S), whose graphs are pair-wise disjoint.

Let
A ∈ ComCoAlg ⋆ (Ran X ). Then, by definition, we have the following map (which is the co-multiplication of the commutative co-algebra structure) . Using the the unit map of the adjunction j * ⊣ j * , we get the following map where for the equivalence, we made use of (2.3.18) and (2.3.20).
Altogether, we get a map A → (union • j) * j ! (A ⊠ · · · ⊠ A) and hence, by adjunction and (2.3.18), we get a map Definition 2.4.10. A is a commutative factorization algebra if the map (2.4.9) is an equivalence for all n's. We use coFact ⋆ (X ) to denote the full subcategory of ComAlg ⋆ (Ran X ) consisting of co-commutative factorization co-algebras.

Let
Then, by definition, we have the following map (which is the multiplication of the commutative algebra structure) This induces the following map of sheaves on (Ran X ) n , and hence, a map of sheaves on (Ran X ) n disj .
Definition 2.4.13. B is a commutative factorization algebra if the map (2.4.12) is an equivalence for all n's. We use Fact ⋆ (X ) to denote the full subcategory of ComAlg ⋆ (Ran X ) consisting of commutative factorization algebras.
2.5. Koszul duality. In this subsection, we will quickly review various concepts and results in the theory of Koszul duality that are relevant to us. This theory, initially developed in [Qui69], illuminates the duality between co-commutative co-algebras and Lie algebras. It was further developed and generalized in the operadic setting in [GK94]. In the chiral/factorizable setting, the paper [FG11] provides us with necessary technical tools and language to carry out many topological arguments in the context of algebraic geometry. The results and definitions we review below could be found in [FG11] and [GR17].
2.5.1. Symmetric sequences. Let Vect Σ denote the category of symmetric sequences. Namely, its objects are collections where each O(n) is an object of Vect, acted on by the symmetric group Σ n . The infinity category Vect Σ is equipped with a natural monoidal structure, which we denote by ⋆, and which makes the functor Vect Σ → Fun(Vect, Vect) given by the following formula 2.5.2. Operads and co-operads. By an operad (resp. co-operad), we will mean an augmented associative algebra (resp. co-algebra) object in Vect Σ , with respect to the monoidal structure described above. We use Op (resp. coOp) to denote the categories of operads (resp. co-operads).
In general, the Bar and coBar construction gives us the following pair of adjoint functors Bar : Op ⇄ coOp : coBar .
Remark 2.5.3. In what follows, we will adopt the following convention: all our operads/co-operads will have the property that the augmentation map is an equivalence when restricted to O(1) (resp. P(1)).
And under this restriction, one can show that the following unit map or in a slightly different notation is an equivalence.

Algebras and co-algebras.
Let C be a stable presentable symmetric monoidal ∞-category compatibly tensored over Vect. Then, an operad O (resp. co-operad P) naturally defines a monad (resp. comonad) on C.
Thus, for an operad O (resp. co-operad P), one can talk about the category of algebras O -alg(C) (resp. co-algebras P -coalg(C)) in C with respect to the operad O (resp. co-operad P).
As usual (as for any augmented monad), one has the following pairs of adjoint functors for an operad O, and similarly, the following pairs of adjoint functors oblv P : P -coalg(C) ⇄ C : coFree P and cotriv P : C ⇄ P -coalg(C) : coBar P for a co-operad P.
2.5.5. Koszul duality. The functors mentioned above could be lifted to get a pair of adjoint functors Turning Koszul duality into an equivalence. In general, the pair of adjoint functors at (2.5.6) is not an equivalence. One of the main achievements of [FG11] is to formulate a precise sufficient condition on the base category C, namely the pro-nilpotent condition, 7 which turns (2.5.6) into an equivalence.
One of the main technical points of our paper is to prove another case where Koszul duality is still an equivalence, even when the categories involved are not pro-nilpotent.
The two main instances of Koszul duality that are important in this paper are the duality between Lie-algebras and ComCoAlg-algebras, and coLie-algebras and ComAlg-algebras.
2.5.8. The case of Lie and ComCoAlg. We have the following equivalence of co-operads (see [FG11]): is equipped with the sign action of the symmetric group Σ n . Here, [n] denotes cohomological shift to the left by n. Equivalently, the functor [1] : C → C gives rise to an equivalence of categories This gives us the following diagram We usually use Chev to denote The interested reader could read more about this in [FG11], since we do not need this fact in the current work.
As above, we usually use coChev to denote and coPrim[1] to denote

TURNING KOSZUL DUALITY INTO AN EQUIVALENCE
The goal of this section is to prove Theorem 1.5.3. We will start with Theorem 3.1.1, which examines the special case where X is just a point, i.e. Shv(Ran X ) ≃ Shv(X ) ≃ Vect, and prove that Koszul duality induces a natural equivalence of categories Note that this is a classical result of Quillen [Qui69], and our proof could be viewed as a recast of his under the light of higher algebra. This point of view allows us to generalize the result to the more general case of interest. Note also that this case is not strictly needed in the proof of the general case. We do, however, recommend the reader to first read it before moving on to the proof of Theorem 3.3.3 since it contains all the essential points without the complicated notation employed in the general case to deal with the combinatorics of the Ran space.
3.1. The case of Lieand ComCoAlg-algebras inside Vect. We will now prove the following Remark 3.1.2. Since Chev is defined as a colimit, it is easy to see that Chev | Lie(Vect ≤−1 ) lands in the correct subcategory cut out by the connectivity assumption Vect ≤−2 (the extra shift to the left is due to (2.5.9)). It is, however, not a priori obvious for Prim[−1], being defined as a limit. Nonetheless, this fact is a direct consequence of Lemma 3.1.10 and Corollary 3.1.11.
Remark 3.1.3. Unless otherwise specified, when it makes sense our functors will be automatically restricted to the subcategories with the appropriate connectivity conditions. For example, we will write Chev instead of Chev | Lie(Vect ≤−1 ) in most cases.
Remark 3.1.4. Note that Theorem 3.1.1 can be proved more generally for a presentable symmetric monoidal stable infinity category with a t-structure satisfying some mild properties. The pair of operad and co-operad Lie and ComCoAlg could also be made more general. See Remarks 3.1.17 and 3.1.18.
3.1.5. To prove that Chev and Prim[−1] are mutually inverse functors, it suffices to show that the left adjoint functor, Chev, is fully-faithful, and the right adjoint functor, Prim[−1] is conservative. We start with the following result, whose proof is carried out in §3.1.13 after some preparation.
As in [FG11, §4.1.8], this immediately implies the following corollary. For the sake of completeness, we include the proof here.
is an equivalence. Since Prim[−1] commutes with sifted colimits by part (i) of Lemma 3.1.6, it suffices to show that the following is an equivalence since any Lie-algebra could be written as a sifted colimit of the free ones. 8 However, we know that (even without the connectivity condition) Chev • Free Lie ≃ triv ComCoAlg and hence, it suffices to show that But now, we are done due to part (ii) of Lemma 3.1.6.
3.1.8. Before proving Lemma 3.1.6, we start with a couple of preliminary observations. In essence, the lemma is a statement about commuting limits and colimits. In a stable infinity category, if, for instance, the limit is a finite one, then one can always do that. In our situation, coBar causes troubles because it is defined as an infinite limit.
The main idea of the proof is that when is computed as an infinite limit, each of its cohomological degrees will be controlled by only finitely many of terms in the limit. 8 This fact applies to the category of algebras over any operad in general.
3.1.9. For brevity's sake, we will use P to denote the co-operad ComCoAlg. Recall that in general, for any c ∈ ComCoAlg(Vect), is a co-simplicial object. Let coBar n P (c) = Tot(coBar • P (c)| ∆ ≤n ) be the limit over the restriction of the co-simplicial object to ∆ ≤n . Then we have the following tower Then, for all n ≥ 0, the following natural map Indeed, this is because of the fact that c ∈ Vect ≤−2 and hence, in the direct sum m = 2 n is the first summand where we have non-degenerate "(co-)cells." The shift to the right by n is due to the fact that we are at level n of the co-simplicial object. As a consequence, tr ≥−2 n+1 +n+1 coBar n P (c) → tr ≥−2 n+1 +n+1 coBar n−1 P (c) is an equivalence and we are done.
Then, for any n, the following natural map tr ≥−n coBar P (c) → tr ≥−n coBar m P (c) is an equivalence for all m ≫ 0, where the bound depends only on n.
Proof. The lemma follows from the general situation considered below. Suppose we have a sequence X 0 ← X 1 ← · · · and integers n, m such that But now, the sequential limit can be computed as the fiber of two infinite products, i.e. we have the following fiber sequence Since the last two terms belong to Vect ≤−n−1 , so is the first term. Therefore, tr ≥−n X ≃ tr ≥−n X m and the proof concludes.
Remark 3.1.12. In the proof above, we use the fact that Vect ≤0 is preserved under countable products in Vect, or equivalently, that countable products are exact with respect to the usual t-structure on Vect. However, since the estimate appearing in (3.1.10) tends to −∞, the conclusion of Corollary 3.1.11 still holds true when countable products are only known to have uniformly bounded cohomological amplitude, i.e. there exists a fixed N such that i V i lives in cohomological degrees ≤ N for any family 3.1.13. We will now complete the proof of Lemma 3.1.6.
Proof of Lemma 3.1.6. The proof is now simple. In fact, we will only prove part (i), as the other one is almost identical. Note that due to (2.5.10), what we prove about coBar P implies the corresponding statement of Prim[−1], up to a shift. It suffices to show that for all n, we have where α runs over some sifted diagram. But now, from Corollary 3.1.11, for all m ≫ 0, we have Remark 3.1.14. The cohomological estimate done above implies that Indeed, from Corollary 3.1.11, we know that for some m ≫ 0, , and moreover, a downward induction using Lemma 3.1.10 shows that Proof. It suffices to show that is conservative, and we will prove that by contradiction. Namely, let is an equivalence, we will derive a contradiction.
Let k be the smallest number such that is not an equivalence. Now, by Corollary 3.1.11, we know that there is some m ≫ 0 such that By an estimate similar to the one at Lemma 3.1.10, we will show that for all n ≥ 1, where F * (−) denotes the fiber as in the proof of Lemma 3.1.10. Indeed, the difference between F n (c 1 ) and F n (c 2 ) lies in cohomological degrees And hence, a downward induction, starting from n = m, using the diagram which contradicts our original assumption. Hence, we are done.
Remark 3.1.17. Note that the proof we gave above could be carried out in a more general setting. Namely, the only properties of Vect that we used are (i) The symmetric monoidal structure is right exact (namely, it preserved Vect ≤0 ).
(ii) The t-structure on Vect is left separated.
Remark 3.1.18. We can also replace the operad Lie by any operad O such that (i) O is classical, i.e. it lies in the heart of the t-structure of Vect.
≃ Λ (as we already assume throughout this paper).
3.2. Higher enveloping algebras. We will briefly explain the topological analogue of the main results in the factorizable setting, proved in the next subsection. In this setting, the result is an immediate consequence of what we already proved above.
The main reference of this part is [GR17].
Then one can form its E n -universal enveloping algebra by applying the following sequence of functors where E n (Lie(Vect)) and E n (ComCoAlg(Vect)) are categories of E n -algebras with respect to the Cartesian monoidal structure on Lie(Vect) and ComCoAlg(Vect) respectively (note that the latter one is just the given by ⊗ in Vect).

It is proved in
Moreover, we know from Theorem 3.1.1 that As a result, we get Proposition 3.2.3. We have the following equivalence of categories 3.2.5. The equivalence (3.2.4) is precisely what we are looking for in the context of factorization algebras on the Ran space in the following subsection. One part of the work is to find connectivity assumptions on Shv(Ran X ) which mirror those appearing in Vect ≤−n−1 and Vect ≤−2 respectively.
3.3. The case of Lie ⋆ -and ComCoAlg ⋆ -algebras on Ran X . We now come to the precise formulation and the proof of Theorem 1.5.3. Throughout this subsection, we will assume that X is smooth over k of dimension d.
Definition 3.3.1. Let Shv(Ran X ) ≤c cA and Shv(Ran X ) ≤c L denote the full subcategory of Shv(Ran X ) consisting of sheaves F such that for all non-empty finite sets I , and respectively, Here, we use the perverse t-structure, and X is a scheme of pure dimension d.
With these connectivity assumptions in mind, the rest of this subsection will be devoted to the proof of the following Theorem 3.3.3. Suppose X is smooth over k of dimension d. We have the following commutative diagram where ≤ c L and ≤ c cA denote the connectivity constraints given in Definition 3.3.1, and where Chev and Prim[−1] are the functors coming from Koszul duality.
Remark 3.3.5. As in the case of Vect, we will in general suppress the distinction between a functor and its restriction to a subcategory cut out by some connectivity condition. For example, we will write Chev instead of Chev | Lie ⋆ (Ran X ) ≤c L unless confusion is likely to occur.
Now, suppose that F 1 , . . . , F k ∈ Shv(Ran X ) ≤c L , then we see that each summand in (3.3.9) lies in perverse cohomological degrees Here, the first inequality is due to the fact that the map is a regular embedding (since X is smooth), and that the (perverse) cohomological amplitude of the !-pullback along a regular embedding is equal to the codimension. The sequence of inequalities above thus implies that Note that for a general operad O, only the first row of (3.3.4) makes sense.
3.3.11. Back to Theorem 3.3.3. First, we will prove the equivalence on the top row of (3.3.4). Then, we will show that it induces an equivalence between the corresponding sub-categories on the bottom row.
As in the case of Vect, to prove that Chev and Prim[−1] are mutually inverse functors, it suffices to show that Chev is fully-faithful, and Prim[−1] is conservative. As above, we start with the following lemma, whose proof, after some preparation, will conclude in §3.3.19. 3.3.14. In essence, the strategy we follow here is identical to that of the Vect case even though the actual execution might seem somewhat more involved. The main observation (which is new compared to the case of Vect) is that to prove the equivalences involved in Lemma 3.3.12, it suffices to prove them after after pulling back to • X I for each non-empty finite set I . 3.3.16. Let F n (A) = Fib(coBar n ComCoAlg (A) → coBar n−1 ComCoAlg (A)), and I a non-empty finite set. Using the same argument as in the case of Vect in combination with the cohomological estimate (3.3.10), we see that F n (A)| • X I lives in cohomological degrees

In general, for any
This gives us the following analog of Lemma 3.1.10.

Lemma 3.3.17. Let
A ∈ ComCoAlg ⋆ (Ran X ) ≤c cA . Then, for any n and I , the following natural map This implies the following result, which is parallel to Corollary 3.1.11. See also Remark 3.1.12, [LZ14, Lemma 3.2.1] and the discussion after it where left-completeness and uniformly bounded cohomological amplitude for countable products are discussed.

Corollary 3.3.18. Let
A ∈ ComCoAlg ⋆ (Ran X ) ≤c cA . Then, for any n and I , the following natural map Proof. It suffices to show that is conservative, and we will do so by contradiction. Namely, let f : A 1 → A 2 be a morphism in ComCoAlg ⋆ (Ran X ) ≤c cA that is not an equivalence. Suppose that is an equivalence, we will derive a contradiction.
Let I the set of smallest cardinality such that the map is not an equivalence. Let k ≥ 0 be the smallest number such that By Corollary 3.3.18, we know that there exists some m ≫ 0 such that for i ∈ {1, 2}. Thus, we get the following equivalence then the difference between F n (A 1 )| • X I and F n (A 2 )| • X I lies in cohomological degrees This implies that for n ≥ 1, lands inside the full-subcategory coFact ⋆ (X ) of factorizable co-algebras. We thus get a functor Chev : Lie ⋆ (X ) ≤c L → coFact ⋆ (X ) ≤c cA , which settles the "if" direction.

For the "only if" direction, let
g ∈ Lie ⋆ (Ran X ) ≤c L whose support does not lie in X . We will show that Chev g is not factorizable.
Using the ass-gr • addFil trick (see §A), it suffices to prove for the case where g is a trivial (i.e. abelian) Lie algebra. In that case, we know that where Sym is taken using the ⊗ ⋆ -monoidal structure.
Let I be the smallest set, with |I | > 1, such that g| • X I ≃ 0. Now, it's easy to see that Sym >0 (g[1]) fails the factorizability condition at • X I , which concludes the "only if" direction.

FACTORIZABILITY OF coChev
In this section, we will prove Theorem 1.5.5, which asserts that when g ∈ coLie ⋆ (X ) satisfies a certain co-connectivity constraint, the commutative algebra Note that an analog of this result, where coChev is replaced by Chev, has been proved in [FG11] (and in fact, we used this result in the previous section). The main difficulties of the coChev case stem from the fact that, unlike Chev, coChev is defined as a limit, and most of the functors that we want it to interact with don't generally commute with limits.
As above, our main strategy is to introduce a certain co-connectivity condition to ensure that when one takes the limit of a diagram involving objects satisfying it, the answer, in some sense, converges instead of running off to infinity, so we still have a good control over it.
We start with the precise statement of the theorem. Then, after a quick digression on the various notions related to the convergence of a limit, we will present the main strategy. Finally, the proof itself will be given.

The statement.
We start with the co-connectivity conditions. Definition 4.1.1. Let Shv(Ran X ) ≥n denote the full subcategory of Shv(Ran X ) consisting of sheaves F such that for all non-empty finite sets I , As before, we use the perverse t-structure.
Our main goal is to prove the following Theorem 4.1.3. Restricted to the full subcategory coLie ⋆ (X ) ≥1 of coLie ⋆ (Ran X ) ≥1 consisting of coLiecoalgebras whose underlying sheaves are supported on the diagonal X , the functor coChev factors through Fact ⋆ , i.e. we have the following commutative diagram In other words, coChev g is factorizable when g ∈ coLie ⋆ (X ) ≥1 .

Stabilizing co-filtrations and decaying sequences (a digression).
We will now describe a condition on co-filtered and graded objects which make them behave nicely with respect to taking limits. Definition 4.2.1. Let C be a stable infinity category equipped with a t-structure. Then, a co-filtered object c ∈ C coFil >0 (see §B) is said to stabilize if for all n, the induced map tr ≤n c m → tr ≤n c m+1 is an equivalence for all m ≫ 0.
A graded object c ∈ C gr >0 is said to be decaying if for all n, we have tr ≤n c m ≃ 0 for all m ≫ 0.
Notation 4.2.2. We use C coFil >0 ,stab and C gr >0 ,decay to denote the subcategories of C coFil >0 and C gr >0 consisting of stabilizing and decaying objects respectively.
We have the following lemmas, whose proofs are straightforward. Proof. By throwing away finitely many terms at the beginning, without loss of generality, we can assume that the natural maps τ ≤n+1 c i → τ ≤n+1 c j , ∀i ≥ j > 0 are all equivalences. Now, it suffices to show that the following map is an equivalence Equivalently, it suffices to show that because Fib(c i → c 1 ) ∈ C ≥n+1 , ∀i. Hence, we are done, since i ≥n+1 : C ≥n+1 → C commutes with limits (see §2.1.3).

Lemma 4.2.5. The natural transformation
→ between functors C gr >0 ,decay → C is an equivalence.
Proof. Note that Moreover, since the sequence we are taking the limit over stabilizes, the result follows as a direct consequence of Lemma 4.2.4. 4.2.6. The various definitions and observations above have straightforward analogues in the case of sheaves on the Ran space.
Definition 4.2.7. A co-filtered sheaf F ∈ Shv(Ran X ) coFil >0 is said to stabilize if for any non-empty finite set I , Similarly, a graded sheaf F ∈ Shv(Ran X ) gr >0 is said to be decaying if for any non-empty finite set I , Notation 4.2.8. We use Shv(Ran X ) coFil >0 ,stab and Shv(Ran X ) gr >0 ,decay to denote the full-subcategories of Shv(Ran X ) coFil >0 and Shv(Ran X ) gr >0 consisting of stabilizing and decaying objects, respectively.
It's straightforward to see that the following analogs of the lemmas above still hold in this setting.

Strategy.
To prove that Chev g is factorizable when g ∈ Lie ⋆ (X ), [FG11] uses the addFil trick (see §A) to reduce to the case where g is a trivial. When g is trivial, we have Chev g ≃ Sym >0 g, and the result can be seen directly.
In the case of coChev, while the core strategy remains the same, it is more complicated to carry out since many commutative diagrams needed for the addFil trick to work (see (A.3.3)) don't commute in general in this new setting. The co-connectivity constraints are what we need to make these diagrams commute and hence, to allow us to reduce to the trivial case. 4.3.1. Let us now sketch the strategy. Suppose for the moment that we have the following commutative diagram, which is analogous to (A.3.3), except for the extra conditions Suppose also that oblv coFil preserves factorizability, and that ass-gr and are conservative with respect to factorizability. 11 Then by the same reasoning as in the addFil trick, to prove that coChev g is factorizable, it suffices to assume that g has a trivial coLie-structure. In that case, and as in the Chev case, we are done.
In §4.4- §4.6, we will carry out the strategy outlined above and conclude the proof of Theorem 4.1.3.

Well-definedness of functors.
Before proving that the diagram commutes, we need to first make sense of it. A priori, the functors written in the diagram are not necessarily well-defined. For instance, we have not shown that all the four instances of coChev land in the correct target categories. Moreover, we also do not know that oblv coFil , ass-gr, and preserve the algebra/co-algebra structures. The latter question is settled by the following observation, whose proof, which makes use of the stability and decaying conditions to commute limits and tensor products, is straight-forward. : Shv(Ran X ) ≥n,gr >0 ,decay → Shv(Ran X ) ≥n are symmetric monoidal with respect to the ⊗ ⋆ -monoidal structure on Ran X . In particular, they automatically upgrade to functors between corresponding categories of algebras/co-algebras.

4.4.2.
We will now tackle the former question: namely, the various instances of the functor coChev appeared in (4.3.2) land in the correct target categories. The top and bottom coChev are the same, and it's easy to see that they land in the correct category using the fact that the shriek-pullback functor is left exact and C ≥n is preserved under limits for any stable infinity category C with a t-structure (since i ≥n commutes with limits, see §2.1.3).
By the same token, we know that the essential images of coChev coFil and coChev gr satisfy the coconnectivity assumption (i.e. live in (perverse) cohomological degree ≥ 1). Thus, it remains to show that they also satisfy the stab and decay conditions respectively. For that, first observe that the assertion about ass-gr in Lemma 4.4.1, combined with the fact that ass-gr commutes with limits, gives us a weakened version of the middle square of (4.3.2).
Now, by Lemma 4.2.9, to show that coChev coFil and coChev gr satisfy the stab and decay conditions respectively, it suffices to show that coChev gr satisfies the decay condition. However, this is also a direct consequence of the fact that the shriek-pullback functor is left exact and C >n is preserved under limits (for any stable infinity category C with a t-structure). Altogether, we have thus proved that all functors in the diagram (4.3.2) above land in the correct categories. 4.5. Commutative diagrams. We will now proceed to prove that the diagram (4.3.2) commutes. First note that we have just settled the commutativity of the middle diagram of (4.3.2) at the end of the previous subsection. 11 Here, by conservativity, we mean that an object satisfies factorizability condition if its image under the functor does.
is the identity functor (see also §A.3.1). However, this is clear since the functor oblv coFil : Shv(Ran X ) ≥n,coFil >0 ,stab → Shv(Ran X ) ≥n commutes with limits for any n, and moreover it is symmetric monoidal with respect to the ⊗ ⋆ -monoidal structure on Shv(Ran X ) by Lemma 4.4.1.
4.6. Relation to factorizability. Using the fact that ass-gr is symmetric monoidal and is a conservative functor, it is easy to see that ass-gr : ComAlg ⋆ (Ran X ) ≥2,coFil >0 ,stab → ComAlg ⋆ (Ran X ) ≥2,gr >0 ,decay reflects factorizability, namely, an object is factorizable if its image is. As we already discussed above, we have an equivalence of functors But now it's clear that reflects factorizability, since does. Finally, since is compatible with ⊠ (for the same reason that it is compatible with ⊗ ⋆ ), and moreover (−) ! commutes with limits (being a right adjoint), we see easily that oblv coFil preserves factorizability. Thus, we conclude the proof of Theorem 4.1.3. 4.7. Relation to coLie ! (X ) and ComAlg ! (X ). In this subsection, we will discuss the various links between objects defined on X such as coLie ! (X ) and ComAlg ! (X ) and objects defined on Ran X such as coLie ⋆ (Ran X ), ComAlg ⋆ (Ran X ) and Fact ⋆ (X ). This subsection is not used anywhere in the paper. We include it here for the sake of completeness. 4.7.1. Recall that on a scheme X , there are two symmetric monoidal structures, ⊗ and ! ⊗. Thus, we could talk about various algebra/co-algebra objects defined on it where Lie * (X ) (not to be confused with Lie ⋆ (X )) is the category of Lie-algebra objects in Shv(X ) with respect to the ⊗-monoidal structure, and coLie ! (X ) (resp. ComAlg ! (X )) is the category of coLie-algebra (resp. commutative algebra) objects in Shv(X ) with respect to the ! ⊗-monoidal structure.
where the LHS denotes the full-subcategory of Lie ⋆ (Ran X ) = Lie(Shv(Ran X ) ⊗ ⋆ ) consisting of Lie-algebras whose underlying sheaves are supported on the diagonal X of Ran X .
where the RHS denotes the full-subcategory of coLie ⋆ (Ran X ) = coLie(Shv(Ran X ) ⊗ ⋆ ) consisting of coLiecoalgebras whose underlying sheaves are supported on the diagonal X of Ran X . 4.7.5. We also have the following functor which commutes with limits. Thus, we get a pair of adjoint functors (4.7.6) ins X ? : ComAlg ! (X ) ⇄ ComAlg ⋆ (Ran X ) : ins ! X .
Theorem 4.7.7. The pair of adjoint functors in (4.7.6) induces an equivalence of categories 4.7.8. The first link between coLie ! (X ), coLie ⋆ (X ), ComAlg ! (X ), ComAlg ⋆ (Ran X ) and Fact ⋆ (X ) is given by the following Proposition 4.7.9. The following diagram commutes Proof. The result is straightforward due to the fact that ins ! X commutes with limits and that it's monoidal. 4.7.11. The second link, and also the more interesting one, is given by the following Proposition 4.7.12. We have the following commutative diagram Proof. By adjunction, for any g ∈ coLie ! (X ), we have a natural map ins X ? • coChev → coChev • ins X ! between objects in ComAlg ⋆ (Ran X ). Now, we know from Theorem 4.7.7 that the LHS is factorizable. Moreover, when g ∈ coLie ! (X ) ≥1 , we know from Theorem 4.1.3 that the RHS is also factorizable. Thus, to show that the map above is an equivalence when g ∈ coLie ! (X ) ≥1 , it suffices to show that they are equivalent on the diagonal. However, that is clear from (4.7.10) and we are done.

INTERACTIONS BETWEEN VARIOUS FUNCTORS ON THE RAN SPACE
In this section, we investigate how the various functors operating on sheaves on the Ran spaces interact with each other. The highlights are Theorem 5.1.2, which says that coChev is compatible with C * c (Ran X , −) under some co-connectivity assumption, and Theorem 5.3.1 which shows how the functor of taking Koszul duality exchanges coChev and Chev under some connectivity assumption. 5.1. C * c (Ran X , −) and coChev. In this subsection, we will prove Theorem 1.5.7, which gives us a criterion for the commutativity of the functor coChev and the functor C * c (Ran X , −). Note that it has been proved in [FG11] that Chev always commutes with C * c (Ran X , −). The main reason is that C * c (Ran X , −) is continuous and monoidal with respect to the ⊗ ⋆ -monoidal structure on Shv(Ran X ) and the usual monoidal structure on Vect. As before, our main difficulty comes from the fact that coChev is defined as a limit, and for that to behave well with respect to C * c (Ran X , −), we need to impose a certain coconnectivity assumption. 5.1.1. Throughout this subsection, X will be assumed to be a proper scheme of pure dimension d.
After some preparation, the actual proof of the theorem will be carried out in §5.1.16. We start with the following elementary lemma whose proof is immediate.
Lemma 5.1.3. Let F : × op → C be a functor. Assume that there exists N ∈ such that for all i, j > N , the following maps assuming that all limits and colimits exist. 12 Since Supp g ⊂ X ⊂ Ran X , we have C * c (Ran X , g) ≃ C * c (X , g) Corollary 5.1.4. Let C be a stable ∞-category equipped with a right-separated t-structure and assume also that filtered colimits are exact with respect to the t-structure. Let such that for any c, the functor tr <c •F satisfies the conditions of Lemma 5.1.3. Then assuming that all limits and colimits make sense.
Proof. The separatedness condition implies that it suffices to prove that for each integer c, the following map is an equivalence Commuting tr <c pass the colimit and limit, the equivalence is a direct consequence of Lemma 5.1.3 above. Note that here, we only use the exactness of filter colimits (tr <0 commutes with limits since it's a right adjoint).
We will apply the discussion above to the situation at hand. 5.1.5. Truncated Ran space. For any scheme X and any positive integer n, we define Ran ≤n X ≃ colim I∈fSet surj |I|≤n X I .
Where (coChev coFil addCoFil g) i is the i-th step in the co-filtration, and moreover where Sym is formed using the ⊗ ⋆ -monoidal structure on Shv(Ran X ).
5.1.8. For brevity's sake, we will denote where Sym is formed using the ⊗ ⋆ -monoidal structure on Shv(Ran X ).
5.1.10. For g ∈ coLie ⋆ (X ) ≥d+1 , consider the following functor F : × op → Vect (5.1.11) (i, j) → C * c (Ran ≤i X , coChev j g) ≃ coChev j C * c (X , g) where the equivalence on the second line is due to the fact that coChev j is computed as a finite limit for each j.
The goal now is to show that F satisfies the conditions stated in Corollary 5.1.4. We start with a couple of cohomological estimates.
Lemma 5.1.12. For any g ∈ coLie ⋆ (X ) ≥d+1 and any non-negative integer i, Supp coChev i g ⊂ Ran ≤i X and for all non-empty finite set I such that |I | ≤ i, (Sym i (g[−1]))| • X I lives in perverse cohomological degrees ≥ i(d + 2).
Proof. This follows directly from (2.4.4) and the fact that !-pullbacks are left exact with respect to the perverse t-structure.
Corollary 5.1.13. For any g ∈ coLie ⋆ (X ) ≥d+1 , any non-empty finite set I , and any positive integer j, (coChev j g)| • X I lives in perverse cohomological degrees ≥ |I |(d + 2). In particular, is t-left exact, the second statement follows from the first. Now, when j < |I |, then there is nothing to prove since everything vanishes. For j ≥ |I |, we have the following sequence of sheaves coChev j g| • X I → coChev j−1 g| • X I → · · · → coChev |I| g| • X I → coChev |I|−1 g| • X I ≃ 0. Inducting on k ∈ {|I |, . . . , j}, using the fact that the k-th fiber of this sequence is Sym k (g[−1])| • X I (see (5.1.9)) and the estimates in Lemma 5.1.12 concludes the proof.
Lemma 5.1.14. For any g ∈ coLie ⋆ (X ) ≥d+1 and any pair of positive integers i, j, Proof. Consider the following sequence of chain complexes , . . . , i} by Lemma 5.1.6. 13 By Lemma 5.1.12, we see that this chain complex lives in cohomological degrees ≥ j(d + 2)− kd when k ≤ j and vanishes otherwise. Thus, in particular, it lives in cohomological degrees ≥ 2 j. Inducting on k ∈ {1, . . . , i}, we conclude the proof.
Proposition 5.1.15. When g ∈ coLie ⋆ (X ) ≥d+1 , the functor F considered at (5.1.11) satisfies the conditions stated in Corollary 5.1.4. In particular, we have a natural equivalence Proof. This is a direct consequence of Corollary 5.1.13 and Lemma 5.1.14. 13 Since X is assumed to be proper throughout this subsection, our statement is valid also for the case k = 1.
Here, (5.1.17) is due to the fact that C * c (Ran ≤i X , −) is a finite colimit of functors of the form C * c (X I , −), each of which commutes with limits since X is proper. Moreover, (5.1.18) is due to Proposition 5.1.15 and (5.1.19) is due to the fact that coChev j is a finite limit and g is supported only on X . Finally, (5.1.20) is obtained by applying the addCoFil trick to the case of Vect.
Remark 5.1.21. In the last step (5.1.20), we need g to live in perverse cohomological degrees ≥ d + 1 so that C * c (X , g) ≃ C * (X , g) lives in cohomological degrees ≥ 1, which is needed to apply the addCoFil trick. Here, X = pt in (4.3.2).

Verdier duality.
Before studying the link between Chev and coChev, we start with a quick recollection of Verdier duality on prestacks along with various useful properties. The main reference is [Gai15]. We only use the very basic properties of D Ran . 5.2.1. Let Y be a prestack such that the diagonal map is pseudo-proper. For F, G ∈ Shv(Y), by a pairing between them, we shall mean a map We define the Verdier dual D Y G of G to be the object representing the functor Namely, we have the following natural equivalence The following lemma is immediate from the definition.
3. We will now study the link between Verdier duality and ⊠.
Proposition 5.2.4. Let Y 1 and Y 2 be finitary pseudo-schemes, and F i ∈ Shv(Y i ) for i ∈ {1, 2}. Then, we have a natural equivalence Proof. First, note that the result holds when both Y 1 and Y 2 are schemes.
For the general case of finitary pseudo-schemes, we write Then, Here, (5.2.6) is due to the fact that the statement we are trying to prove holds for the case of schemes, (5.2.7) is due to the fact that the limits we are taking are all finite (due to the finitary assumption), and finally, both (5.2.5) and (5.2.8) are due to Lemma 5.2.2 and Proposition 5.2.9 below.
Remark 5.2.10. One direct corollary of this proposition is the fact that for any sheaf F ∈ Shv(X ), we have the following natural equivalence Corollary 5.2.11. Let F 1 , F 2 , · · · , F k ∈ Shv(Ran X ) with finite supports, i.e. there exists an n such that all the F i 's are !-pushforward of sheaves on Ran ≤n X . Then, we have the following natural equivalence Proof. Since the sheaves involved have finite supports, their box-tensor commutes with Verdier duality on Ran ≤n X , by Proposition 5.2.4. Since Ran ≤n X → Ran X is finitary pseudo-proper, Proposition 5.2.9 implies that their box-tensor also commutes with Verdier duality on Ran X . Finally, using the fact that the union map is finitary pseudo-proper, Proposition 5.2.4 then implies that ⊗ ⋆ of these sheaves also commutes with Verdier duality on the Ran space. 5.3. Chev, coChev, and D Ran X . We will now turn to Theorem 1.5.9, which provides a link between the two functors Chev and coChev via the functor of taking Verdier duality on the Ran space. Note that this is the only place we use Verdier duality on the Ran space. However, we essentially use it in a rather minimal way: not much besides the definition itself.
Proof. We will employ ideas originated from the addFil and addCoFil tricks (see also §A). First, observe that for any g ∈ Lie ⋆ (X ), we have a canonical equivalence addCoFil D Ran X g ≃ D Ran X addFil g.
We use Chev i g and coChev i D Ran X g to denote the i-th piece in the filtration/co-filtration of Chev(addFil g) and coChev(addCoFil D Ran X g) respectively.
From §A and the top part of the commutative diagram (4.3.2), we have the following natural equivalences At the same time, by Lemma 5.2.2, we know that Thus, it suffices to show that Now, it's an immediate consequence of Corollary 5.2.11. Corollary 5.3.2. Let g ∈ Lie ⋆ (X ) ≤−1 . Then D Ran X Chev(g) is a factorizable commutative algebra on Ran X .
Proof. This is a direct consequence of Theorem 5.3.1 and Theorem 4.1.3.

coChev and open embeddings.
We end the section with the following simple observation. Then for any g ′ ∈ coLie ⋆ (X ′ ), we have the following natural equivalence Proof (Sketch). The result is a direct consequence of the fact that ( j Ran ) * is symmetric monoidal and commutes with limits. The latter is due to the fact that it is a right adjoint. The former is due to the fact that for any open embeddings of prestacks f i : X ′ i → X i and any F i ∈ Shv(X ′ i ) for i = 1, 2, we have a natural equivalence This is in turn a consequence of (2.3.16) and the corresponding fact for schemes.

AN APPLICATION TO THE ATIYAH-BOTT FORMULA
We will now give an application of the results proved so far to the Atiyah-Bott formula. As mentioned in the introduction, these results allow us to simplify the second of the two main steps in the original proofs given in [GL14] and [Gai15]. In what follows, §6.1- §6.4 are intended to orient the readers with the existing results proved in [GL14] and [Gai15], 14 whereas the purpose of the last part, §6.5, is to explain how the results we've proved so far fit in with the rest. 14 Namely, all the results stated in these subsections could be found in [GL14] or [Gai15]. The readers should be warned that we provide a mere overview of the development given in these two papers, with many technical points elided. 6.1. The statement. From now on, X is a smooth and complete curve over an algebraically closed field k, and G a smooth, fiber-wise connected group-scheme over X , whose generic fiber is semi-simple simply connected. Due to [GL14, Lem. 7.1.1 and Prop A.3.11], we can (and from now on we will) assume that G is semi-simple simply connected over an open dense subset j : X ′ → X , and moreover, the fibers of G over any point in X − X ′ are homologically trivial.
We will also use j Ran : Ran X ′ → Ran X to denote the corresponding open embedding on the Ran space and Γ j Ran : Ran X ′ → Ran X ′ × Ran X to denote its graph.
6.1.1. Let G 0 be the split form of G. Then it is well-known that is a free commutative algebra, for some M 0 ∈ Vect. In the case of ℓ-adic sheaves in positive characteristic setting, this equivalence is compatible with the geometric Frobenius action, where and e's are the exponents of G 0 . The assignment G 0 → M 0 is functorial with respect to automorphisms of G 0 , and hence, for a general G (subject to the assumptions mentioned above), we get a local system M ∈ Shv(X ′ ), whose !-fiber at each geometric point x ∈ X is equivalent to M 0 .
Below is the statement of the Atiyah-Bott formula.
(b) When k = q , and X and G are defined over q , the above equivalence can be chosen to be compatible with the Frobenius actions.
6.2. BG and the sheaf B.
6.2.1. The sheaf B that we will now describe encodes the reduced cohomology of BG, the relative (over X ) classifying stack of G. For each I ∈ Ran X (S), let D I ⊂ S × X be the corresponding Cartier divisor. Let BG I denote the Artin stack classifying G-bundles over D I and f I : BG I → S the forgetful map. Then, we define , where D S is the functor of taking Verdier duality on S. These sheaves, assembled together, give rise to a sheaf (see also [GL14,Prop. 5
6.2.2. Note that for any finite set of points {x 1 , . . . , x n } ∈ (Ran X )(k), the !-fiber of B at this point is However, we see easily from (6.2.3) that B is not factorizable. The functor TakeOut developed in [Gai15] allows us to remove all the extra components in it and construct out of it a new object B ∈ Fact ⋆ (X ) with the correct !-fibers at a point {x 1 , . . . , Moreover, B has the same cohomology along Ran X as the original sheaf B (see also [Gai15, Cor. 5.3.5]) 6.2.5. B and Bun G . For every S ∈ Sch and I ∈ (Ran X )(S), we have a map of prestacks over S by restricting the bundle to the divisor D I This induces a map B S,I → ω S ⊗ C * red (Bun G ) and hence, also a map Applying the functor C * c (Ran X , −) and using the fact that Ran X is homologically contractible, we get a map (6.2.7) C * c (Ran X , B) ≃ C * c (Ran X , B) → C * red (Bun G ).
6.2.8. Using (6.1.2) and the assumption we have on G, i.e. it has homologically contractible fibers outside of X ′ , one gets an equivalence where B ′ is the restriction of B to Ran X ′ and, the symmetric algebra is taken inside Shv(Ran X ) using the ⊗ ⋆ -monoidal structure.
6.2.10. Using the equivalence (6.2.9) and the fact that C * c (Ran X , −) commutes with Sym, 15 we get an explicit presentation of the LHS of (6.2.7) ), which appears in the statement of the Atiyah-Bott formula as stated in Theorem 6.1.3. 6.2.12. Now, we are done if we could show that the map in (6.2.7) is an equivalence. 6.3. Affine Grassmannian and the sheaf A. Unfortunately, one does not know how to directly prove that (6.2.7) is an equivalence. Instead, [GL14] proceeds with an equivalence of a dual nature, which we will now briefly recall. 15 Note that this is a special case of the fact that C * c (Ran X , −) commutes with Chev. And in fact, both are due to the same reasons: that C * c (Ran X , −) is continuous and that it's symmetric monoidal.
6.3.1. The main player in this step is the affine Grassmannian, or more precisely, a factorizable version thereof. Let G and X be as above. The factorizable affine Grassmannian of G, denoted by Gr Ran X ′ , is the prestack whose S-points are given by where (i) P is a G-bundle over S × X , (ii) I is a non-empty finite subset of X ′ (S), (iii) α is a trivialization of P on the complement of the graph of I .
6.3.2. From the definition, we have the following natural morphism where we remember only the set I , and similarly another natural morphism where we remember only the bundle P.
6.3.3. The map g allows us to define and the map u induces a map at the homology level, namely . Together, we get the following map (6.3.5) C * c (Ran X ′ , A ′ ) → C red * (Bun G ).
6.3.6. Note that since Gr Ran X ′ → Ran X ′ is pseudo-proper, A ′ is easy to describe. Namely for any finite set of points {x 1 , x 2 , . . . , x n } ⊂ X (k), the !-fiber of A ′ at this point is 6.3.8. Using a variant of the diagonal map Gr → Gr × Gr, one can equip A ′ with the structure of an object in However, note that the sheaf A ′ is not factorizable, since its !-fiber, as described in (6.3.7), is too big, i.e. it's not equivalent to (6.3.9) Using a similar reasoning as in the case of B and B, we can construct an object A ′ ∈ coFact ⋆ (X ′ ) with the correct !-fiber as given in (6.3.9), and moreover, A ′ has the property that (6.3.10) This theorem is essentially a result about the homological contractibility of the space of rational map (maps that are defined only on an open subset) from X to G. An earlier version of this was proved in [Gai12]. Together with (6.3.10) we have the following Proposition 6.3.13. We have a natural equivalence . 6.4. Pairing. We will now describe how the equivalence given by Proposition 6.3.13 helps us show that (6.2.7) is an equivalence.
6.4.1. For any schemes S, S ′ ∈ Sch and any non-empty finite subsets I ⊂ X (S) and I ′ ⊂ X ′ (S ′ ), we have a natural map (which is just a more elaborate variant of (6.2.6))

which induces a map
A ′ ⊠ B → ω Ran X ′ ×Ran X , and hence, a pairing (using TakeOut) 6.4.2. Restricting this map to Ran X ′ × Ran X ′ gives us the following map and hence, using the definition of Verdier duality, a map between objects in ComAlg ⋆ (Ran X ′ ).
6.4.4. It is proved, in fact twice (using different methods), in §17 and §18 of [Gai15], that the restriction of (6.4.3) to the diagonal X ′ of Ran X ′ is an equivalence. Namely, we have (6.4.5) 6.5. The last steps. The results that we have just proved in this paper appear in two places in the concluding steps, which are given by Proposition 6.5.1 and 6.5.4. Together, they imply the Atiyah-Bott formula.
Proof. It is well-known that for a split semi-simple simply connected group G 0 , C red * (Gr G 0 , Λ) lives in cohomological degrees ≤ −2. Using the fact that Gr Ran X ′ → Ran X ′ is pseudo-proper and that A ′ is factorizable, we see that for each non-empty finite set I , A ′ | • X ′ I lives in (perverse) cohomological degrees ≤ −3|I |. Now, by Theorem 3.3.3, we know that there exists an object a ′ ∈ Lie ⋆ (X ′ ) ≤c L such that A ′ ≃ Chev(a ′ ). Theorem 5.3.1 then implies that D Ran X ′ Chev(a ′ ) ≃ coChev(D X ′ a ′ ), which is known to be factorizable by Theorem 4.1.3 Corollary 6.5.2. The map given in (6.4.3) is an equivalence, i.e.
Proof. The first statement is a direct consequence of the proposition above and the equivalence (6.4.5), where as the second statement is the result of Proposition 5.4.1.

APPENDIX A. THE addFil TRICK
In this appendix, we will quickly recall, without proof, a useful construction taken from [GR17, §IV.2], which allows us to reduce many statements about P-algebras to trivial P-algebras, where P is an operad in Vect. Throughout this subsection, all categories without any further description will be assumed to be presentable, symmetric monoidal stable infinity over a field k of characteristic 0. Moreover, functors between these categories are assumed to be continuous.
All such categories, along with continuous functors between them, form a category, which we will use DGCat SymMon pres,cont , to denote, or for simplicity DGCat SymMon .
A.1. Notations. For a symmetric monoidal category C, we denote the category of filtered objects in C C Fil = Fun( , C), the category of functors from to C. Here, is a ordered set, viewed as a category. Similarly, we denote the category of graded objects C gr = Fun( set , C), where set is a the discrete category, whose underlying underlying objects are the integers. 16
A.2. Functors. Now, we will recall several familiar functors between C, C Fil , and C gr .
A.2.1. Let V = · · · → V n−1 → V n → V n+1 → · · · , be an object in C Fil . Then, we define ass-gr : C Fil → C gr to be the functor of taking the associated graded object ass-gr(V ) n = coFib(V n−1 → V n ), A.2.3. Note that the categories C Fil and C gr are equipped with a natural symmetric monoidal structure coming from C, and moreover, the functors ass-gr, oblv Fil , gr → Fil, and are naturally symmetric monoidal.
A.2.4. Adding a filtration. Let addFil : C → C Fil be the functor defined as follows: for an object V in C, addFil(V ) n = V, when n ≥ 1, 0, otherwise.
A.3. Interactions with algebras over an operad. Let P be an operad in Vect. Then we have the following pair of functors addFil : P -alg(C) → P -alg(C Fil >0 ) and oblv Fil : P -alg(C Fil >0 ) → P -alg(C). Then from what we've discussed above, we have the following commutative diagram P -alg(C) Or more concretely, we have a compatible family of morphisms in F (C) Φ 1 (c) → Φ 2 (c) parametrized by pairs (C, c) where c ∈ C and C ∈ DGCat SymMon , and we want to prove that α is an equivalence.
A.3.5. The top square of the commutative diagram above implies that it suffices to show that Φ Fil 1 • addFil → Φ Fil 2 • addFil is an equivalence. But since ass-gr and are conservative, it suffices to show that • ass-gr •Φ Fil 1 • addFil → • ass-gr •Φ Fil 2 • addFil is an equivalence, which, due to the commutativity of the diagrams, is equivalent to • ass-gr • addFil : P -alg(C) → F (C) is canonically equivalent to triv P •oblv P , i.e.
A.3.8. This implies that it suffices to prove that is an equivalence only for the case where c is a trivial algebra.
A.4. A general principle. More generally, suppose we want to prove a property of Φ(c) for some c ∈ P -alg(C). Moreover, suppose this property is preserved under under oblv Fil , and is conservative under and ass-gr. Then, it suffices to prove the case where c has a trivial algebra structure.

APPENDIX B. CO-FILTRATION AND addCoFil
In this appendix, we will collect various notions that are dual to the one in §A. These are used in the body of the paper to give a proof of the addCoFil trick in a special case.
B.1. Notations. For a symmetric monoidal category C, we denote the category of co-filtered objects C coFil = Fun( op , C).
We will also use C coFil >0 to denote the full-subcategory of C coFil consisting of objects supported in positive degrees. Similarly for graded objects C gr and C gr >0 . B.2. Functors. As in the case of filtration, there are several familiar functors between C, C coFil , and C gr .
be an object in C coFil . Then we define ass-gr : C coFil → C gr to be the functor of taking the associated graded object ass-gr(V ) n = Fib(V n → V n−1 ), and oblv coFil : C coFil → C to be the right Kan extension along op → pt.
B.2.2. Note that the category C coFil naturally inherits the monoidal structure coming from C. Moreover, the functor ass-gr is monoidal.
B.2.3. We also use : C gr → C to denote the right Kan extension along set → pt. Namely B.2.4. Adding a co-filtration. We will use addCoFil : C → C coFil to denote a functor defined as follows: for an object V in C, addCoFil(V ) n = V, when n ≥ 1, 0, otherwise.