SHIFTED POISSON AND SYMPLECTIC STRUCTURES ON DERIVED N-STACKS

J.P.PRIDHAM

Abstract. We show that on a derived Artin N-stack, there is a canonical equivalence between the spaces of n-shifted symplectic structures and non-degenerate n-shifted Poisson structures.

Introduction

In [PTVV], the notion of an n-shifted symplectic structure for a derived Artin stack was introduced. The definition of an n-Poisson structure was also sketched, and an approach suggested to prove equivalence of n-shifted symplectic structures and nondegenerate n-Poisson structures, by first establishing deformation quantisation for symplectic structures. The Darboux theorems of [BBBBJ, BG] imply the existence of shifted Poisson structures locally on derived Deligne–Mumford stacks, but not globally.

We prove existence of shifted Poisson structures by a direct approach, not involving deformation quantisation or Darboux theorems. The key new notion is that of compatibility between an n-shifted pre-symplectic (i.e. closed, possibly degenerate) structure ω and an n-Poisson structure π. In the case of unshifted structures on underived manifolds, compatibility simply says that the associated maps between tangent and cotangent bundles satisfy

π^{♯} ∘ ω^{♯} ∘ π^{♯} = π^{♯},

which ensures that ω^{♯} = (π^{♯})^{−1} whenever π is non-degenerate. We demonstrate that this notion admits a natural generalisation incorporating shifts and higher coherence data. We expect that this notion of compatibility between Poisson and pre-symplectic structures is the same as in Definition 1.4.14 of the contemporaneous treatment [CPT^{+}], whose proof also does not involve deformation quantisation. Our notion of compatibility is extended in [Pri4] from Poisson structures to quantisations.

Our first main observation is that there is a canonical global section σ of the tangent space T P(X, n) of the space P(X, n) of n-Poisson structures, given by differentiating the 𝔾_{m}-action on the differential graded Lie algebra of polyvectors. For an unshifted Poisson structure on a smooth underived scheme, this just maps π to itself. In general, if we write π = ∑_{r≥2} π_{r} with π_{r} an r-vector, then

σ(π) = ∑_{r≥2}(r − 1)π_{r}.

For any n-shifted Poisson structure π, contraction gives a multiplicative map μ(−, π) from de Rham to Poisson cohomology, so any n-shifted pre-symplectic structure ω defines another global section μ(ω, −) of the tangent space of the space of Poisson structures. In the unshifted case on a smooth scheme, the associated map from the cotangent space to the tangent space is simply given by

This work was supported by the Engineering and Physical Sciences Research Council [grant number EP/I004130/2].

μ(ω, π)^{♯} = π^{♯} ∘ ω^{♯} ∘ π^{♯}.

We then say that a pair (ω, π) is compatible if μ(ω, π) ≃ σ(π). More formally, the space of compatible pairs is the homotopy limit of the diagram

\xymatrix{PreSp(X, n) × P(X, n) \ar[r]^{μ} \ar[r]_{σ} & T P(X, n) \ar[r] & P(X, n)}.

In the unshifted case, this amounts to seeking compatible pairs in the sense above.

Poisson structures are functorial with respect to étale morphisms, so the notions above are readily defined for derived DM N-stacks. For derived Artin N-stacks, the formulation or Poisson structures is more subtle: we show that an Artin N-stack admits an étale cover by stacks coming from bi-CDGAs, and put Poisson algebra structures on the bi-CDGAs. These bi-CDGAs perform the same role as the formal affine derived stacks of [CPT^{+}].

We then show that:

(1) If π is non-degenerate, then every compatible pre-symplectic structure ω is also non-degenerate, hence symplectic, and the space of such structures is contractible.

(2) If ω is symplectic, then the space of non-degenerate compatible Poisson struc-

tures is contractible. Thus the spaces of n-shifted symplectic structures and non-degenerate n-Poisson structures are both weakly equivalent to the space of non-degenerate compatible pairs (ω, π). These results have recently been obtained in [CPT^{+}], using a very different method to formulate compatibility.

The structure of the paper is as follows.

Section 1 addresses the case of a single, fixed, derived affine scheme. The compatibility operator μ is introduced in Definition 1.16, and the key technical result is Lemma 1.17, which shows how μ interpolates between the de Rham differential and the Schouten– Nijenhuis bracket. §1.4 shows how to realise the spaces of pre-symplectic structures, Poisson structures and compatible pairs as towers of homotopy fibres. This uses the obstruction theory associated to pro-nilpotent L_{∞}-algebras, regarding the construction as a generalised deformation problem. The correspondence between symplectic and non-degenerate Poisson structures in the affine case is then given in Corollary 1.38.

Section 2 then applies these results to derived stacks. To date, the only proof that the cotangent complex governs deformations of a (derived) algebraic stack (rather than just morphisms to it from a fixed object) uses simplicial resolutions by affine schemes (cf. [Aok, Theorem 1.2], [Pri3, Theorem 8.8]). Since the construction of Poisson structures is a kind of generalised deformation problem, it is unsurprising that they can be studied using such resolutions. For derived Deligne–Mumford N-stacks, we look at spaces of structures on the diagrams given by suitable simplicial resolutions, and show that they are independent of the resolution; the key being étale functoriality. Theorem 2.19 then establishes an equivalence between the spaces of n-symplectic and non-degenerate n-Poisson structures.

In section 3, these results are extended to derived Artin N-stacks. Given a simplicial resolution by derived affines, we can form an associated bi-CDGA, and indeed a simplicial resolution by bi-CDGAs (Corollary 3.13). The morphisms in this resolution resemble étale maps, allowing a suitably functorial generalisation of the results of §1 to bi-CDGAs. The main result is then Theorem 3.31, which establishes an equivalence between the spaces of n-symplectic and non-degenerate n-Poisson structures on derived Artin N-stacks.

I would like to thank Victor Ginzburg for suggesting a simplification to the proofs of Lemma 1.17 and Proposition 1.26.

Contents

Introduction 1 1. Compatible structures on derived affines 3 1.1. Shifted Poisson structures 3 1.2. Shifted pre-symplectic structures 6 1.3. Compatible pairs 7 1.4. The obstruction tower 10 1.5. The equivalence 13 2. Compatible structures on derived DM stacks 15 2.1. Compatible structures on diagrams 15 2.2. Derived N-hypergroupoids 17 3. Compatible structures on derived Artin stacks 20 3.1. Bi-CDGAs 21 3.2. Comparing DG Artin hypergroupoids and bi-CDGAs 23 3.3. Poisson and symplectic structures 27 3.4. Diagrams and functoriality 29 References 32

1. Compatible structures on derived affines

For the purposes of this section, we will fix a commutative DG algebra R = R^{•} over ℚ, and a cofibrant commutative DG R^{•}-algebra A = A^{•}. We will denote the differential on A by δ.

In particular, the cofibrancy condition holds whenever the underlying morphism R^{#} → A^{#} of graded commutative algebras is freely generated in non-positive cochain degrees. It ensures that the module Ω_{A/R}^{1} of Kähler differentials is a model for the cotangent complex of A over R. For the purposes of this section, this latter property is all we need, so we could relax the cofibrancy condition slightly to include morphisms which are ind-smooth rather than freely generated.

We define Ω_{A/R}^{p} := Λ_{A}^{p}Ω_{A/R}^{1}, and we denote its differential (inherited from A) by δ.

There is also a de Rham differential Ω_{A/R}^{p} → Ω_{A/R}^{p+1}, which we denote by d.

Given A-modules M, N in cochain complexes, we write \underline{Hom}_{A}(M, N) for the cochain complex given by

\underline{Hom}_{A}(M, N)^{i} = Hom_{A^{#}}(M^{#}, N^{#}[i]),

with differential δf = δ_{N} ∘ f ± f ∘ δ_{M}, where V^{#} denotes the graded vector space underlying a cochain complex V.

1.1. Shifted Poisson structures.

1.1.1. Polyvector fields.

Definition 1.1. Define the complex of n-shifted polyvector fields on A by

P̂ol(A/R, n) := \underline{Hom}_{A}(CoSymm_{A}(Ω_{A/R}^{1}[n + 1]), A),

with graded-commutative multiplication following the usual conventions for symmetric powers. [Here, CoS_{A}^{p}(M) = CoSymm_{A}^{p}(M) = (M^{⊗_{A}p})^{Σ_{p}} and CoSymm_{A}(M) = ⨁_{p≥0} CoS_{A}^{p}(M).]

The Lie bracket on Hom_{A}(Ω_{A/R}^{1}, A) then extends to give a bracket (the Schouten– Nijenhuis bracket)

[−, −]: P̂ol(A/R, n) × P̂ol(A/R, n) → P̂ol(A/R, n)[−1 − n],

determined by the property that it is a bi-derivation with respect to the multiplication operation.

Thus P̂ol(A/R, n) has the natural structure of a P_{n+2}-algebra (i.e. an (n + 1)-shifted Poisson algebra), and in particular P̂ol(A/R, n)[n+1] is a differential graded Lie algebra (DGLA) over R.

Note that the differential δ on P̂ol(A/R, n) can be written as [δ, −], where δ ∈ P̂ol(A/R, n)^{1} is the element defined by the derivation δ on A.

Strictly speaking, P̂ol is the complex of multiderivations, as polyvectors are usually defined as symmetric powers of the tangent complex. The two definitions agree (modulo completion) whenever the tangent complex is perfect, and Definition 1.1 is the more natural object when the definitions differ.

Definition 1.2. Define a decreasing filtration F on P̂ol(A/R, n) by

F^{i}P̂ol(A/R, n) := \underline{Hom}_{A}(⨁_{j≥i} CoS_{A}^{j}(Ω_{A/R}^{1}[n + 1]), A);

this has the properties that P̂ol(A/R, n) = \underleftarrow{lim}_{i} P̂ol(A/R, n)/F^{i}, with [F^{i}, F^{j}] ⊂ F^{i+j−1}, δF^{i} ⊂ F^{i}, and F^{i}F^{j} ⊂ F^{i+j}.

Observe that this filtration makes F^{2}P̂ol(A/R, n)[n + 1] into a pro-nilpotent DGLA.

1.1.2. Poisson structures.

Definition 1.3. Given a DGLA L, define the the Maurer–Cartan set by

MC(L) := \{ω ∈ L^{1} | dω + \frac{1}{2}[ω, ω] = 0 ∈ ⨁_{n} L^{2}\}.

Following [Hin], define the Maurer–Cartan space \underline{MC}(L) (a simplicial set) of a nilpotent DGLA L by

\underline{MC}(L)_{n} := MC(L ⊗_{ℚ} Ω^{•}(Δ^{n})),

where

Ω^{•}(Δ^{n})) = ℚ[t_{0}, t_{1}, …, t_{n}, δt_{0}, δt_{1}, …, δt_{n}]/(∑ t_{i} − 1, ∑ δt_{i})

is the commutative dg algebra of de Rham polynomial forms on the n-simplex, with the t_{i} of degree 0.

Definition 1.4. Given an inverse system L = {L_{α}}_{α} of nilpotent DGLAs, define

MC(L) := \underleftarrow{lim}_{α} MC(L_{α}) \underline{MC}(L) := \underleftarrow{lim}_{α} \underline{MC}(L_{α}).

Note that MC(L) = MC(\underleftarrow{lim}_{α} L_{α}), but \underline{MC}(L) ≠ \underline{MC}(\underleftarrow{lim}_{α} L_{α}).

Definition 1.5. Define an R-linear n-shifted Poisson structure on A to be an element of

MC(F^{2}P̂ol(A/R, n)[n + 1]),

and the space P(A/R, n) of R-linear n-shifted Poisson structures on A to be given by the simplicial set

P(A/R, n) := \underline{MC}(\{F^{2}P̂ol(A/R, n)[n + 1]/F^{i+2}\}_{i}).

Also write P(A/R, n)/F^{i+2} := \underline{MC}(F^{2}P̂ol(A/R, n)[n + 1]/F^{i+2}), so P(A/R, n) = \underleftarrow{lim}_{i} P(A/R, n)/F^{i+2}.

Remark 1.6. Observe that elements of P_{0}(A/R, n) = MC(F^{2}P̂ol(A/R, n)[n+1]) consist of infinite sums π = ∑_{i≥2} π_{i} with

π_{i}: CoS_{A}^{i}(Ω_{A/R}^{1}[n + 1]) → A[n + 2]

satisfying δ(π_{i}) + \frac{1}{2} ∑_{j+k=i+1}[π_{j}, π_{k}] = 0. This is precisely the condition which ensures that π defines an L_{∞}-algebra structure on A[n]. This then makes A into a P̂_{n+1}-algebra in the sense of [Mel, Definition 2.9]. Equivalently, this is an algebra for the operad Com ∘ (L_{∞}[−n]) via a distributive law (L_{∞}[−n]) ∘ Com → Com ∘ (L_{∞}[−n]).

It is important to remember that P(−/R, n) is just one explicit presentation of the right-derived functor of P_{0}(−/R, n).

Definition 1.7. We say that an n-shifted Poisson structure π = ∑_{i≥2} π_{i} is nondegenerate if π_{2} : CoS_{A}^{2}(Ω_{A/R}^{1}[n + 1]) → A[n + 2] induces a quasi-isomorphism

π_{2}^{♯}: Ω_{A/R}^{1}[n] → \underline{Hom}_{A}(Ω_{A/R}^{1}, A)

and Ω_{A/R}^{1} is perfect.

Define P(A/R, n)^{nondeg} ⊂ P(A/R, n) to consist of non-degenerate elements — this is a union of path-components.

1.1.3. The canonical tangent vector σ. The space P(A/R, n) admits a scalar multiplication, which is inherited from the action on P̂ol(A/R, n) in which Hom_{A}(CoS_{A}^{p}(Ω_{A/R}^{1}[n+ 1]), A) is given weight p − 1. Differentiating this action gives us a global tangent vector on P(A/R, n), as follows. Take ϵ to be a variable of degree 0, with ϵ^{2} = 1.

Definition 1.8. Define the tangent spaces

T P(A/R, n) := \underline{MC}(\{F^{2}P̂ol(A/R, n)[n + 1] ⊗_{ℚ} ℚ[ϵ]/F^{i+2}\}_{i})
T P(A/R, n)/F^{i+2} := \underline{MC}(F^{2}P̂ol(A/R, n)[n + 1] ⊗_{ℚ} ℚ[ϵ]/F^{i+2}).

These are simplicial sets over P(A/R, n) (resp. P(A/R, n)/F^{i+2}), fibred in simplicial abelian groups.

Definition 1.9. Given π ∈ P_{0}(A/R, n), observe that δ + [π, −] defines a square-zero derivation on P̂ol(A/R, n), and denote the resulting complex by

P̂ol_{π}(A/R, n).

The product and bracket on polyvectors make this a P_{n+2}-algebra, and it inherits the filtration F.

Given π ∈ P_{0}(A/R, n)/F^{p}, we define P̂ol_{π}(A/R, n)/F^{p} similarly. This is a CDGA, and F^{1}P̂ol_{π}(A/R, n)/F^{p} is a P_{n+2}-algebra, because F^{i} · F^{j} ⊂ F^{i+j} and [F^{i}, F^{j}] ⊂ F^{i+j−1}.

Note that P̂ol_{π}(A/R, n) is just the natural n-shifted analogue of the complex computing Poisson cohomology.

The following is an instance of a standard result on square-zero extensions of DGLAs:

Lemma 1.10. The fibre T_{π}P(A/R, n) of T P(A/R, n) over π ∈ P(A/R, n) is canonically homotopy equivalent to the Dold–Kan denormalisation of the good truncation τ^{≤0}(F^{2}P̂ol_{π}(A/R, n)[n + 2]). In particular, its homotopy groups are given by

π_{i}T_{π}P(A/R, n) = H^{n+2−i}(F^{2}P̂ol(A/R, n), δ + [π, −]).

The corresponding statements for T_{π}P(A/R, n)/F^{i+2} also hold.

As a consequence of this lemma, there is a morphism

σ: F^{2}P̂ol(A/R, n)[n + 1] → F^{2}P̂ol(A/R, n)[n + 1] ⊗_{ℚ} ℚ[ϵ]

of filtered DGLAs sending α = ∑_{i≥2} α_{i} to ∑_{i≥2}(i − 1)α_{i}, for α_{i}: CoS_{A}^{i}(Ω_{A/R}^{1}[n + 1]) → A.

Definition 1.11. Define the canonical tangent vector σ: P(A/R, n) → T P(A/R, n) on the space of n-shifted Poisson structures by applying \underline{MC} to the morphism σ of DGLAs.

Explicitly, this sends π = ∑ π_{i} to σ(π) = ∑_{i≥2}(i − 1)π_{i}, which thus has the property that δσ(π) + [π, σ(π)] = 0.

This preserves the cofiltration in the sense that it comes from a system of maps σ: P(A/R, n)/F^{i+2} → T P(A/R, n)/F^{i+2}.

1.2. Shifted pre-symplectic structures.

Definition 1.12. Define the de Rham complex DR(A/R) to be the product total complex of the bicomplex

A ⟶^{d} Ω_{A/R}^{1} ⟶^{d} Ω_{A/R}^{2} ⟶^{d} …,

so the total differential is d ± δ.

We define the Hodge filtration F on DR(A/R) by setting F^{p}DR(A/R) ⊂ DR(A/R) to consist of terms Ω_{A/R}^{i} with i ≥ p.

Properties of the product total complex ensure that a map f: A → B induces a quasi-isomorphism DR(A/R) → DR(B/R) whenever the maps Ω_{A/R}^{p} → Ω_{B/R}^{p} are quasiisomorphisms, which will happen whenever f is a weak equivalence between cofibrant R-algebras.

The complex DR(A/R) has the natural structure of a commutative DG algebra over R, filtered in the sense that F^{i}F^{j} ⊂ F^{i+j}.

Definition 1.13. Define an n-shifted pre-symplectic structure ω on A/R to be an element

ω ∈ Z^{n+2}F^{2}DR(A/R).

Explicitly, this means that ω is given by an infinite sum ω = ∑_{i≥2} ω_{i}, with ω_{i} ∈ (Ω_{A/R}^{i})^{n+2−i} and dω_{i} = δω_{i+1}.

Definition 1.14. Define an n-shifted symplectic structure ω on A/R to be an n-shifted pre-symplectic structure ω for which the component ω_{2} ∈ Z^{n}Ω_{A/R}^{2} induces a quasiisomorphism

ω_{2}^{♯}: Hom_{A}(Ω_{A/R}^{1}, A) → Ω_{A/R}^{1}[n].

and Ω_{A/R}^{1} is perfect as an A-module.

Now, we can regard F^{2}DR(A/R)[n + 1] as a filtered DGLA with trivial bracket. This has the property that MC(F^{2}DR(A/R)[n+1]) = Z^{n+2}F^{2}DR(A/R). We therefore make the following definition:

Definition 1.15. Define the space of n-shifted pre-symplectic structures on A/R to be the simplicial set

PreSp(A/R, n) := \underline{MC}(\{F^{2}DR(A/R)[n + 1]/F^{i+2}\}_{i})

Also write PreSp(A/R, n)/F^{i+2} := \underline{MC}(F^{2}DR(A/R)[n + 1]/F^{i+2}), so PreSp(A/R, n) = \underleftarrow{lim}_{i} PreSp(A/R, n)/F^{i+2}.

Set Sp(A/R, n) ⊂ PreSp(A/R, n) to consist of the symplectic structures — this is a union of path-components.

Note that PreSp(A/R, n)/F^{i+2} is canonically weakly equivalent to the Dold–Kan denormalisation of the complex τ^{≤0}(F^{2}DR(A/R)[n + 2]/F^{i+2}) (and similarly for the limit PreSp(A/R, n)), but the description in terms of \underline{MC} will simplify comparisons.

1.3. Compatible pairs. We will now develop the notion of compatibility between a pre-symplectic structure and a Poisson structure. Analogous results for the unshifted, underived C^{∞} context can be found in [KSM, Proposition 6.4].

Definition 1.16. Given π ∈ (F^{2}P̂ol(A, n)/F^{p})^{n+2}, define

μ(−, π): DR(A/R)/F^{p} → P̂ol(A, n)/F^{p}

to be the morphism of graded A-algebras given on generators Ω_{A/R}^{1} by

μ(adf, π) := π⌟(adf) = a[π, f].

Given b ∈ (F^{2}P̂ol(A, n)/F^{p}), we then define

ν(−, π, b): DR(A/R)/F^{p} → P̂ol(A, n)/F^{p}

to be the A-linear derivation with respect to the ring homomorphism μ(−, π) given on generators Ω_{A/R}^{1} by

ν(adf, π, b) := b⌟(adf) = a[b, f].

To see that these are well-defined in the sense that they descend to the quotients by F^{p}, observe that because π ∈ F^{2}, contraction has the property that

μ(Ω^{1}, π) ⊂ F^{1}, ν(Ω^{1}, π, b) ⊂ F^{1},

so μ(F^{p}, π), ν(F^{p}, π, b) ⊂ F^{p} by multiplicativity.

Explicitly, when ϕ = adf_{1} ∧ … ∧ df_{p},

μ(ϕ, π) = a[π, f_{1}] … [π, f_{p},],
ν(ϕ, π, b) = ∑_{i} ±a[π, f_{1}] … [b, f_{i}] … [π, f_{p}].

Lemma 1.17. For ω ∈ DR(A)/F^{p} and π ∈ F^{2}P̂ol(A, n)^{n+2}/F^{p}, we have

[π, μ(ω, π)] = μ(dω, π) + \frac{1}{2}ν(ω, π, [π, π]),

δ_{π}μ(ω, π) = μ(Dω, π) + ν(ω, π, κ(π)),

where δ_{π} = [δ + π, −] is the differential on T_{π}P(A/R, n)/F^{p} and D = d ± δ is the total differential on (F^{2}DR(A)/F^{p}).

Proof. For fixed π and varying ω, all expressions are derivations with respect to μ(−, π), so it suffices to check this expression in the cases p = 0 and p = 1, ω = df. In these cases, we have

[π, μ(a, π)] = [π, a] = μ(da, π),
δ_{π}μ(a, π) = μ(Da, π),
[π, μ(df, π)] = [π, [π, f]] = \frac{1}{2}ν(df, π, [π, π]),
δ_{π}μ(df, π) = δ_{π}[π, f] = ν(df, π, [δ, π] + \frac{1}{2}[π, π]).

Because ν(a, π, [π, π]) = 0 (ν being A-linear) and ddf = 0, this gives the required results. □

In particular, this implies that when π is Poisson, μ(−, π) defines a map from de Rham cohomology to Poisson cohomology.

Lemma 1.18. There are maps

(pr_{2} + μϵ): PreSp(A/R, n)/F^{p} × P(A/R, n)/F^{p} → T P(A/R, n)/F^{p}

over P(A/R, n)/F^{p} for all p, compatible with each other. In particular, we have

(pr_{2} + μϵ): PreSp(A/R, n) × P(A/R, n) → T P(A/R, n).

Proof. For ω ∈ PreSp(A/R, n)_{0}, π ∈ P(A/R, n)_{0}, Lemma 1.17 shows that μ(ω, π) ∈ T_{π}P(A/R, n)/F^{p}. Replacing A, R with A ⊗ Ω^{•}(Δ^{m}), R ⊗ Ω^{•}(Δ^{m}) then shows that the statement also holds on the mth level of the simplicial set. □

An alternative approach to proving Lemma 1.18 is to observe that μ defines a filtered L_{∞}-morphism

F^{2}DR(A/R)[n + 1] × F^{2}P̂ol(A/R, n)[n + 1] ⟶^{pr_{2}+μϵ} F^{2}P̂ol(A/R, n)[n + 1][ϵ]

(with respect to the filtration F), and then to apply the functor \underline{MC}.

Example 1.19. If ω and π are pre-symplectic and Poisson, with ω_{i} = 0 for i > 2 and π_{i} = 0 for i > 2, then observe that μ(ω, π) induces the morphism

μ(ω, π)^{♯}: Ω_{A/R}^{1}[n] → Hom_{A}(Ω_{A/R}^{1}, A)

given by

μ(ω, π)^{♯} = π^{♯} ∘ ω^{♯} ∘ π^{♯}.

Definition 1.20. We say that an n-shifted pre-symplectic structure ω and an n-Poisson structure π are compatible (or a compatible pair) if

[μ(ω, π)] = [σ(π)] ∈ H^{n+2}(F^{2}P̂ol_{π}(A/R, n)) = π_{0}T_{π}P(A/R, n),

where σ is the canonical tangent vector of Definition 1.11.

Example 1.21. Following Example 1.19, if ω_{i} = 0 for i > 2 and π_{i} = 0 for i > 2, then (ω, π) are a compatible pair if and only if the map

π^{♯} ∘ ω^{♯} ∘ π^{♯}: Ω_{A/R}^{1}[n] → Hom_{A}(Ω_{A/R}^{1}, A)

is homotopic to π^{♯}, because σ(π) = π in this case.

In particular, if π is non-degenerate, this means that ω and π determine each other up to homotopy.

Lemma 1.22. If (ω, π) is a compatible pair and π is non-degenerate, then ω is symplectic.

Proof. Even when the vanishing conditions of Example 1.19 are not satisfied, we still have

π_{2}^{♯} ∘ ω_{2}^{♯} ∘ π_{2}^{♯} ≃ π_{2}^{♯},

so if π_{2}^{♯} is a quasi-isomorphism, then ω_{2}^{♯} must be its homotopy inverse. □

Definition 1.23. Given a simplicial set Z, an abelian group object A in simplicial sets over Z, and a section s: Z → A, define the homotopy vanishing locus of s to be the homotopy limit of the diagram

\xymatrix{Z \ar[r]^{s} \ar[r]_{0} & A \ar[r] & Z}.

We can write this as a homotopy fibre product Z ×_{(s,0),A×A}^{h} A, for the diagonal map A → A × A. When A is a trivial bundle A = Z × V, for V a simplicial abelian group, note that the homotopy vanishing locus is just the homotopy fibre of s: Z → V over 0.

Definition 1.24. Define the space Comp(A/R, n) of compatible n-shifted pairs to be the homotopy vanishing locus of

μ−σ: PreSp(A/R, n)×P(A/R, n) → pr_{2}^{∗}T P(A/R, n) = PreSp(A/R, n)×T P(A/R, n).

We define a cofiltration on this space by setting Comp(A/R, n)/F^{p} to be the homotopy vanishing locus of

μ − σ: PreSp(A/R, n)/F^{p} × P(A/R, n)/F^{p} → pr_{2}^{∗}T P(A/R, n)/F^{p}.

We can rewrite Comp(A/R, n) as the homotopy limit of the diagram

\xymatrix{PreSp(A/R, n) × P(A/R, n) \ar[r]^{(pr_{2}+μϵ)} \ar[r]_{pr_{2}+σpr_{2}ϵ} & T P(A/R, n) \ar[r] & P(A/R, n)}

of simplicial sets.

In particular, an object of this space is given by a pre-symplectic structure ω, a Poisson structure π, and a homotopy h between μ(ω, π) and σ(π) in T_{π}P(A/R, n).

Definition 1.25. Define Comp(A/R, n)^{nondeg} ⊂ Comp(A/R, n) to consist of compatible pairs with π non-degenerate. This is a union of path-components, and by Lemma 1.22 has a natural projection

Comp(A/R, n)^{nondeg} → Sp(A/R, n)

as well as the canonical map

Comp(A/R, n)^{nondeg} → P(A/R, n)^{nondeg}.

Proposition 1.26. The canonical map

Comp(A/R, n)^{nondeg} → P(A/R, n)^{nondeg}

is a weak equivalence.

Proof. For any π ∈ P(A/R, n), the homotopy fibre of Comp(A/R, n)^{nondeg} over π is just the homotopy fibre of

μ(−, π): PreSp(A/R, n) → T_{π}P(A/R, n)

over σ(π).

The map μ(−, π): DR(A/R) → (P̂ol(A, n), δ_{π}) is a morphism of complete filtered CDGAs by Lemma 1.17, and non-degeneracy of π_{2} implies that we have a quasiisomorphism on the associated gradeds gr_{F}. We therefore have a quasi-isomorphism of filtered complexes, so we have isomorphisms on homotopy groups:

π_{j}PreSp(A/R, n) → π_{j}T_{π}P(A/R, n)
H^{n+2−j}(F^{2}DR(A/R)) → H^{n+2−j}(F^{2}P̂ol(A, n), δ_{π}).
□

1.4. The obstruction tower. We will now show that the tower Comp(A/R, n) → … → Comp(A/R, n)/F^{i+2} → … → Comp(A/R, n)/F^{2} does not contain nearly as much information as first appears.

1.4.1. Small extensions and obstructions.

Definition 1.27. Say that a surjection L → M of DGLAs is small if the kernel I satisfies [I, L] = 0.

Lemma 1.28. Given a small extension e: L → M of DGLAs with kernel I, there is an exact sequence

π_{0}\underline{MC}(L) → π_{0}\underline{MC}(M) ⟶^{o_{e}} H^{2}(I)

of sets.

Proof. This is well-known. The obstruction map o_{e} is given by

o_{e}(ω) := d_{L}ω̃ + \frac{1}{2}[ω̃, ω̃]

for any lift ω̃ ∈ L^{1} of ω ∈ MC(M). □

In particular, if I is acylic, note that these lemmas combine to give us a trivial fibration

\underline{MC}(L) → \underline{MC}(M).

Proposition 1.29. For any small extension e: L → M of DGLAs with kernel I, there is an obstruction map ob_{e}: \underline{MC}(M) → \underline{MC}(I[1]) in the homotopy category of simplicial sets, with homotopy fibre \underline{MC}(L).

Proof. This is essentially the same as [Pri1, Theorem 1.45]. Set M′ to be the mapping cone of I → L, with DGLA structure given by setting [M′, I[1]] = 0. Then we have a surjection M′ → M ⊕ I[1] of DGLAs with kernel I. Since the map M′ → M is a small extension with acyclic kernel, it follows that \underline{MC}(M′) → \underline{MC}(M) is a trivial fibration, hence a weak equivalence.

The obstruction map is then given by \underline{MC}(M′) → \underline{MC}(I[1]), the homotopy fibre being \underline{MC}(L), as required. □

Proposition 1.30. Let A be an abelian group object over a simplicial set Z, with homotopy fibre A_{z} over z ∈ Z. If s: Z → A is a section with homotopy vanishing locus Y, there is a long exact sequence

\xymatrix{⋯ \ar[r] & π_{i}(Y, z) \ar[r] & π_{i}(Z, z) \ar[r]^{s} & π_{i}(A_{z}, 0) \ar[r] & π_{i−1}(Y, z) \ar[r] & ⋯ \\ ⋯ \ar[r] & π_{0}Y \ar[r] & π_{0}Z \ar[r]^{s} & π_{0}(A_{?}),}

of groups and sets, where the final map sends z to s(z) ∈ π_{0}(A_{s})

Proof. Since Y is the homotopy fibre product

Z ×_{(s,0),(A×_{Z}^{h}A)} A,

the long exact sequence of homotopy gives

… → π_{i}(Y, z) → π_{i}(Z, z) × π_{i}(A, z) → π_{i}(A ×_{Z}^{h} A, z) → ….

Since (A ×_{Z}^{h} A) ×_{A}^{h} {z} = A_{z}, this simplifies to

… → π_{i}(Y, z) → π_{i}(Z, z) → π_{i}(A_{z}, z).
□

Combining Propositions 1.29 and 1.30 gives:

Corollary 1.31. Given a small extension e: L → M of DGLAs with kernel I, there is a canonical long exact sequence

\xymatrix{⋯ \ar[r]^{e_{∗}} & π_{i}\underline{MC}(M) \ar[r]^{o_{e}} & H^{2−i}(I) \ar[r] & π_{i−1}\underline{MC}(L) \ar[r]^{e_{∗}} & ⋯ \\ ⋯ \ar[r]^{e_{∗}} & π_{1}\underline{MC}(M) \ar[r]^{o_{e}} & H^{1}(I) \ar[r] & π_{0}\underline{MC}(L) \ar[r]^{e_{∗}} & π_{0}\underline{MC}(M) \ar[r]^{o_{e}} & H^{2}(I)}

1.4.2. Towers of obstructions. The following is the long exact sequence of cohomology:

Proposition 1.32. For each p, there is a canonical long exact sequence

… → H^{n+2−p−i}(Ω_{A/R}^{p}) → π_{i}(PreSp/F^{p+1}) → π_{i}(PreSp/F^{p}) → H^{n+3−p−i}(Ω_{A/R}^{p}) → …

of homotopy groups, where PreSp = PreSp(A/R, n).

Definition 1.33. Given a compatible pair (ω, π) ∈ Comp(A/R, n)/F^{3} and p ≥ 0, define the complex M(ω, π, p) to be the cocone of the map

Ω_{A/R}^{p}[n − p + 1] ⊕ \underline{Hom}_{A}(CoS_{A}^{p}(Ω_{A/R}^{1}[n + 1]), A)[n + 1]
→ \underline{Hom}_{A}(CoS_{A}^{p}(Ω_{A/R}^{1}[n + 1]), A)[n + 1]

given by combining

Symm^{p}(π^{♯}): Ω_{A/R}^{p}[p] → \underline{Hom}_{A}(CoS_{A}^{p}(Ω_{A/R}^{1}[n + 1]), A)

with the map

ν(ω, π) − (p − 1): \underline{Hom}_{A}(CoS_{A}^{p}(Ω_{A/R}^{1}[n + 1]), A) → \underline{Hom}_{A}(CoS_{A}^{p}(Ω_{A/R}^{1}[n + 1]), A),

where

ν(ω, π)(b) := ν(ω, π, b).

Because ω and π lie in gr_{F}^{2} , the description of Definition 1.16 simplifies to give

ν(adf_{1} ∧ df_{2}, π)(b) = ±a[π, f_{1}][b, f_{2}] ± a[b, f_{1}][π, f_{2}].

Remark 1.34. On \underline{Hom}_{A}(Ω_{A/R}^{1}[n + 1], A) (the case p = 1), observe that the map ν(ω, π) is just given by π^{♯} ∘ ω^{♯}. Moreover, contraction with a 1-form defines a derivation with respect to the commutative multiplication on polyvectors, so ν(ω, π) is the derivation

ν(ω, π): P̂ol(A/R, n) → P̂ol(A/R, n)

given on generators by π^{♯} ∘ ω^{♯}.

If π is non-degenerate, then (by compatibility) π^{♯} ∘ ω^{♯} is homotopic to the identity map, and thus gr_{F}^{p} ν(ω, π) is homotopic to p. In particular, gr_{F}^{p} ν(ω, π) − (p − 1) is a quasi-isomorphism in this case, so the projection

M(ω, π, p) → Ω_{A/R}^{p}[n − p + 1]

is a quasi-isomorphism.

If we apply Corollary 1.31 to the small extensions gr_{F}^{p} L → L/F^{p+1} → L/F^{p} for the DGLA of polyvectors, and taking the long exact sequence of homotopy groups, we get:

Proposition 1.35. For each p ≥ 3, there is a canonical long exact sequence

[figure: ⋮; e_{∗}; ⋮; e_{∗}; π_{i}(Comp(A/R, n)/F^{p}); o_{e}; π_{1}(Comp(A/R, n)/F^{p}); o_{e}; π_{0}(Comp(A/R, n)/F^{p}); o_{e}; H^{2−i}M(ω_{2}, π_{2}, p); H^{1}M(ω_{2}, π_{2}, p); H^{2}M(ω_{2}, π_{2}, p); π_{i−1}(Comp(A/R, n)/F^{p+1}); e_{∗}; π_{0}(Comp(A/R, n)/F^{p+1}); e_{∗}; ⋮]

of homotopy groups and sets, where π_{i} indicates the homotopy group at basepoint (ω, π), and the target of the final map is understood to mean

o_{e}(ω, π) ∈ H^{2}M(ω_{2}, π_{2}, p).

Proof. Proposition 1.29 gives fibration sequences

P(A/R, n)/F^{p+1} →P(A/R, n)/F^{p} → MC(gr_{F}^{p} P̂ol(A/R, n)[n + 2])
T P(A/R, n)/F^{p+1} →T P(A/R, n)/F^{p} → MC(gr_{F}^{p} P̂ol(A/R, n)[n + 2][ϵ])
PreSp(A/R, n)/F^{p+1} →PreSp(A/R, n)/F^{p} → MC(Ω_{A/R}^{p}[n − j]),

for ϵ^{2} = 0.

We can regard these as homotopy vanishing loci for sections of trivial bundles. Combined with the description of Comp(A/R, n) as a homotopy vanishing locus, this gives Comp(A/R, n)/F^{p+1} as a homotopy vanishing locus on Comp(A/R, n)/F^{p}.

In more detail, we pull the sequences above back along

(i_{1}, i_{2}): Comp(A/R, n)/F^{p} → PreSp(A/R, n)/F^{p} × P(A/R, n)/F^{p},

giving a morphism

Comp(A/R, n)/F^{p} × MC(gr_{F}^{p} P̂ol(A/R, n)[n + 2] ⊕ Ω_{A/R}^{p}[n − j])

⟶^{gr_{F}^{p} (μ−σ)} Comp(A/R, n)/F^{p} × MC(gr_{F}^{p} P̂ol(A/R, n)[n + 2])

of trivial bundles on Comp(A/R, n)/F^{p}. If we write N for the homotopy kernel of this map, we obtain a bundle over Comp(A/R, n)/F^{p} equipped with a section s whose homotopy vanishing locus is Comp(A/R, n)/F^{p+1}.

It therefore remains to describe the tangents of the maps μ, σ. Since σ is linear on gr_{F}^{p}, it is its own tangent, equal to (p − 1). To calculate the tangent of μ, take e with e^{2} = 0, a ∈ F^{p}DR(A/R) and b ∈ F^{p}P̂ol(A/R, n). Then

μ(ω + ae, π + be) − μ(ω, π) = μ(ae, π + be) + μ(ω, π + be) − μ(ω, π)
= μ(a, π)e + ν(ω, π, b)e,

because e^{2} = 0.

Since terms in F^{p+1} vanish, and μ preserves the filtration F^{∗+2}, the only contributions left are

μ(a, π_{2})e + ν(ω_{2}, π_{2}, b)e.

At a point (ω, π) ∈ Comp(A/R, n)/F^{p}, the homotopy fibre N_{ω,π} of the bundle N above is therefore just MC(M(ω_{2}, π_{2}, p)[1]). Then Comp(A/R, n)/F^{p+1} is the homotopy vanishing locus of the obstruction map (which takes values in N), giving the long exact sequence of homotopy groups from Proposition 1.30. □

1.5. The equivalence.

Corollary 1.36. The canonical map

Comp(A/R, n)^{nondeg} → (Comp(A/R, n)/F^{3})^{nondeg} ×_{(PreSp(A/R,n)/F^{3})}^{h} PreSp(A/R, n)

is a weak equivalence.

Proof. Remark 1.34 shows that for non-degenerate ω_{2}, the map

M(ω_{2}, π_{2}, p) → Ω_{A/R}^{p}[n − j − 1]

is a quasi-isomorphism. Propositions 1.32 and 1.35 thus combine to show that the maps

Comp(A/R, n)^{nondeg}/F^{p+1} → (Comp(A/R, n)^{nondeg}/F^{p}) ×_{(PreSp(A/R,n)/F^{p})}^{h} (PreSp(A/R, n)/F^{p+1})

are weak equivalences for all p ≥ 3. We then just take the limit over all p. □

Proposition 1.37. The canonical map

Comp(A/R, n)^{nondeg}/F^{3} → Sp(A/R, n)/F^{3}

is a weak equivalence.

Proof. We may apply Proposition 1.30 to the definition of Comp as a homotopy vanishing locus, giving a long exact sequence

[figure: ⋮; π_{i}(Comp(A/R, n)/F^{3}, (ω, π)); π_{0}(Comp(A/R, n)/F^{3}); π_{i}(PreSp(A/R, n)/F^{3}, ω) × π_{i}(P(A/R, n)/F^{3}, π); μ−σ; π_{0}(PreSp(A/R, n)/F^{3}) × π_{0}(P(A/R, n)/F^{3}); μ−σ; π_{i}(T P(A/R, n)/F^{3}, 0); π_{0}(T_{ω,π}P(A/R, n)/F^{3})]

of groups and sets.

We just have

π_{i}(PreSp(A/R, n)/F^{3}, ω) = H^{n−i}(Ω_{A/R}^{2})
π_{i}(P(A/R, n)/F^{3}) = H^{n+2−i}(gr_{F}^{2} P̂ol(A/R, n))
π_{i}(T_{ω,π}P(A/R, n)/F^{3}) = H^{n+2−i}(gr_{F}^{2} P̂ol(A/R, n)),

with the map μ − σ given as in Definition 1.33 by combining Symm^{2}(π^{♯}): Ω_{A/R}^{2}[−2] → \underline{Hom}_{A}(CoS_{A}^{2}(Ω_{A/R}^{1}[n + 1]), A) with the map ν(ω, π) − 1.

As in Remark 1.34, when π is non-degenerate, the map gr_{F}^{2} ν(ω, π) − 1 is a quasiisomorphism, inducing isomorphisms

π_{i}(Comp(A/R, n)/F^{3}, (ω, π)) ≅ π_{i}(PreSp(A/R, n)/F^{3}, ω)

for all i > 0. For i = 0, the argument of Example 1.21 works equally well modulo F^{3}; combined with the exact sequence above, it shows that the locus

π_{0}(Comp(A/R, n)/F^{3})^{nondeg} ↪ π_{0}(PreSp(A/R, n)/F^{3}) = H^{n}(Ω_{A/R}^{2})

corresponds to the non-degenerate elements π_{0}(Sp(A/R, n)/F^{3}). □

Corollary 1.38. The canonical maps

Comp(A/R, n)^{nondeg} → Sp(A/R, n)
Comp(A/R, n)^{nondeg} → P(A/R, n)^{nondeg}

are weak equivalences.

Proof. For the first map, just combine Corollary 1.36 with Proposition 1.37. The second map is given by Proposition 1.26. □

2. Compatible structures on derived DM stacks

We will now study symplectic and Poisson structures on derived N-stacks. Rather than studying a single CDGA as in Corollary 1.38, will will establish a similar result for strings of étale morphisms between non-degenerate structures. Following the philosophy of [Pri3], we can represent a derived Artin or DM N-stack 𝔛 as a simplicial derived affine scheme X_{•} satisfying various conditions. Thus X_{•} is Spec O(X)^{•} for a cosimplicial diagram O(X)^{•} of CDGAs. For a derived DM stack, the morphisms in this diagram are étale, and descent results will show that Poisson structures are independent of the choice of resolution X_{•}. A modified construction will be given for derived Artin stacks in §3.

From now on, we will regard the CDGAs encountered so far as chain complexes … ⟶^{δ} A_{1} ⟶^{δ} A_{0} ⟶^{δ} … rather than cochain complexes — this will enable us to distinguish easily between derived (chain) and stacky (cochain) structures in §3.

Definition 2.1. Write dgCAlg(ℚ) for the category of chain complexes over ℚ with graded-commutative multiplication. We will refer to its objects as chain CDGAs (in contrast with the cochain CDGAs encountered thus far). Given R ∈ dgCAlg(ℚ), we write dgCAlg(R) for the category of graded-commutative R-algebras in chain complexes.

Defining the functor u from cochain complexes to chain complexes by (uV)_{i} := V^{−i}, observe that u is an isomorphism from the category of ℚ-CDGAs to dgCAlg(ℚ), with inverse u^{−1}.

Throughout this section, we now fix R ∈ dgCAlg(ℚ). Given A ∈ dgCAlg(R), we will simply write DR(A/R) := DR(u^{−1}A/u^{−1}R).

2.1. Compatible structures on diagrams.

2.1.1. Definitions. Given a small category I, an I-diagram A of chain CDGAs over R, and A-modules M, N in I-diagrams of chain complexes, we can define the cochain complex \underline{Hom}_{A}(M, N) to be the equaliser of the obvious diagram

∏_{i∈I} \underline{Hom}_{A(i)}(M(i), N(i)) ⟹ ∏_{f: i→j in I} \underline{Hom}_{A(i)}(M(i), f_{∗}N(j)).

All the constructions of §1.1 then adapt immediately; in particular, we can define

P̂ol(A/R, n) := \underline{Hom}_{A}(CoSymm_{A}(Ω_{A/R}^{1}[n + 1]), A),

leading to a space P(A/R, n) of Poisson structures.

In order to ensure that this has the correct homological properties, we now consider categories of the form [m] = (0 → 1 → … → m).

Lemma 2.2. If A is an [m]-diagram in chain CDGAs over R which is cofibrant and fibrant for the injective model structure (i.e. each A(i) is cofibrant and the maps A(i) → A(i+1) are surjective), then \underline{Hom}_{A}(CoS_{A}^{k}Ω_{A}^{1}, A) is a model for the derived Hom-complex R\underline{Hom}_{A}(LCoS_{A}^{k}LΩ_{A}^{1}, A), and \underline{Hom}_{A}(A, Ω_{A}^{p}) ≃ ho\underleftarrow{lim}_{i} LΩ_{A(i)}^{p}.

Proof. Because A(i) is cofibrant, Ω_{A(i)}^{p} is cofibrant as an A(i)-module, so the complex R\underline{Hom}_{A}(LCoS_{A}^{k}LΩ_{A}^{1}, A) is the homotopy limit of the diagram

\underline{Hom}_{A(0)}(C(0), A(0)) → \underline{Hom}_{A(0)}(C(0), A(1)) ← \underline{Hom}_{A(1)}(C(1), A(1)) → …,

where we write C(i) := CoS_{A(i)}^{k}Ω_{A(i)}^{1}. Since the maps A(i) → A(i + 1) are surjective, the maps \underline{Hom}_{A(i)}(CoS_{A(i)}^{k}Ω_{A(i)}^{1}, A(i)) → \underline{Hom}_{A(i)}(Ω_{A(i)}^{p}, A(j)) are so, and thus the homotopy limit is calculated by the ordinary limit, which is precisely \underline{Hom}_{A}(CoS_{A}^{k}Ω_{A}^{1}, A).

Because [m] has initial object 0, \underline{Hom}_{A}(A, Ω_{A}^{p}) ≅ \underline{Hom}_{A(0)}(A(0), Ω_{A(0)}^{p}) = Ω_{A(0)}^{p}. For the same reason, ho\underleftarrow{lim}_{i} Ω_{A(i)}^{p} ≃ Ω_{A(0)}^{p}. □

For such diagrams, we can also adapt all the constructions of §1.2 immediately by taking inverse limits, so setting

PreSp(A/R, n) := PreSp(A(0)/R, n) = \underleftarrow{lim}_{i∈[m]} PreSp(A(i)/R, n)

for any [m]-diagram A of chain CDGAs over R.

We then adapt the constructions of §1.3, defining

μ: PreSp(A/R, n) × P(A/R, n) → T P(A/R, n)

by setting μ(ω, Δ)(i) := μ(ω(i), Δ(i)) ∈ T P(A(i)/R, n) for i ∈ [m], and letting Comp(A/R, n) be the homotopy vanishing locus of

(μ − σ): PreSp(A/R, n) × P(A/R, n) → pr_{2}^{∗}T P(A/R, n).

The obstruction functors and their towers from §1.4 also adapt immediately to [m]- diagrams, giving the obvious analogues of the obstruction spaces defined in terms of

\underline{Hom}_{A}(CoS_{A}^{p}(Ω_{A/R}^{1}[n + 1]), A) and Ω_{A(0)}^{p}.

2.1.2. Functors and descent.

Lemma 2.3. If D = (A → B) is a fibrant cofibrant [1]-diagram in dgCAlg(R) which is formally étale in the sense that the map

Ω_{A}^{1} ⊗_{A} B → Ω_{B}^{1}

is a quasi-isomorphism, then the map

\underline{Hom}_{D}(CoS_{D}^{k}Ω_{D}^{1}, D) → \underline{Hom}_{A}(CoS_{A}^{k}Ω_{A}^{1}, A),

is a quasi-isomorphism.

Proof. This follows immediately from the proof of Lemma 2.2, using the quasiisomorphism

\underline{Hom}_{B}(CoS_{B}^{k}Ω_{B}^{1}, B) → \underline{Hom}_{A}(CoS_{A}^{k}Ω_{A}^{1}, B).
□

Definition 2.4. Write dgCAlg(R)_{c,↠} ⊂ dgCAlg(R) for the subcategory with all cofibrant chain CDGAs over R as objects, and only surjective morphisms.

We already have functors PreSp(−, n) and Sp(−, n) from dgCAlg(R) to sSet, the category of simplicial sets, mapping quasi-isomorphisms in dgCAlg(R)_{c} to weak equivalences. Poisson structures are only functorial with respect to formally étale morphisms, in an ∞-functorial sense which we now make precise.

Observe that, writing F for any of the constructions P(−, n), Comp(−, n), PreSp(−, n), and the associated filtered and graded functors, applied to [m]-diagrams in dgCAlg(R)_{c,↠}, we have:

Properties 2.5. (1) the maps from F (A(0) → … → A(m)) to

F (A(0) → A(1)) ×_{F (A(1))}^{h} F (A(1) → A(2)) ×_{F (A(2))}^{h} … ×_{F (A(n−1))}^{h} F (A(n − 1) → A(n))

are weak equivalences;

(2) if the [1]-diagram A → B is a quasi-isomorphism, then the natural maps from F (A → B) to F (A) and to F (B) are weak equivalences.

(3) if the [1]-diagram A → B is formally étale, then the natural map from F (A → B) to F (A) is a weak equivalence.

These properties follow from Lemmas 2.2 and 2.3, together with the obstruction calculus of §1.4 extended to diagrams.

Property 2.5.1 ensures that the simplicial classes ∐_{A∈B_{m}dgCAlg(R)_{c,↠}} F (A) fit together to give a complete Segal space ∫ F over the nerve BdgCAlg(R)_{c,↠}. Taking complete Segal spaces [Rez, §6] as our preferred model of ∞-categories:

Definition 2.6. Define LdgCAlg(R)_{c,↠}, LdgCAlg(R), L ∫ F, and LsSet to be the ∞- categories obtained by localising the respective categories at quasi-isomorphisms or weak equivalences.

Property 2.5.2 ensures that the homotopy fibre of L ∫ F → LdgCAlg(R)_{c,↠} over A is still just the simplicial set F (A), regarded as an ∞-groupoid.

Since the surjections in dgCAlg(R) are the fibrations, the inclusion LdgCAlg(R)_{c,↠} → LdgCAlg(R) is a weak equivalence, and we may regard L ∫ F as an ∞-category over LdgCAlg(R).

Furthermore, Property 2.5.3 implies that the ∞-functor L ∫ F → LdgCAlg(R) is a co-Cartesian fibration when we restrict to the subcategory LdgCAlg(R)^{ét} of homotopy formally étale morphisms, giving:

Definition 2.7. When F is any of the constructions above, we define

RF: LdgCAlg(R)^{ét} → LsSet

to be the ∞-functor whose Grothendieck construction is L ∫ F.

In particular, the observations above ensure that

(RF)(A) ≃ F (A)

for all cofibrant chain CDGAs A over R.

An immediate consequence of Corollary 1.38 is that the canonical maps

RComp(−/R, n)^{nondeg} → RSp(−/R, n)
RComp(−/R, n)^{nondeg} → RP(−/R, n)^{nondeg}

are natural weak equivalences of ∞-functors on LdgCAlg(R)^{ét}.

2.2. Derived N-hypergroupoids.

2.2.1. Background. We now require our chain CDGA R over ℚ to be concentrated in non-negative chain degrees, and write dg_{+}CAlg(R) ⊂ dgCAlg(R) for the full subcategory of chain CDGAs which are concentrated in non-negative chain degrees. We denote the opposite category to dg_{+}CAlg(R) by DG^{+}Aff_{R}. Write sDG^{+}Aff_{R} for the category of simplicial diagrams in DG^{+}Aff_{R}. A morphism in DG^{+}Aff_{R} is said to be a fibration if it is given by a cofibration in the opposite category dg_{+}CAlg(R) (which in turn just means that it is a retract of a quasi-free map).

Definition 2.8. For m ≥ 0, the combinatorial m-simplex Δ^{m} ∈ sSet is characterised by the property that Hom_{sSet}(Δ^{m}, K) ≅ K_{m} for all simplicial sets K. Its boundary ∂Δ^{m} ⊂ Δ^{m} is given by ⋃_{i=0}^{m} ∂^{i}(Δ^{m−1}) (taken to include the case ∂Δ^{0} = ∅), and for m ≥ 1 the kth horn Λ^{m,k} is given by ⋃\begin{matrix}0≤i≤m \\ i≠k \end{matrix} ∂^{i}(Δ^{m−1}).

Definition 2.9. Given a simplicial set K and a simplicial object X_{•} in a complete category C, we follow [GJ, Proposition VII.1.21] in defining the K-matching object in C by

M_{K}X := Hom_{sSet}(K, X).

Note that for finite simplicial sets K, the matching object M_{K}X still exists even if C only contains finite limits.

Explicitly, the matching object M_{∂Δ^{m}}(X) is given by the equaliser of a diagram

∏_{0≤i≤m} X_{m−1} ⟹ ∏_{0≤i<j≤m} X_{m−2},

while the partial matching object M_{Λ_{k}^{m}}(X) is given by the equaliser of a diagram

∏_{\begin{matrix}0≤i≤m \\ i≠k \end{matrix}} X_{m−1} ⟹ ∏_{\begin{matrix}0≤i<j≤m \\ i,j≠k \end{matrix}} X_{m−2}.

We now recall some results from [Pri3].

Definition 2.10. Given Y ∈ sDG^{+}Aff_{R}, a DG Artin (resp. DM) N-hypergroupoid X over Y is a morphism X → Y in sDG^{+}Aff_{R} for which:

(1) the matching maps

X_{m} → M_{∂Δ^{m}}(X) ×_{M_{∂Δ^{m}}(Y)} Y_{m}

are fibrations for all m ≥ 0;

(2) the partial matching maps

X_{m} → M_{Λ_{k}^{m}}(X) ×_{M_{Λ^{m}_{k}}^{h}(Y)}^{h} Y_{m}

are smooth (resp. étale) for all k, m, and are weak equivalences for all m > N and all k.

A morphism X → Y in sDG^{+}Aff_{R} is a trivial DG Artin (resp. DM) N-hypergroupoid if and only if the matching maps

X_{m} → M_{∂Δ^{m}}(X) ×_{M_{∂Δ^{m}}(Y)} Y_{m}

of Definition 2.9 are smooth (resp. étale) fibrations for all m, and are weak equivalences for all m ≥ n.

The following is [Pri3, Theorem 4.15 and Corollary 6.35], as spelt out in [Pri2, Theorem 5.11]:

Theorem 2.11. The ∞-category of strongly quasi-compact N-geometric derived Artin (resp. DM) stacks over R is given by localising the category of DG Artin (resp.) N-hypergroupoids over R at the class of trivial relative DG Artin (resp. DM) Nhypergroupoids.

Given a DG Artin (resp. DM) N-hypergroupoid, we denote the associated Ngeometric derived Artin (resp. DM) stack by X^{♯}.

Definition 2.12. Given X ∈ sDG^{+}Aff_{R}, define cdgMod(X) to be the category of O(X)-modules in cosimplicial diagrams of chain complexes. We say that a morphism M → N in cdgMod(X) is a weak equivalence if it induces quasi-isomorphisms M^{i} → N^{i} of chain complexes in each cosimplicial level.

The following is [Pri3, Proposition 5.12 and Remarks 5.24]:

Proposition 2.13. For a DG Artin N-hypergroupoid X, the ∞-category of quasicoherent complexes (in the sense of [Lur, §5.2]) on the n-geometric derived stack X^{♯} is equivalent to the localisation at weak equivalences of the full subcategory dgMod(X)_{cart} of cdgMod(X) consisting of modules M which are homotopy-Cartesian in the sense that the maps

∂^{i}: L∂_{i}^{∗}M^{m−1} → M^{m}

are quasi-isomorphisms for all i and m.

Definition 2.14. We make cdgMod(X) into a simplicial category by setting (for K ∈ sSet)

(M^{K})^{n} := (M^{n})^{K_{n}},

as an O(X)^{n}-module in chain complexes. This has a left adjoint M ↦ M ⊗ K.

Given M ∈ cdgMod(X), define \underline{M} ∈ (cdgMod(X))^{Δ} to be the cosimplicial diagram given in cosimplicial level n by M ⊗ Δ^{n}.

Combining [Pri3, Definition 7.7 and Corollary 7.19] gives:

Definition 2.15. For a DG Artin N-hypergroupoid X, define the cotangent complex 𝕃^{X/S} ∈ cdgMod(X) by 𝕃^{X/S} := Tot N_{c}^{≤N} \underline{Ω(X/S)}, where N_{c} denotes cosimplicial conormalisation, and Tot is the direct sum total functor from cochain chain complexes to chain complexes.

By [Pri3, Lemma 7.17 and Corollary 7.14], this is homotopy-Cartesian and recovers the usual cotangent complexes in derived algebraic geometry under the correspondence of Proposition 2.13.

2.2.2. Poisson and symplectic structures. Take a DG Deligne–Mumford Nhypergroupoid X over R. Because the face maps ∂^{i}: Δ^{m−1} → Δ^{m} are weak equivalences, the morphisms ∂_{i}: X_{m} → X_{m−1} are all étale. Since σ_{i}: X_{m−1} → X_{m} has left inverse ∂_{i}, it follows that σ_{i} is also étale.

Therefore O(X) is a functor from Δ^{opp} to dgCAlg(R)^{ét}.

Definition 2.16. For any of the functors F in Definition 2.7, write

F (X) := ho\underleftarrow{lim}_{j∈Δ} RF (O(X_{j})).

This gives us spaces P(X/R, n), PreSp(X/R, n) and Comp(X/R, n) of Poisson structures, pre-symplectic structures and compatible pairs. Explicitly, strictification theorems then imply that an element of P(X/R, n) can be represented by a cosimplicial diagram P of P̂_{n+1}-algebras equipped with a weak equivalence from O(X) to the cosimplicial chain CDGA underlying P.

Proposition 2.17. If Y → X is a trivial DG DM hypergroupoid, then the morphism

F (X) → F (Y)

is an equivalence for any of the constructions F = P, Comp, PreSp.

Proof. The morphism F (X) → F (Y) exists because the maps Y_{j} → X_{j} are all étale. By Propositions 1.32, 1.35 and the corresponding statement for P, it suffices to prove that the morphisms

ho\underleftarrow{lim}_{j∈Δ} Ω_{X_{j}/R}^{p} → ho\underleftarrow{lim}_{j∈Δ} Ω_{Y_{j}/R}^{p},
ho\underleftarrow{lim}_{j∈Δ} \underline{Hom}_{O(X_{j})}(CoS^{p}(Ω_{X_{j}/R}^{1}[n + 1]), O(X_{j})) → ho\underleftarrow{lim}_{j∈Δ} \underline{Hom}_{O(Y_{j})}(CoS^{p}(Ω_{Y_{j}/R}^{1}[n + 1]), O(Y_{j}))

are quasi-isomorphisms. This follows by faithfully flat descent for quasi-coherent sheaves, since 𝔛 := X^{♯} ≃ Y^{♯}, the expressions reducing to RΓ(𝔛, LΩ_{𝔛/R}^{p}) and R\underline{Hom}_{𝒪_{𝔛}} (CoS^{p}(𝕃_{𝔛/R}[n + 1]), 𝒪_{𝔛}), respectively. □

Thus the following is well-defined:

Definition 2.18. Given a strongly quasi-compact DG DM N-stack 𝔛, define the spaces P(𝔛/R, n), Comp(𝔛/R, n), Sp(𝔛, n) to be the spaces P(X/R, n), Comp(X/R, n), Sp(X, n) for any DG DM N-hypergroupoid X with X^{♯} ≃ 𝔛.

By the results of §2.1.2, an immediate consequence of Corollary 1.38 is then:

Theorem 2.19. There are natural weak equivalences

Sp(𝔛/R, n) ← Comp(𝔛/R, n)^{nondeg} → P(𝔛/R, n)^{nondeg}.

3. Compatible structures on derived Artin stacks

We now show how to extend the comparisons above to derived Artin N-stacks over a fixed chain CDGA R ∈ dg_{+}CAlg(ℚ). The main difficulty is in establishing an appropriate notion for shifted Poisson structures in this setting. It is tempting just to consider DG Artin N-hypergroupoids equipped with Poisson structures levelwise, with weak equivalences generated by trivial DG Artin N-hypergroupoids. With this approach, it is not easy to compute the obstructions arising, and it seems unlikely that they will be related to the cotangent complex in the desired way.

Remark 3.1. By contrast, deformations of a DG Artin hypergroupoid X do recover all deformations of the associated derived Artin stack, at least on the level of objects (cf. [Pri3, Theorem 8.8]). The key in that case was the comparison between Ω^{1}(X/R) and the cotangent complex 𝕃_{X/R} in [Pri3, Lemma 8.4], which relies on strong boundedness properties. The shifts involved in the definition of Poisson structures preclude any analogous result for the obstruction tower associated to a semi-strict Poisson structure.

3.1. Bi-CDGAs. Our solution is to replace Artin N-hypergroupoids in CDGAs with DM hypergroupoids in a suitable category of bi-CDGAs. The idea is to globalise one of the intermediate steps [Pri1, Theorem 4.26] in the proof of the equivalence between formal moduli problems and DGLAs.

Given a bi-CDGA A, the definition of an n-shifted Poisson structure is fairly obvious: it is a Lie bracket of total degree n + 1, or rather an L_{∞}-structure in the form of a sequence [−]_{m} of m-ary operations of chain degree n − m + 1. However, the precise formulation (Definition 3.19) is quite subtle, involving lower bounds on the cochain degrees of the operations.

Definition 3.2. Define a chain cochain complex V over ℚ to be a bigraded ℚ-vector space V = ⨁_{i,j} V_{j}^{i}, equipped with square-zero linear maps ∂: V_{j}^{i} → V_{j}^{i+1} and δ: V_{j}^{i} → V_{j−1}^{i} such that ∂δ + δ∂ = 0.

There is an obvious tensor product operation ⊗ on this category, and a bi-CDGA is then defined to be a chain cochain complex A equipped with a commutative product A ⊗ A → A and unit k → A.

We regard all chain complexes as chain cochain complexes V = V_{•}^{0}. Given a chain CDGA R, a bi-CDGA over R is then a morphism R → A of bi-CDGAs. We write DGdgCAlg(R) for the category of bi-CDGAs over R, and DG^{+}dgCAlg(R) for the full subcategory consisting of objects A concentrated in non-negative cochain degrees.

Definition 3.3. Say that a morphism U → V of chain cochain complexes is a levelwise quasi-isomorphism if U^{i} → V^{i} is a quasi-isomorphism for all i ∈ ℤ. Say that a morphism of bi-CDGAs is a levelwise quasi-isomorphism if the underlying morphism of chain cochain complexes is so.

Lemma 3.4. There is a cofibrantly generated model structure on bi-CDGAs over R in which fibrations are surjections and weak equivalences are levelwise quasi-isomorphisms.

Proof. We first prove the corresponding statement for chain cochain complexes over ℚ. Let D^{i} be the chain complex k_{[−i]} → k_{[1−i]} and S^{i} := k_{[−i]}, so there is an obvious map S^{i} → D^{i}. Write D_{i} for the cochain complex k^{[1−i]} → k^{[−i]}. We can then take the set I of generating cofibrations to consist of the morphisms

{D_{i} ⊗ S^{j} → D_{i} ⊗ D^{j}}_{i,j∈ℤ},

with the set J of generating trivial cofibrations given by

{0 → D_{i} ⊗ D^{j}}_{i,j∈ℤ}.

It is straightforward to verify that this satisfies the conditions of [Hov, Theorem 2.1.19]. The forgetful functor from bi-CDGAs to chain cochain complexes has a left adjoint V ↦ R ⊗ Symm_{ℚ}V, which transfers the model structure by [Hir, Theorem 11.3.2]. □

There is a denormalisation functor D from non-negatively graded CDGAs to cosimplicial algebras, with left adjoint D^{∗} as in [Pri1, Definition 4.20]. Given a cosimplicial chain CDGA A, D^{∗}A is then a bi-CDGA, with (D^{∗}A)_{j}^{i} = 0 for i < 0.

Lemma 3.5. The functor D^{∗} is a left Quillen functor from the Reedy model structure on cosimplicial chain CDGAs to the model structure of Lemma 3.4.

Proof. The denormalisation functor D on non-negatively graded cochain complexes extends to all cochain complexes by composing with brutal truncation. The right adjoint to D^{∗} is given by applying D to the cochain index of a bi-CDGA A. Since fibrations and weak equivalences in the Reedy model structure are levelwise surjections and levelwise quasi-isomorphisms, it follows immediately that D is right Quillen, so D^{∗} is left Quillen. □

To any DG Artin N-hypergroupoid X over R, we can then associate the bi-CDGA D^{∗}O(X). This behaves well because DG Artin hypergroupoids are Reedy fibrant, so there is no need to replace D^{∗} with an associated left-derived functor. We cannot recover X from the levelwise quasi-isomorphism class of D^{∗}O(X), but we can recover X_{0} and the formal completion of X along X_{0}. Thus D^{∗}O(X) will play the same role as the formal affine derived stacks of [CPT^{+}].

For any DG Artin N-hypergroupoid X, the diagram X^{Δ^{m}} is another DG Artin Nhypergroupoid resolving the same stack, and we will then consider the cosimplicial bi-CDGA m ↦ D^{∗}O(X^{Δ^{m}}), which we can think of as a kind of DM N-hypergroupoid in bi-CDGAs.

3.1.1. Modules over bi-CDGAs. For now, we fix a cofibrant bi-CDGA A over a ℚ-CDGA R.

Definition 3.6. Given a chain cochain complex V, define the cochain complex T̂ot V ⊂ Tot^{Π}V by

(T̂ot V)^{m} := (⨁_{i<0} V_{i−m}^{i}) ⊕ (∏_{i≥0} V_{i−m}^{i})

with differential ∂ ± δ.

An alternative description of T̂ot V is as the completion of Tot V with respect to the filtration {Tot σ^{≥m}V }_{m}, where σ^{≥m} denotes brutal truncation in the cochain direction. In fact, we can write

\underleftarrow{lim}_{m} \underrightarrow{lim}_{n} Tot ((σ^{≥−n}V)/(σ^{≥m}V)) = T̂ot V = \underrightarrow{lim}_{n} \underleftarrow{lim}_{m} Tot ((σ^{≥−n}V)/(σ^{≥m}V)).

The latter description also shows that there is a canonical map (T̂ot U) ⊗ (T̂ot V) → T̂ot (U ⊗ V) — the same is not true of Tot^{Π} in general.

Write DGdgMod(A) for the category of A-modules in chain cochain complexes. When A ∈ DG^{+}dgCAlg(ℚ), write DG^{+}dgMod(A) ⊂ DGdgMod(A) for the full subcategory of objects concentrated in non-negative cochain degrees.

Definition 3.7. Given A-modules M, N in chain cochain complexes, we define internal Homs Hom_{A}(M, N) by

Hom_{A}(M, N)_{j}^{i} = Hom_{A_{#}^{#}}(M_{#}^{#}, N_{#[j]}^{#[i]}),

with differentials ∂f := ∂_{N} ∘ f ± f ∘ ∂_{M} and δf := δ_{N} ∘ f ± f ∘ δ_{M}, where V_{#}^{#} denotes the bigraded vector space underlying a chain cochain complex V.

We then define the Hom complex ˆ \underline{Hom}_{A}(M, N) by

ˆ\underline{Hom}_{A}(M, N) := T̂ot Hom_{A}(M, N).

Observe that there is a multiplication ˆ \underline{Hom}_{A}(M, N) ⊗ ˆ \underline{Hom}_{A}(N, P) → ˆ \underline{Hom}_{A}(M, P)

— the same is not true for Tot^{Π}Hom_{A}(M, N) in general.

Lemma 3.8. For A ∈ DG^{+}dgCAlg(ℚ), the denormalisation functor D defines an equivalence between DG^{+}dgMod(A), and the category cdgMod(DA) of DA-modules in cosimplicial chain complexes.

Proof. The inverse functor is simply given by the cosimplicial conormalisation functor N_{c}, together with the Alexander–Whitney cup product. For an arbitrary cosimplicial chain CDGA B, the cup product on N_{c}B need not be commutative, but because the shuffle product is left inverse to the cup product, N_{c}DA = A as a bi-DGA. □

We now write D: DGdgMod(A) → cdgMod(DA) for the composition of D with the brutal truncation functor σ^{≥0}. We call a DA-module M levelwise cofibrant if each M^{i} is cofibrant in dgMod(D^{i}A).

Lemma 3.9. For A ∈ DG^{+}dgCAlg(ℚ), a levelwise cofibrant DA-module M in cdgMod(DA), and P ∈ DGdgMod(A), there is a canonical quasi-isomorphism

R\underline{Hom}_{DA}(M, DP) ≃ Tot^{Π}σ^{≥0}Hom_{A}(N_{c}M, P).

Proof. The equivalence of Lemma 3.8 gives us an isomorphism

\underline{Hom}_{DA}(M, DP) ≅ Z^{0}Hom_{A}(N_{c}M, P).

There is a model structure on cdgMod(DA) in which cofibrations and and weak equivalences are defined levelwise in cosimplicial degrees. Similarly, there is a model structure on DGdgMod(A) in which weak equivalences are defined levelwise in cochain degrees, and a morphism M → N is a cofibration if M^{#} → N^{#} has the left lifting property with respect to all surjections of A^{#}-modules in graded chain complexes. These are essentially the injective diagram model structures (somewhat confusingly, the diagrams take values in the projective model structure on chain complexes). The functors N_{c} ⊣ D then form a Quillen adjunction, so to calculate R\underline{Hom}_{DA} it suffices to take a fibrant replacement for P.

Write S for the chain cochain complex S_{#}^{#} = ⨁_{i≥0}(ℚ_{[−i]}^{[−i]} ⊕ ℚ_{[−i]}^{[1−i]}), with all possible differentials in S being the identity. Thus S looks like a staircase in the second quadrant, with all columns acyclic. A fibrant replacement of P is given by Hom_{ℚ}(S, P), so

R\underline{Hom}_{DA}(M, DP) ≃ \underline{Hom}_{A}(M, DHom_{ℚ}(S, P))
≅ Z^{0}Hom_{A}(N_{c}M, DHom_{ℚ}(S, P))
≅ Z^{0}Hom_{ℚ}(S, Hom_{A}(N_{c}M, P))
≅ Tot^{Π}σ^{≥0}Hom_{A}(N_{c}M, P).
□

Definition 3.10. Given a bi-CDGA A, say that an A-module M in chain cochain complexes is homotopy-Cartesian if the maps

A^{i} ⊗_{A^{0}}^{L} M^{0} → M^{i}

are quasi-isomorphisms for all i.

3.2. Comparing DG Artin hypergroupoids and bi-CDGAs. Given a DG Artin N-hypergroupoid X, and an O(X)-module M, we may pull M back along the unit η: O(X) → DD^{∗}O(X) of the adjunction D^{∗} ⊣ D, and then apply Lemma 3.8 to obtain a D^{∗}O(X)-module N_{c}η^{∗}M.

Definition 3.11. As in [Pri3, Definition 2.15], define almost cosimplicial diagrams to be functors on the subcategory Δ_{∗} of the ordinal number category Δ containing only those morphisms f with f(0) = 0; define almost simplicial diagrams dually. Thus an almost simplicial diagram X_{∗} in C consists of objects X_{n} ∈ C, with all of the operations ∂_{i}, σ_{i} of a simplicial diagram except ∂_{0}, and satisfying the usual relations.

Given a simplicial (resp. cosimplicial) diagram X, we write X_{#} (resp. X^{#}) for the underlying almost simplicial diagram (resp. almost cosimplicial) diagram.

The denormalisation functor D descends to a functor from graded-commutative algebras to almost cosimplicial algebras, with D^{∗} thus descending to a functor in the opposite direction. In other words, (D^{∗}B)^{#} does not depend on ∂_{B}^{0}, and ∂_{DA}^{0} is the only part of the structure on DA to depend on ∂_{A}.

From this, it can be shown that for any DG Artin N-hypergroupoid X, the gradedcommutative algebra H_{0}D^{∗}O(X)^{#} is freely generated over H_{0}O(X)^{0} = H_{0}D^{∗}O(X)^{0} by a graded projective module, and that

H_{i}^{∗}O(X)^{#} ≅ H_{0}D^{∗}O(X)^{#} ⊗_{H_{0}O(X)^{0}} H_{i}O(X)^{0}.

If M is a homotopy-Cartesian O(X)-module, the map

(η^{∗}M)^{0} ⊗_{D^{0}D^{∗}O(X)} D^{#}D^{∗}O(X) → (η^{∗}M)^{#}

of almost cosimplicial chain complexes is a levelwise quasi-isomorphism, from which it follows that N_{c}^{0}(η^{∗}M) ⊗_{D^{∗}O(X)^{0}} D^{∗}O(X)^{#} → N_{c}(η^{∗}M)^{#} is a levelwise quasiisomorphism, so N_{c}(η^{∗}M)^{#} is also homotopy-Cartesian.

3.2.1. Equivalences of hypersheaves. Given a simplicial presheaf F on DGAff(R), there is an induced simplicial presheaf F D on DGdgAlg(R) given by

(F D)(A) := ho\underleftarrow{lim}_{i∈Δ} F (D^{i}A).

In the case of a simplicial derived affine X, regarded as a functor from dgAlg(R) to simplicial sets, XD is represented by the cosimplicial chain CDGA D^{∗}O(X^{Δ^{•}}), given in cosimplicial level m by D^{∗}O(X^{Δ^{m}}). Applying D then gives a bisimplicial derived affine Spec DD^{∗}O(X^{Δ^{•}}), and it is natural to consider the diagonal simplicial object diag Spec DD^{∗}O(X^{Δ^{•}}).

Proposition 3.12. For any Reedy fibrant simplicial derived affine X, the morphism

X → diag Spec DD^{∗}O(X^{Δ^{•}})

of simplicial derived affines, coming from the isomorphisms D^{∗}O(X^{Δ^{m}})^{0} = O(X_{m}), induces a weak equivalence on the associated simplicial presheaves on DGAff(R).

Proof. We can rewrite the morphism above as

ho\underrightarrow{lim}_{i∈Δ} X_{i} → holim_{\overrightarrow{i,j∈Δ×Δ}} Spec D^{j}D^{∗}O(X^{Δ^{i}}).

It will therefore suffice to show that for all j, the maps η^{j}: X → Spec D^{j}D^{∗}O(X^{Δ^{•}}) give weak equivalences of the associated simplicial presheaves. For this it is enough to show that each η^{j} is a simplicial deformation retract.

Now, the bisimplicial derived affine j ↦ X^{Δ^{j}} admits a map from the bisimplicial object j ↦ X, which we denote by cX. Forgetting the coface maps ∂^{0} : Δ^{j} → Δ^{j+1} in Δ^{•} gives us an almost cosimplicial simplicial set Δ^{#}, and hence an almost simplicial simplicial derived affine X^{Δ^{#}}. Inclusion of the 0th vertex makes Δ^{0} a deformation retract of Δ^{j}, with obvious contracting homotopy Δ^{1} × Δ^{j} → Δ^{j}. These homotopies combine to give a homotopy Δ^{1} × Δ^{#} → Δ^{#} of almost cosimplicial simplicial sets.

This homotopy makes O(cX_{#}) a cosimplicial deformation retract of O(X^{Δ^{#}}) (as cosimplicial almost cosimplicial rings). Since DD^{∗} descends to a functor on almost cosimplicial algebras, applying DD^{∗} then makes O(cX_{#}) a cosimplicial deformation retract of DD^{∗}O(X^{Δ^{•}})^{#}, so in particular the maps η^{j}: X → Spec D^{j}D^{∗}O(X^{Δ^{•}}) are all simplicial deformation retracts. □

If X is a DG Artin N-hypergroupoid, with associated N-stack 𝔛, this means that we can recover 𝔛 from the cosimplicial bi-CDGA D^{∗}O(X^{Δ^{•}}), just as well as from the cosimplicial CDGA O(X).

Proposition 3.12 has the following immediate consequence:

Corollary 3.13. For any simplicial presheaf F on DGAff(R) and any Reedy fibrant simplicial derived affine X, there is a canonical weak equivalence

ho\underleftarrow{lim}_{j∈Δ} map(Spec DD^{∗}O(X^{Δ^{j}}), F) → map(X, F).

3.2.2. Tangent and cotangent complexes. Now consider a DG Artin N-hypergroupoid X over R. Combining the adjunction D^{∗} ⊣ D with Lemma 3.8 and the universal property of Kähler differentials, observe that there is an isomorphism

N_{c}(Ω_{O(X)/R}^{1} ⊗_{O(X)} DD^{∗}O(X)) ≅ Ω_{D^{∗}O(X)/R}^{1}.

Since O(X) is Reedy cofibrant, Lemma 3.5 ensures that D^{∗}O(X) is cofibrant in the model structure of Lemma 3.4. If we write cone_{h} and cone_{v} for cones in the chain and cochain directions respectively, then for any D^{∗}O(X)-module M, the map cone_{h}cone_{v}(M)_{[i]}^{[m−1]} ↠ cone_{h}(M)_{[i]}^{[m]} is levelwise acyclic, so cofibrancy of D^{∗}O(X) implies lifting of the respective spaces of derivations, or equivalently surjectivity of

Hom_{D^{∗}O(X)}(Ω_{D^{∗}O(X)}^{1}, M)_{i}^{m−1} ↠ Z^{m}Hom_{D^{∗}O(X)}(Ω_{D^{∗}O(X)}^{1}, M)_{i}.

In other words, the columns of Hom_{D^{∗}O(X)}(Ω_{D^{∗}O(X)}^{1}, M) are acyclic.

Because X is an Artin N-hypergroupoid, there are in fact other restrictions on Ω_{D^{∗}O(X)}^{1}. The results below (or an argument adapted from [Pri3, Lemma 2.16]) show that for M ∈ DG^{+}dgMod(D^{∗}O(X)), the rows Hom_{D^{∗}O(X)}(Ω_{D^{∗}O(X)}^{1}, M)^{i} are acyclic for i < −N. Moreover, the argument of [Pri1, Corollary 1.46] shows that when M is concentrated in degree (0, 0), we have H_{j}Hom_{D^{∗}O(X)}(Ω_{D^{∗}O(X)}^{1}, M)^{i} = 0 for i, j ≠ 0, although we will not need this.

Lemma 3.14. If, for A ∈ DG^{+}dgCAlg(R), an A-module M ∈ DG^{+}dgMod(R) admits an A-linear contracting homotopy, then the map

ho\underrightarrow{lim}_{i∈Δ} Spec D^{i}A → ho\underrightarrow{lim}_{i∈Δ} Spec D^{i}(A ⊕ M)

of simplicial presheaves on DG^{+}Aff(R) is a weak equivalence.

Proof. A contracting homotopy is the same as a section of the canonical map cocone_{v}(M) → M of A-modules, where the vertical cocone cocone_{v}(M) is given by M ⊗ cocone_{v}(ℚ) for cocone_{v}(ℚ) = ℚ ⊕ ℚ^{[−1]} with ∂ the identity.

On applying D, we then have a section s of Dcocone_{v}(M) → DM as DA-modules in cosimplicial chain complexes. The Alexander–Whitney cup product gives a canonical morphism Dcocone_{v}(M) → (DM) ⊗ Dcocone_{v}(ℚ). Since ℚ^{Δ^{1}} = ℚ ⊕ Dcocone_{v}(ℚ), our section s then gives a contracting homotopy DM → (DM)^{Δ^{1}}, so evaluation at 0 ∈ Δ^{1} is the zero map, and evaluation at 1 ∈ Δ^{1} the identity.

We therefore have a homotopy Δ^{1} × Spec D(A ⊕ M) → Spec D(A ⊕ M) of simplicial derived affine schemes (and hence of simplicial presheaves) realising Spec DA as a deformation retract of Spec D(A ⊕ M). □

From now on, we will simply write (Spec DA)^{♯} := ho\underrightarrow{lim}_{i∈Δ}(Spec D^{i}A)^{♯} for the étale hypersheafification of the simplicial derived affine Spec DA.

Corollary 3.15. For A ∈ DG^{+}dgCAlg(R), M ∈ DG^{+}dgMod(A) and m > 0, there is a weak equivalence

(Spec D(A ⊕ M^{[−m]}))^{♯} ≃ (Spec D(A ⊕ M) × S^{m})^{♯} ∪_{Spec D(A)^{♯}×S^{m}}^{L} (Spec DA)^{♯}

of étale hypersheaves on DGAff(R), where S^{m} ≃ Δ^{m}/∂Δ^{m} is the m-sphere.

Proof. We have an exact sequence 0 → M[−1] → cocone_{v}(M) → M → 0. Since cocone_{v}(M) has a contracting homotopy, Lemma 3.14 shows that (Spec D(A ⊕ cocone_{v}(M))^{♯} ≃ (Spec DA)^{♯}. Thus (Spec D(A ⊕ M[−1]))^{♯} is the homotopy pushout of the nilpotent maps

(Spec DA)^{♯} ← (Spec D(A ⊕ M))^{♯} → (Spec DA)^{♯},

so is the suspension of (Spec D(A ⊕ M))^{♯} over (Spec DA)^{♯}. Replacing M with M[−j] for 0 ≤ j ≤ m then gives the desired result by induction. □

Definition 3.16. For any bi-CDGA A over R, the module Ω_{A/R}^{1} of Kähler differentials is an A-module in chain cochain complexes, and we define Ω_{A/R}^{p} := Λ_{A}^{p}Ω_{A/R}^{1}, denoting its differentials (inherited from A) by (∂, δ). There is also a de Rham differential d: Ω_{A/R}^{p} →

Ω_{A/R}^{p+1}.

Proposition 3.17. For a DG Artin N-hypergroupoid X over R, and any M ∈ DG^{+}dgMod(D^{∗}O(X)), the cotangent complex 𝕃^{X/R} satisfies

R\underline{Hom}_{O(X)}(𝕃^{X/R}, DM) ≃ Tot^{Π}σ^{≥−N}Hom_{D^{∗}O(X)}(Ω_{D^{∗}O(X)/R}^{1}, M)
≃ ˆ \underline{Hom}_{D^{∗}O(X)}(Ω_{D^{∗}O(X)/R}^{1}, M).

The same is true for any homotopy-Cartesian module M ∈ DGdgMod(D^{∗}O(X)).

Proof. If 𝔛 = X^{♯} is the derived N-stack associated to X, then for f: X → 𝔛, Lemma 3.15 gives an equivalence

R\underline{Hom}_{𝒪_{𝔛}} (Lc^{∗}𝕃^{𝔛^{S^{N}}} , Rf_{∗}DM) ≃ R\underline{Hom}_{O(X)}(Ω_{X/R}^{1}, D(M^{[−N]})),

for the map c: 𝔛 → 𝔛^{S^{N}}. Since 𝔛 is an N-stack, there is a canonical equivalence 𝕃^{𝔛} ≃ 𝕃_{[N]}^{𝔛^{S^{N}}} (cf. [Pri3, Corollary 7.19]), so Lemma 3.9 gives

R\underline{Hom}_{O(X)}(𝕃^{X/R}, DM) ≃ Tot^{Π}σ^{≥−N}Hom_{D^{∗}O(X)}(Ω_{D^{∗}O(X)/R}^{1}, M).

Since X is a fortiori an (N + r)-hypergroupoid for all r ≥ 0, we may replace σ^{≥−N} with σ^{≥−N−r}. Taking the filtered colimit over all r then replaces Tot^{Π}σ^{≥−N}Hom with T̂ot Hom = ˆ \underline{Hom}, yielding the second quasi-isomorphism. Finally, observe that for M homotopy-Cartesian, we must have M^{i} acyclic for i < 0, because A^{i} = 0. Thus we can replace M with σ^{≥0}M, and the results apply.

□

Proposition 3.18. For a DG Artin N-hypergroupoid X over R, any M ∈ DG^{+}dgMod(D^{∗}O(X)), and any r ≥ 0, we have

R\underline{Hom}_{O(X)}((𝕃^{X/R})^{⊗r}, DM) ≃ ˆ \underline{Hom}_{D^{∗}O(X)}((Ω_{D^{∗}O(X)/R}^{1})^{⊗r}, M).

Proof. There is a natural map 𝕃^{X/R} → Ω_{X/R}^{1} coming from Definition 2.15. Corollary 3.9 thus ensures that we have a natural map from right to left. The spectral sequence for the cochain brutal truncation filtration for M allows us to reduce to the case where M = M^{i}[−i] is an O(X_{0})-module, since O(X)^{0} = D^{∗}O(X)^{0}. Writing L for the pullback of 𝕃^{X/R} to X_{0}, and Ω̄ for the pullback of Ω_{D^{∗}O(X)/R}^{1} to O(X_{0}), we therefore wish to show that the map

T̂ot Hom_{O(X_{0})}(Ω̄^{⊗r}, M) → R\underline{Hom}_{O(X_{0})}(L^{⊗r}, M)

is a quasi-isomorphism.

Proposition 3.17 implies that Ω̄ is levelwise quasi-isomorphic to the brutal cotruncation σ^{≤N} Ω̄ := Ω̄/σ^{≥N+1}, so we may replace Ω̄^{⊗r} with a bicomplex in cochain degrees [0, Nr]. The resulting Hom bicomplex is bounded below, so T̂ot is equivalent to Tot^{Π}, and we need only show a quasi-isomorphism

R\underline{Hom}_{O(X_{0})}((Tot σ^{≤N} Ω̄)^{⊗r}, M) ≃ R\underline{Hom}_{O(X_{0})}(L^{⊗r}, M),

which follows because L ≃ Tot σ^{≤N} Ω̄ by Proposition 3.17. (We could also take the limit over all N to give L ≃ Tot^{Π}Ω̄.) □

3.3. Poisson and symplectic structures. To allow for some flexibility in computations and resolutions, we will not just consider bi-CDGAs of the form D^{∗}O(X) for DG Artin N-hypergroupoids X, but will fix A ∈ DG^{+}dgCAlg(R) satisfying:

(1) for any cofibrant replacement Ã → A in the model structure of Lemma 3.4, the morphism Ω_{Ã/R}^{1} → Ω_{A/R}^{1} is a levelwise quasi-isomorphism,

(2) the A^{#}-module (Ω_{A/R}^{1})^{#} in graded chain complexes is cofibrant (i.e. it has the left lifting property with respect to all surjections of A^{#}-modules in graded chain complexes),

(3) there exists N for which the chain complexes (Ω_{A/R}^{1} ⊗_{A} A^{0})^{i} are acyclic for all i > N.

Observe that these conditions are satisfied by D^{∗}O(X), by the results above, but they are also satisfied by more general bi-CDGAs. The first two conditions do not require A to be cofibrant in the model structure of Lemma 3.4; it is enough for A^{#} to be cofibrant as a graded chain CDGA over R, or for A ∈ DG^{+}dg_{+}CAlg(R) with A_{#}^{#} ind-smooth as a bigraded commutative algebra over R_{#}.

3.3.1. Polyvectors. All the definitions and properties of §1 now carry over. In particular:

Definition 3.19. Given a bi-CDGA A over R as above, define the complex of n-shifted polyvector fields on A by

P̂ol(A/R, n) := ∏_{j≥0} ˆ \underline{Hom}_{A}(CoS_{A}^{j}((Ω_{A/R}^{1})_{[−n−1]}), A),

with graded-commutative multiplication following the usual conventions for symmetric powers. This has a filtration by complexes

F^{p}P̂ol(A/R, n) := ∏_{j≥p} ˆ \underline{Hom}_{A}(CoS_{A}^{j}((Ω_{A/R}^{1})_{[−n−1]}), A),

with [F^{i}, F^{j}] ⊂ F^{i+j−1} and F^{i}F^{j} ⊂ F^{i+j}, where the commutative product and Schouten–Nijenhuis bracket are defined as before.

We now define the space P(A/R, n) of Poisson structures and its tangent space T P(A/R, n) by the formulae of Definitions 1.5, 1.8. As in Definition 1.11, there is a canonical tangent vector σ: P(A/R, n) → T P(A/R, n).

As well as composition for internal Hom’s, we have maps Hom(M_{1}, P_{1}) ⊗ Hom(M_{2}, P_{2}) → Hom(M_{1} ⊗ M_{2}, P_{1} ⊗ P_{2}), and hence

Hom_{A}(Ω_{A}^{1}, Hom_{A}(M, A))^{⊗p} → Hom_{A}((Ω_{A}^{1})^{⊗p}, Hom_{A}(M, A)^{⊗p})
→ Hom_{A}((Ω_{A}^{1})^{⊗p}, Hom_{A}((M^{⊗p}, A)).

Substituting M = ⨁_{j=2}^{r−1} CoS_{A}^{j}((Ω_{A/R}^{1})_{[−n−1]}), taking shifts and S_{p}-coinvariants, and

applying T̂ot , we see that an element

π ∈ F^{2}P̂ol(A/R, n)^{n+2}/F^{r}

thus defines a contraction morphism

μ(−, π): Tot^{Π}Ω_{A}^{p} → F^{p}P̂ol(A/R, n)^{n+2}/F^{p(r−1)}

of bigraded vector spaces, noting that Tot^{Π}Ω_{A}^{p} = T̂ot Ω_{A}^{p}. When r = 3 and p = 1, we simply denote this morphism by π_{2}^{♯}: Tot^{Π}Ω_{A}^{1} → ˆ \underline{Hom}_{A}(Ω_{A}^{1}, A)[−n].

Definition 3.20. We say that a Poisson structure π ∈ P(A/R, n)/F^{p} is non-degenerate if Tot^{Π}(Ω_{A/R}^{1} ⊗_{A} A^{0}) is a perfect complex over A^{0} and the map

π_{2}^{♯}: Tot^{Π}(Ω_{A/R}^{1} ⊗_{A} A^{0}) → ˆ \underline{Hom}_{A}(Ω_{A}^{1}, A^{0})[−n]

is a quasi-isomorphism.

Lemma 3.21. If π_{2} ∈ P(A/R, n)/F^{3} is non-degenerate, then the maps

μ(−, π_{2}): Tot^{Π}(Ω_{A}^{p} ⊗_{A} M) → ˆ \underline{Hom}_{A}(CoS_{A}^{p}((Ω_{A/R}^{1})_{[−n−1]}), M)

are quasi-isomorphisms for all M ∈ DG^{+}dgMod(A).

Proof. We may proceed as in Proposition 3.18. By hypothesis, (Ω_{A/R}^{1} ⊗_{A} A^{0})^{i} is acyclic for all i > N, so Hom_{A}(CoS_{A}^{p}((Ω_{A/R}^{1})_{[−n−1]}), A)^{i} is acyclic for all i < −Np, meaning that

we may replace ˆ \underline{Hom} with Tot^{Π}Hom. Since M = \underleftarrow{lim}_{i}(M/σ^{≥i}M), a spectral sequence argument allows us to reduce to the case where M ∈ dgMod(A^{0}).

Me may also replace (Ω_{A/R}^{1}⊗_{A}A^{0}) with its brutal truncation σ^{≤N}(Ω_{A/R}^{1}⊗_{A}A^{0}), so all complexes are bounded in the cochain direction, and then observe that non-degeneracy of π gives us quasi-isomorphisms

(Tot σ^{≤N}(Ω_{A/R}^{1} ⊗_{A} A^{0}))^{⊗_{(A^{0})}p} ⊗_{A^{0}} M → \underline{Hom}_{A^{0}}(σ^{≤N}(Ω_{A/R}^{1} ⊗_{A} A^{0})^{⊗_{(A^{0})}p}, M)[−np];

the required result follows on taking S_{p}-coinvariants. □

3.3.2. The de Rham complex.

Definition 3.22. Define the de Rham complex DR(A/R) to be the product total complex of the bicomplex

Tot^{Π}A ⟶^{d} Tot^{Π}Ω_{A/R}^{1} ⟶^{d} Tot^{Π}Ω_{A/R}^{2} ⟶^{d} …,

so the total differential is d ± δ ± ∂.

We define the Hodge filtration F on DR(A/R) by setting F^{p}DR(A/R) ⊂ DR(A/R) to consist of terms Tot^{Π}Ω_{A/R}^{i} with i ≥ p.

The definitions of shifted symplectic structures from §1.2 now carry over:

Definition 3.23. Define the space PreSp(A/R, n) of n-shifted pre-symplectic structures on A/R by writing PreSp(A/R, n)/F^{i+2} := \underline{MC}(F^{2}DR(A/R)[n + 1]/F^{i+2}), and setting PreSp(A/R, n) := \underleftarrow{lim}_{i} PreSp(A/R, n)/F^{i+2}.

Say that an n-shifted pre-symplectic structure ω is symplectic if Tot^{Π}(Ω_{A/R}^{1} ⊗_{A} A^{0}) is a perfect complex over A^{0} and the map

ω_{2}^{♯}: ˆ \underline{Hom}_{A}(Ω_{A}^{1}, A^{0})[−n] → Tot^{Π}(Ω_{A/R}^{1} ⊗_{A} A^{0})

is a quasi-isomorphism. Let Sp(A/R, n) ⊂ PreSp(A/R, n) consist of the symplectic structures — this is a union of path-components.

Lemma 3.24. For a DG Artin N-hypergroupoid X, the space of closed p-forms of degree n from [PTVV] is given by

A_{R}^{p,cl}(X^{♯}, n) ≃ ho\underleftarrow{lim}_{j∈Δ} \underline{MC}(F^{p}DR(D^{∗}O(X^{Δ^{j}})/R)[n + p − 1]).

Proof. This follows by combining Lemma 3.13 and Proposition 3.18. □

3.3.3. Compatible pairs. The proof of Lemma 1.18 adapts to give maps

(pr_{2} + μϵ): PreSp(A/R, n)/F^{p} × P(A/R, n)/F^{p} → T P(A/R, n)/F^{p}

over P(A/R, n)/F^{p} for all p, compatible with each other.

We may now define the space Comp(A/R, n) of compatible pairs as in Definition 1.24, with the results of §1 all adapting to show that the maps Sp(A/R, n) ← Comp(A/R, n)^{nondeg} → P(A/R, n)^{nondeg} are weak equivalences.

3.4. Diagrams and functoriality. We now extend the constructions of §2.1 to bi-CDGAs.

3.4.1. Definitions. Given a small category I, an I-diagram A of bi-CDGAs over R, and A-modules M, N in I-diagrams of chain cochain complexes, we can define the cochain complex ˆ \underline{Hom}_{A}(M, N) to be the equaliser of the obvious diagram

∏_{i∈I} ˆ \underline{Hom}_{A(i)}(M(i), N(i)) ⟹ ∏_{f: i→j in I} ˆ \underline{Hom}_{A(i)}(M(i), f_{∗}N(j)).

All the constructions of §3.3 then adapt immediately; in particular, we can define

P̂ol(A/R, n) := ∏_{j≥0} ˆ \underline{Hom}_{A}(CoS_{A}^{j}((Ω_{A/R}^{1})_{[−n−1]}), A),

leading to a space P(A/R, n) of Poisson structures.

We can now adapt the formulae of §2.1 to this setting, defining pre-symplectic structures by

PreSp(A/R, n) := PreSp(A(0)/R, n) = \underleftarrow{lim}_{i∈[m]} PreSp(A(i)/R, n)

for any [m]-diagram A of bi-CDGAs over R, and setting Comp(A/R, n) be the homotopy vanishing locus of the obvious maps

(μ − σ): PreSp(A/R, −1) × P(A/R, n) → T P(A/R, n).

over P(A/R, n).

The obstruction functors and their towers from §1.4 also adapt immediately, giving the obvious analogues of the obstruction spaces defined in terms of

ˆ\underline{Hom}_{A}(CoS_{A}^{p}(Ω_{A/R}^{1}[n + 1]), A), Tot^{Π}Ω_{A(0)}^{p}.

3.4.2. Functors and descent. For [m]-diagrams in DG^{+}dgCAlg(R), we will consider the injective model structure, so an [m]-diagram A is cofibrant if each A(i) is cofibrant for the model structure of Lemma 3.4, and is fibrant if the maps A(i) → A(i + 1) are all surjective.

Lemma 3.25. If D = (A → B) is a fibrant cofibrant [1]-diagram in DG^{+}dgCAlg(R) which is formally étale in the sense that the map

{Tot σ^{≤q}(Ω_{A}^{1} ⊗_{A} B^{0})}_{q} → {Tot σ^{≤q}(Ω_{B}^{1} ⊗_{B} B^{0})}_{q}

is a pro-quasi-isomorphism, then the map

ˆ\underline{Hom}_{D}(CoS_{D}^{k}Ω_{D}^{1}, D) → ˆ \underline{Hom}_{A}(CoS_{A}^{k}Ω_{A}^{1}, A),

is a quasi-isomorphism for all k.

Proof. This follows by reasoning as in the proof of Proposition 3.18. □

Definition 3.26. Write DG^{+}dgCAlg(R)_{c,↠} ⊂ DG^{+}dgCAlg(R) for the subcategory with all cofibrant bi-CDGAs (in the model structure of Lemma 3.4) over R as objects, and only surjections as morphisms.

For the notion of formally étale from Lemma 3.25, we may extend the conditions of Properties 2.5 to constructions on DG^{+}dgCAlg(R)_{c,↠}, with quasi-isomorphisms taken levelwise. The constructions P(−, n), Comp(−, n), PreSp(−, n), and their associated filtered and graded functors, all satisfy these properties. The first two properties follow from the right lifting property for fibrations (in the injective model structure on diagrams), and the third from Lemma 3.25.

Thus the simplicial classes ∐_{A∈B_{m}DG^{+}dgCAlg(R)_{c,↠}} F (A) fit together to give a complete Segal space ∫ F over the nerve BDG^{+}dgCAlg(R)_{c,↠}.

Definition 2.6 then adapts to give us an ∞-category L ∫ F, and Definition 2.7 adapts to give an ∞-functor

RF: LDG^{+}dgCAlg(R)^{ét} → LsSet

with (RF)(A) ≃ F (A) for all cofibrant bi-CDGAs A over R, where DG^{+}dgCAlg(R)^{ét} ⊂ DG^{+}dgCAlg(R) is the subcategory of homotopy formally étale morphisms.

An immediate consequence of §3.3.3 is that the canonical maps

RComp(−/R, n)^{nondeg} → RSp(−/R, n)
RComp(−/R, n)^{nondeg} → RP(−/R, n)^{nondeg}

are weak equivalences of ∞-functors on LDG^{+}dgCAlg(R)^{ét}.

Lemma 3.13 and Proposition 3.18 ensure that if a morphism X → Y of DG Artin Nhypergroupoids becomes an equivalence on hypersheafifying, then D^{∗}O(Y) → D^{∗}O(X) is formally étale in the sense of Lemma 3.25. In particular this means that the maps ∂^{i}: D^{∗}O(X^{Δ^{j}}) → D^{∗}O(X^{Δ^{j+1}}) and σ^{i}: D^{∗}O(X^{Δ^{j+1}}) → D^{∗}O(X^{Δ^{j}}) are formally étale. Thus D^{∗}O(X^{Δ^{•}}) can be thought of as a DM hypergroupoid in bi-CDGAs, and we may make the following definition:

Definition 3.27. Given a DG Artin N-hypergroupoid X over R and any of the functors F above, write

F (X) := ho\underleftarrow{lim}_{j∈Δ} RF (D^{∗}O(X^{Δ^{j}})).

Example 3.28. If R is a field k, and Y/k is a smooth affine scheme equipped with an action of an affine group scheme G, then the nerve X := B[Y/G] is a hypergroupoid resolving the stack [Y/G]. Since B[Y/G]_{i} = Y × G^{i}, the simplicial scheme X^{Δ^{j}} is given by B[Y × G^{j}/G^{j+1}], with action

(y, h_{1}, …, h_{n})(g_{0}, …, g_{n}) = (yg_{0}, g_{0}^{−1}h_{1}g_{1}, g_{1}^{−1}h_{2}g_{2}, …, g_{n−1}^{−1}h_{n}g_{n}).

Taking Reedy fibrant replacement of X gives us a DG Artin hypergroupoid X′ and a levelwise quasi-isomorphism X → X′. Since X is levelwise smooth, the map D^{∗}O((X′)^{Δ^{•}}) → D^{∗}O(X^{Δ^{•}}) is a levelwise quasi-isomorphism of cosimplicial bi-CDGAs. Thus Poisson structures on the stack [Y/G]^{♯} can be defined in terms of polyvectors on the cosimplicial CDGA D^{∗}O(X^{Δ^{•}}).

The completion O(X̂) of O(X) over O(X_{0}) is a cosimplicial ring with O(X̂)^{i} = O(Y)⟦(𝔤^{∨})^{⨁ i}⟧, for 𝔤 the Lie algebra of 𝔤. Thus the CDGA D^{∗}O(X) has underlying graded ring O(Y)[𝔤[1]^{∨}], with differential given by combining the bar construction for 𝔤 with the action on Y.

The same calculation for [Y × G^{j}/G^{j+1}] gives that D^{∗}O(X^{Δ^{j}}) has underlying graded ring O(Y ×G^{j})[𝔤^{⊕(j+1)}[1]^{∨}]. Thus D^{∗}O(X^{Δ}) has underlying cosimplicial graded algebra

O(B[Y/G])[E𝔤[1]^{∨}],

where the simplicial vector space E𝔤 = B[𝔤/𝔤] is the 0-coskeleton of 𝔤.

In particular, the CDGA D^{∗}O((BG)^{Δ}) is given by O(BG)[E𝔤[1]^{∨}] with differential twisted by the bar construction. It is important to note that although the bar construction of a simple Lie algebra is quasi-isomorphic to k as a cochain complex, it is not levelwise quasi-isomorphic as a cochain chain complex, because 𝔤 is not acyclic as a DGLA.

Proposition 3.29. If Y → X is a trivial DG Artin hypergroupoid, then the morphism

F (X) → F (Y)

is an equivalence for any of the constructions F = P, Comp, PreSp.

Proof. The proof of Proposition 2.17 adapts, replacing Propositions 1.32 and 1.35 with Corollary 3.13 and Proposition 3.18. □

Thus the following is well-defined:

Definition 3.30. Given a strongly quasi-compact DG Artin N-stack 𝔛, define the spaces P(𝔛/R, n), Comp(𝔛/R, n), Sp(𝔛, n) to be the spaces P(X/R, n), Comp(X/R, n), Sp(X, n) for any DG Artin N-hypergroupoid X with X^{♯} ≃ 𝔛.

Combined with the results of above, an immediate consequence of §3.3.3 is:

Theorem 3.31. For any strongly quasi-compact DG Artin N-stack 𝔛 over R, there are natural weak equivalences

Sp(𝔛/R, n) ← Comp(𝔛/R, n)^{nondeg} → P(𝔛/R, n)^{nondeg}.

For an alternative proof of this result, see [CPT^{+}, Theorem 3.2.4].

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