Linear Batalin-Vilkovisky quantization as a functor of $\infty$-categories

We study linear Batalin-Vilkovisky (BV) quantization, which is a derived and shifted version of the Weyl quantization of symplectic vector spaces. Using a variety of homotopical machinery, we implement this construction as a symmetric monoidal functor of ∞-categories. We also show that this construction has a number of pleasant properties: It has a natural extension to derived algebraic geometry, it can be fed into the higher Morita category of En-algebras to produce a “higher BV quantization” functor, and when restricted to formal moduli problems, it behaves like a determinant. Along the way we also use our machinery to give an algebraic construction of En-enveloping algebras for shifted Lie algebras.


Introduction
A well-known adage in mathematical physics is that "quantization is not a functor," but with suitable restrictions, there are situations where quantization is functorial. Our goal here is to articulate the simplest piece of the Batalin-Vilkovisky formalism as a functor, using modern machinery of higher categories and derived geometry.
The most fundamental case of quantization assigns to the vector space R 2n the Weyl algebra, which is the associative algebra on 2n generators p 1 , . . . , p n , q 1 , . . . , q n with relations [p i , p j ] = 0 = [q i , q j ] and [p i , q j ] = δ ij . Here R 2n should be thought of as the cotangent bundle of R n , equipped with its standard symplectic structure, which is the arena for classical mechanics on R n . This assignment can formulated as a functor, known as Weyl quantization, from symplectic vector spaces (or more generally, vector spaces with a skew-symmetric pairing) to assocative algebras. For us this is the model case of functorial quantization. This construction naturally breaks up into three steps: (1) To a vector space V with skew-symmetric pairing ω we associate its Heisenberg Lie algebra Heis(V, ω), which is the direct sum V ⊕ R equipped with the Lie bracket where [x, y] = ω(x, y) for x, y in V , and all other brackets are zero.
On the other hand, if we set = 0 we get Sym(V ), equipped with the Poisson bracket {x, y} = lim →0 [x, y]/ = ω(x, y), for x, y ∈ V , which is an algebraic version of the Poisson algebra of classical observables. The universal enveloping algebra U Heis(V, ω) can thus be viewed as a deformation quantization of the Poisson algebra Sym(V ). This procedure is at the core of all approaches to "free theories," and hence the base case for the more challenging and more interesting interacting theories.
Our main object of study in this paper is a derived and shifted version of this constructionderived in the sense that we replace vector spaces by cochain complexes (or more generally modules over a commutative differential graded algebra) and shifted in the sense that we consider skewsymmetric pairings of degree 1. We will construct a functorial quantization of these objects to E 0 -algebras (which are just pointed cochain complexes), using shifted versions of the Heisenberg Lie algebra construction and of the universal enveloping algebra. (In general, we expect that there is a functorial quantization of cochain complexes with (1 − n)-shifted skew-symmetric pairings to E n -algebras, and we discuss below how we believe this arises naturally from the case n = 0 that we consider.) Our construction produces the simplest possible examples of Batalin-Vilkovisky quantization. This homological approach to quantization of field theories was introduced by Batalin and Vilkovisky [BV81,BV83,BV85] as a generalization of the BRST formalism, in an effort to deal with complicated field theories such as supergravity. Their formalism for field theory, both classical and quantum, has broad application and conceptual depth. (For recent work on these issues, see [Cat06,CMR15,CMR12,Cos11,CG16].) For brevity, we will talk about the BV formalism and BV quantization.
(We should point out that we mean here the Lagrangian formulation, whose quantum aspect is focused on a homological approach to the path integral.) In this introduction, we begin by describing our main results in §1.1. Next we describe some consequences in the setting of derived geometry in §1.2 and then discuss an extension to "higher BV quantization" in §1.3, where we also sketch how we expect our results to relate to the simplest examples of AKSZ theories.
Afterwards, in §1.4, we discuss BV quantization from the perspective of physics -notably, how it is a homological version of integration -and explain how the standard approach relates to our work here. (A reader coming from a field theory setting might prefer to read that discussion before the preceding sections; on the other hand, readers without such background should feel free to skip it, as nothing in the rest of the paper depends on it.) 1.1. Our Main Results. As in the case of Weyl quantization, our construction naturally breaks up into three steps: first we apply an analogue of the Heisenberg Lie algebra construction, then an enveloping algebra functor, and finally we "set = 1." These constructions are certainly well-known among those who work with BV quantization, although rarely articulated in this way, and they can be found, in a slightly different form, in [BD04] and later in [CG16].
In the first step we start with a cochain complex V over the base field k, equipped with a 1-shifted antisymmetric pairing ω : Λ 2 V → k[1] -we call such objects 1-shifted quadratic modules. We then define a 1-shifted Heisenberg Lie algebra Heis 1 (V, ω) by equipping the cochain complex V ⊕ kc, where c is an added central element, with the bracket [v, w] = ω(v, w)c for v, w ∈ V . Unfortunately, this simple construction is not homotopically meaningful, which requires us to do a bit of work. In §3, we show: In the second step we apply an enveloping algebra functor. However, this no longer produces an associative algebra, but rather a BD-algebra in the following sense: BD-algebras can be encoded as algebras for an operad, and since we are working over a base field of characteristic zero, there are well-behaved model categories of such operad algebras. Using this machinery we explicitly describe the BD-enveloping algebra of a 1-shifted Lie algebra and show that it gives a symmetric monoidal functor of ∞-categories from Lie 1 (A) to the ∞-category Alg BD (A[ ]) of BD-algebras in differential graded modules over A[ ]. The second claim is not entirely obvious, since the model categories in question are not compatible with the tensor products in the usual sense.
In the Weyl quantization story, we produced an associative algebra with a parameter . Setting = 0 this algebra reduced to a Poisson algebra, and setting = 1 it was the Weyl algebra. Hence the parameter explicitly describes a deformation quantization. Interpreting the symplectic vector space as the phase space of a classical system, the Weyl quantization procedure gave us both the classical observables -the Poisson algebra of functions on the phase space, when = 0 -and the quantum observables, when = 1.
In the BV formalism, the classical observables form a 1-shifted Poisson algebra, and the quantum observables are just a pointed cochain complex. Starting with a BD-algebra M over A, we can recover both of these structures by taking to be 0 or 1. If we set to 0, i.e. we pass to the quotient M/( ), then we obtain a shifted Poisson algebra structure, which we interpret as the dequantization of the BD-algebra M . If we set to 1, i.e. we pass to the quotient M/( − 1), then the differential is not a derivation and so up to quasi-isomorphism, the only remaining algebraic structure is the unit. That is, the reduction M/( − 1) is essentially just a pointed A-module. More precisely, a pointed A-module is the same thing as an algebra in A-modules for an operad E 0 , and the structure we have on M/( −1) is encoded by an operad E 0 ; these operads are weakly equivalent, and so they encode the same kind of information.
The abstract problem of BV quantization is: given a shifted Poisson algebra R, produce a BDalgebra R whose dequantization is quasi-isomorphic to R. Composing our shifted Heisenberg functor with the BD-enveloping algebra, we thus get a functorial quantization procedure that abstracts and encodes the usual approach to BV quantization for linear systems. The shifted Poisson algebra obtained from this quantization by setting = 0 can be identified with the enveloping shifted Poisson algebra of the shifted Lie algebra we started with. Moreover, the E 0 -algebra we get from taking = 1 is also an enveloping algebra. More formally, we can sum up our work on operads and enveloping algebras in §2 as: that is natural in the commutative differential graded k-algebra A.
Although not strictly needed for our main results, we take the time to prove some further interesting results concerning the symmetric monoidal enveloping functor U E0 : Lie 1 (A) → Alg E0 (A). Firstly, in §2.5 we show that (at the model category level) it can be identified with (a shifted version of) the Chevalley-Eilenberg chains, which describe Lie algebra homology. Secondly, in §2.7 we use it to construct an enveloping E n -algebra functor Lie 1−n (A) → Alg En (A), by a different approach than those of Fresse [Fre14] and Knudsen [Knu16]. Let us briefly sketch the idea: By using ∞-operads, we get from U E0 a functor between ∞-categories of E n -algebras Alg En (Lie 1 (A)) → Alg En (Alg E0 (A)) ≃ Alg En (A) (this approach does not work on the model category level); using the bar/cobar adjunction, we then show that the ∞-category Alg En (Lie 1 (A)) is equivalent to Lie 1−n (A), by an argument we learned from Nick Rozenblyum.
1.2. Extension to Derived Algebraic Geometry. In §4.1 we show that our functors all have natural extensions to derived stacks: for instance, given a quasi-coherent sheaf of 1-shifted quadratic modules on a derived stack, there is a functorial way to quantize it to a quasi-coherent sheaf of E 0algebras. This result is essentially a formal consequence of the naturality of our constructions in the ∞-categorical setting, together with Lurie's descent theorem for ∞-categories of modules.
This result begs the question of what such a quantization means in natural geometric examples, which we hope to explore in future work. For example, in light of [CPT + 15,Hen13,GR16], for a well-behaved 0-shifted symplectic stack X, its relative tangent complex T X/X dR can be input to our construction. What does this quantization mean?
In § §4.2-4.4 we will focus on cotangent quantization, which sends a graded vector bundle V (i.e. a finite direct sum of shifts of vector bundles) to the quantization of V ⊕ V ∨ [1]. This case has striking behaviour: the cotangent quantization of a graded vector bundle is a line bundle (up to a shift). In other words, this functor factors through Pic, the stack of invertible sheaves, and hence it behaves likes a determinant functor. (It does not possess all the properties of the determinant, however.) This feature demonstrates a sense in which BV quantization is a kind of homological encoding of the path integral, since the determinant line of a vector space is the natural home for translation-invariant volume forms on the vector space.
Remark 1.2.1. From the viewpoint of physics, more specifically the divergence complex perspective that we discuss in Section 1.4 below, this behavior is not too surprising. Indeed, the standard toy example of BV quantization produces a cochain complex that is isomorphic to the polynomial de Rham complex on a vector space V , shifted down by the dimension of V . Poincaré's lemma then tells us that we get a one-dimensional vector space in degree − dim(V ). We leverage this example as far as it can easily go.
We expect this result to be true in somewhat greater generality, likely for 1-shifted symplectic vector bundles, which are quadratic modules whose pairing is non-degenerate and whose underlying module is a sum of shifts of vector bundles. For more general quasicoherent sheaves, however, the quantization is not invertible. On the other hand, we show it is constructibly invertible in a certain sense. Interpreting the meaning of this behavior is an intriguing question. We expect that it is closely related to recent work [BF09, BBBBJ15, Pri15] on vanishing cycles on stacks. (In a sense, the BV formalism is an obfuscated version of the twisted de Rham complex, as explained in Section 1.4, and hence closely related to vanishing cycles.) 1.3. Higher BV Quantization and AKSZ Theories. As we discussed above, we construct, for any derived stack X, a sequence of symmetric monoidal functors Using ∞-operads, this sequence immediately induces functors between ∞-categories of E n -algebras: We mentioned above that E n -algebras in Lie 1 (X) are equivalent to (1 − n)-shifted Lie algebras, and heuristically it looks like there is an analogous description of E n -algebras in Quad 1 (X). More precisely, we expect the following: Conjecture 1.3.1. There is a natural equivalence Alg En (Quad 1 (X)) ≃ Quad 1−n (X). The induced functor Quad 1−n (X) ≃ Alg En (Quad 1 (X)) → Alg En (Lie 1 (X)) ≃ Lie 1−n (X) is a (1 − n)-shifted version of the Heisenberg Lie algebra, so the composite functor Quad 1−n (X) → Alg En (X) is the E n -enveloping algebra of the shifted Heisenberg Lie algebra.
In fact, this construction has a further interesting extension. Recall that given a nice symmetric monoidal ∞-category C, the higher Morita category of E n -algebras Alg n (C), as constructed in [Hau14], is a symmetric monoidal (∞, n)-category with • objects E n -algebras in C, • 1-morphisms E n−1 -algebras in bimodules in C, • 2-morphisms E n−2 -algebra in bimodules in bimodules in C, • . . .
• n-morphisms bimodules in . . . in bimodules in C. In §4.5 we show that the ∞-categories we work with satisfy the requirements for these higher Morita categories to exist, and our functors give symmetric monoidal functors between them. In particular, we get higher BV quantization functors Alg n (Quad 1 (X)) → Alg n (Alg E0 (QCoh(X))).
We expect that the i-morphisms in Alg n (Quad 1 (X)) have an interesting interpretation, in the same way as we conjectured above for the objects. Specifically, for i = 1 they should be a cospan analogue of the linear version of the Poisson morphisms studied by [CPT + 15,Saf15], and for i > 1 we should have iterated versions of this notion. We have learned from Nick Rozenblyum that such results should be provable using the same techniques as in his unpublished proof that E n -algebras in P k -algebras are P k+n -algebras.
The (∞, n)-categories Lag n−1,lin (∞,n) (A), or rather the more general version Lag k (∞,n) whose objects are k-shifted symplectic derived Artin stacks, will be constructed in forthcoming work of the second author together with Damien Calaque and Claudia Scheimbauer. Moreover, it will be shown there that the AKSZ construction, implemented in this algebro-geometric setting in [PTVV13], gives for every k-symplectic derived Artin stack X an extended oriented topological quantum field theory (TQFT) (where the dimension n is arbitrary). Combining this with our hypothetical dualizing functor, we would have for every (n − 1)-shifted symplectic cochain complex X a chain of symmetric monoidal functors of (∞, n)-categories Bord or (∞,n) → Lag n−1,lin (∞,n) (k) → Alg n (Quad 1 (k)) → Alg n (Alg E 0 (k)). The resulting TQFT can be interpreted as a quantization of the AKSZ field theory with target X.
1.4. Linear BV Quantization and Integration. Let's turn here to a more traditional presentation of linear BV quantization and interpret it as a hidden version of the de Rham complex providing the connection to homological perspectives on integration. (For further discussion of BV quantization, we recommend [Fio03,CMR16,Wit90,Cos11,CG16].) Let V be a finite-dimensional vector space over, say, the real numbers. A standard way to encode the algebraic relations among integrals (more accurately, integrands) is via the de Rham complex. In particular, if K ⊂ V is a smooth compact region -like a closed ball of codimension 0 -then two top forms α and α ′ = α + dβ satisfy Hence top forms modulo exact terms encodes integrals modulo boundary integrals. We want to see what this perspective tells us about the integrals that represent toy models of the path integral.
The model case for physics is to fix a quadratic form Q on V with a global minimum at the origin and consider the integrand e −Q(x) d n x. Up to scale, this provides a probability measure on V whose average is the origin and which extends around it as a "Bell curve". (On R, this might be Q(x) = x 2 , which gives the Gaussian measure as the integrand.) It is a toy model of a free theory in physics, with Q the action functional. (As Q is quadratic, its equations of motion are linear and so "free.") It is reasonable, in order to explore this measure, to focus on its moments, i.e. to understand the integrals with p ∈ Sym(V * ) a polynomial. Indeed, the perturbative machinery of Feynman diagrams can be understood as formally extending these computations to formal power series. Our BV approach to this free case then similarly extends and provides another perspective on the origin of Feynman diagrams as "homotopy transfer". (See [GJF12,Gwi12] for more.) In other words, we want to understand the expected value map Note that these integrands decay very fast at infinity and hence are integrable.
Observe that this map factors as a composition The kernel of E can be identified with those integrands p(x)e −Q(x) d n x that are exact, i.e. in the image of the de Rham differential d. In fact, the kernel of E is the image of the "divergence against the volume form µ = e −Q(x) d n x" operator div µ : sending a vector field X with polynomial coefficients to the Lie derivative of µ along X. More generally we can use the volume form µ = e −Q(x) d n x to produce an injection from the polyvector fields on V with polynomial coefficients into de Rham forms. The de Rham differential preserves the image (just note that the derivative of e −Q(x) is a polynomial times itself) and hence pulls back to a "divergence operator" div µ on PV * poly (V ). By construction, this divergence complex Div = (PV * poly (V ), div µ ) encodes the moments of the measure µ in the map . (In this situation, the rest of the cohomology vanishes, by a Poincaré lemma argument.) Hence it captures the information we most want from the measure. But this construction has several features that make it possible to generalize this approach to infinite-dimensional vector spaces and manifolds (i.e. to actual field theories) and also to derived settings, where the usual approaches to integration do not always work. Two aspects are: • It replaces the measure by the divergence operator, and so one can try to axiomatize the properties of divergence complexes and then search for new examples. In particular, it replaces questions about integration by examining relations between integrands. • It focuses on functions and their expected values -i.e. integration against a fixed volume form -rather than a general theory of integration. Thus, by contrast to the de Rham complex, it makes sense in infinite dimensions, whereas top forms make no sense there. Here we will focus on the first aspect, using an operad introduced by Beilinson and Drinfeld [BD04] for the axiomatization, and introduce a wealth of examples from higher algebra and derived geometry. The second aspect is pursued wherever BV quantization is used in field theory, such as [CMR16,Cos11,CG16]. Of course, this BV approach to the path integral does not resolve all challenges! Viewing integration this way loses some of the advantages of other perspectives and introduces new puzzles and challenges.
Let us now rapidly sketch the algebraic features of the divergence complex that we will focus on. First, polyvector fields have a natural graded-commutative product by wedging (in parallel with de Rham forms, but not preserved by the map ι µ !). Second, polyvector fields have a shifted Poisson bracket, known as the Schouten bracket, which is defined by extending the natural action of vector fields on functions and vector fields. Explicitly, we define for f a function and X, Y vector fields. (In general, there are signs to keep track of, due to the Koszul sign rule, but we will not focus on that in this introduction.) Finally, the divergence operator is a derivation with respect to the bracket (i.e. with respect to the shifted Lie algebra structure) but it is not a derivation with respect to the commutative product. Instead, it satisfies the relation div(αβ) = div(α)β + (−1) α α div(β) + {α, β} for any α, β polyvector fields. (These features do not depend on the coefficients being polynomial and hold for holomorphic or smooth coefficients too.) This relation says that the bracket encodes the failure of div to be a derivation. It should be seen as analogous to deformation quantization, where the failure to be commutative is encoded in the commutator bracket and, to first order in , this failure is the Poisson bracket. This perspective leads directly to the definition of a Beilinson-Drinfeld algebra (see Definition 1.1.2).
In our model case, the shifted Poisson algebra is (Note that the zeroth cohomology is precisely functions on the critical set of Q, which fits nicely with the fact that observables in a classical theory should be functions on the critical points of the action.) The BV quantization is where △ = div Leb is divergence against the Lebesgue measure d n x. (In formulas, one usually sees where the x i are a basis for V * and the ξ i are the dual basis for V [1].) In this example, we explicitly see that the deformation of the differential amounts to taking into account the relations among integrands. There is one final thing to note about this model example, which makes manifest the analogy with Weyl quantization. Observe that the shifted Poisson bracket {−, −} is linear in nature. If we fix a basis {x i } for V * and a dual basis {ξ i } for V , then with dim(V ) = n and the x i 's in degree zero and the ξ i 's in degree one. The bracket is which looks just like a shifted version of the Poisson bracket on the symplectic vector space T * R n . Indeed, we can view this shifted bracket as arising from a shifted skew-symmetric pairing which is simply the restriction of {−, −} to the linear space generating the graded-symmetric algebra of polyvector fields. There is then a shifted Lie algebra g given by centrally extending the abelian Lie algebra V * ⊕ V [1] by Rc, i.e.
Thus g is clearly a kind of shifted Heisenberg Lie algebra. To obtain the BV quantization, we do not take the universal enveloping algebra, which would produce an associative algebra, but instead take the enveloping BD algebra U BD (g). (We construct this enveloping algebra functor in the text.) The quotient U BD (g)/(c = ) recovers the standard BV quantization on the nose.
1.5. Notations and Conventions. Throughout this paper, we work in the setting of cochain complexes over a field k of characteristic zero. In other words, everything is differential graded, aside from the occasional motivational remark. Hence, when we speak about an algebra, we always mean an algebra object in some category (or higher category) with a forgetful functor to cochain complexes. After the introduction, we will simply speak about commutative or Lie algebras and not differential graded commutative algebras or differential graded Lie algebras. Notationally, A typically denotes a commutative algebra in cochain complexes over the field k (i.e. a cdga), and g typically denotes a Lie algebra in cochain complexes over k (i.e. a dgla). A module over an algebra always means a module object and we will not use the term differential graded module. Thus we write A-module rather that differential graded A-module and so on.
To construct our ∞-categories and functors we will also need to work with both model categories and simplicial categories. To distinguish the three kinds of mapping objects that arise we adopt the convention that for an ordinary category C we write Hom C (x, y) for the set of maps between objects x and y, for a simplicial category C we write Hom C (x, y) for the simplicial set of maps, and for an ∞-category C we write Map C (x, y) for the space of maps.
In many cases, we will have to work with a model category, a simplicial category and an ∞category that encode the same homotopy theory, and we use a typographical convention to distinguish these. For instance, there is a category Mod(A) of A-modules in Mod(k), the category of cochain complexes over k. There is also a simplicial category Mod(A) of (cofibrant) A-modules, and there is an ∞-category There are two exceptions to the convention we just described. When O is the commutative operad Comm, we use the abbreviated notations Comm(A), Comm(A), and Comm(A), and when O is the Lie operad Lie, or more generally the n-shifted Lie operad Lie n , we use Lie n (A), Lie n (A), and Lie n (A).
1.6. Acknowledgments. OG thanks Kevin Costello for teaching him about the BV formalism and pointing out that it behaves like a determinant, an idea he pursued in his thesis and that prompted this collaboration. He also thanks Nick Rozenblyum and Toly Preygel for many helpful conversations around quantization and higher categories.
RH thanks Irakli Patchkoria for help with model-categorical technicalities and Dieter Degrijse for some basic homological algebra.
We thank Theo Johnson-Freyd, David Li-Bland, and Claudia Scheimbauer for stimulating conversations around these topics, particularly the possibility of higher Weyl quantization.

Operads and Enveloping Algebras
Our goal in this section is to introduce the operads that play a central role in BV quantization and to construct a collection of functors between their ∞-categories of algebras. To do so, we first explain what we mean by the ∞-category of algebras over a k-linear operad O, as there is not yet available a theory of enriched ∞-operads. Thus, the beginning of this section is devoted to higher-categorical machinery: we draw together results from the literature in order to With these tools available, we turn to our problem of interest.
The main result of this section can then be summarized in the following commuting diagram of symmetric monoidal functors: forget forget which says in essence that (1) every shifted Lie algebra g in Mod(A) generates a shifted Poisson algebra U P0 (g) = Sym(g) that admits a natural BV quantization by its BD-enveloping algebra U BD (g[ ]), and (2) when is specialized to 1, this quantization reduces to C L (g) = C Lie (g[−1]) (i.e. the derived coinvariants, or Chevalley-Eilenberg chains, of the unshifted Lie algebra) . These relationships certainly seem to be folklore among the community who work with the BV formalism, but we need the result in this higher-categorical setting and so provide proofs. (See, for instance, [BD04,BV14,BL13].) We will begin by proving everything in the setting of model categories and then apply our machinery to obtain the desired statements for ∞-categories.
2.1. Model Categories of Modules and Operad Algebras. Let k be a field of characteristic 0. We write Mod(k) for the category of (unbounded) cochain complexes of k-modules, equipped with the standard projective model structure: If A is a commutative algebra over k, i.e. a commutative algebra object in Mod(k), then we can lift this model structure to the category Mod(A) of A-modules in Mod(k): Proposition 2.1.2 (Hinich, Schwede-Shipley). Let A be a commutative algebra over k. Then the category Mod(A) has a left proper combinatorial model structure where the weak equivalences and fibrations are the maps whose underlying maps of cochain complexes are weak equivalences and fibrations in Mod(k). If the underlying cochain complex of A is cofibrant, then the forgetful functor also preserves cofibrations. Moreover, this is a symmetric monoidal model category with respect to ⊗ A .
Proof. This is [Hin97,§3]  This fact is standard; we include a short proof for completeness.
Proof. Mod(A) is a cofibrantly generated model category, with the set I of generating cofibrations being S n is the cochain complex with k in degree −n and 0 elsewhere, and D n+1 k is that with k in degrees −n and −n − 1, with differential id k , and 0 elsewhere (cf. [BMR14, Theorem 3.3]). It follows that the cofibrant A-modules are the objects that are retracts of I-cell complexes, where the latter are the objects X that can be written as colimits of a sequence of maps 0 = F 0 → F 1 → F 2 → · · · , with each F n−1 → F n obtained as a pushout where T n is a set. Note in particular that each filtration quotient F n /F n−1 is of the form i∈Tn S n+1 A , i.e. it is a sum of copies of shifts of A.
Now suppose X is a cofibrant A-module, and f : M → M ′ is a quasi-isomorphism. We wish to prove that X ⊗ f is a quasi-isomorphism. Since quasi-isomorphisms are closed under retracts, it suffices to prove this under the assumption that X is an I-cell complex; we therefore fix a filtration F n of X as above. We claim that the induced maps F n ⊗ A M → F n+1 ⊗ A M are injective, so that we get a filtration of X ⊗ A M . Assuming this, we have short exact sequences of cochain complexes over k, , and using the associated long exact sequence, we see by induction that To prove injectivity for F n ⊗ A M → F n+1 ⊗ A M , observe that we can prove this on the level of underlying graded k-modules. The freeness of F n+1 /F n implies that we can choose a splitting of For later use, we note a useful consequence of this: Lemma 2.1.5. Suppose A is a commutative algebra over k. Then the nth symmetric power functor Sym n A : Mod(A) → Mod(A) preserves quasi-isomorphisms between cofibrant A-modules. Proof. The functor Sym n A is defined by the tensor product A ⊗ A[Σn] (-) ⊗An where A has the trivial Σ n -action and (-) ⊗An has the obvious action by permuting the factors. Since k is a field of characteristic zero, every module over k[Σ n ] is projective. In particular, k is a projective k[Σ n ]-module, and hence it is cofibrant in Mod(k[Σ n ]). Since A[Σ n ] ⊗ k[Σn] -is a left Quillen functor, this implies that A is cofibrant in Mod(A[Σ n ]). It therefore follows from Lemma 2.1.4 that the functor A ⊗ A[Σn] (-) preserves quasi-isomorphisms. We are left with showing that if M → N is a quasi-isomorphism of cofibrant A-modules, then M ⊗An → N ⊗An is a quasi-isomorphism, which follows from -⊗ A -being a left Quillen bifunctor.
It will also be useful to know that in the case of a field we can relax the assumption that M is cofibrant: Lemma 2.1.6. For every X ∈ Mod(k), the functor X ⊗ -preserves quasi-isomorphisms.
Proof. By [Hov99, Lemma 2.3.6], any bounded-above cochain complex of k-modules is cofibrant, so the result holds in this case by Lemma 2.1.4. But any cochain complex X is a filtered colimit of bounded-above cochain complexes. Since the tensor product commutes with colimits in each variable and quasi-isomorphisms are closed under filtered colimits, we obtain the result.
Proposition 2.1.7. Any map of commutative algebras φ : A → B induces a Quillen adjunction If φ is a quasi-isomorphism, then this adjunction is a Quillen equivalence.
Proof. It is a Quillen adjunction because weak equivalences and fibrations are detected in Mod(k). It is a Quillen equivalence for φ a quasi-isomorphism by [ If f is a weak equivalence then this is a Quillen equivalence. (iv) Any map of commutative algebras φ : A → B gives rise to a Quillen adjunction This adjunction is a Quillen equivalence if φ is a quasi-isomorphism and one of the following holds: As special cases, we have the model categories Comm(A) of commutative algebras and Lie n (A) of n-shifted Lie algebras in Mod(A).
Proof. We use results from [PS14], whose hypotheses hold for Mod(k) by [ Since k is a field, Lemma 2.1.6 tells us that R ⊗ r : for any commutative algebra R. By (iii) this implies that both vertical morphisms are left Quillen equivalences. We also know that the top horizontal map is a left Quillen equivalence by (iv)(a), so it follows that the bottom horizontal map must be one too. Case (c) is proved similarly, taking a cofibrant replacement O ′′ → O ′ in operads in Mod(R) and using Lemma 2.1.4 to conclude that Remark 2.1.10. The results of Pavlov and Scholbach encompass a broader class of examples, including model categories of cochain complexes of bornological and convenient vector spaces constructed in [Wal15]. These would form a natural context for many examples coming from field theory where our formulation of functorial BV quantization would apply, but we will restrict our efforts here to an algebraic setting.

∞-Categories of Modules and Operad
Algebras. From the model categories discussed in §2.1, we can obtain ∞-categories by inverting the weak equivalences, i.e. the quasi-isomorphisms. Let A be a commutative algebra over k. We write Here we will use the results of §A.2 to show that these ∞-categories can alternatively be described using the standard simplicial category structures defined by tensoring with the algebras of polynomial differential forms: Let Ω(∆ n ) denote the commutative differential graded k-algebra of polynomial differential forms on ∆ n . That is, where each x i has degree 0 and dx i has degree 1 and the differential is the derivation determined by d(x j ) = dx j . This construction extends to a unique limit-preserving functor Ω : Set op ∆ → Comm(k). For all the categories C considered above, we can use the simplicial object Ω(∆ • ) to define a simplicial enrichment, by taking the mapping spaces to be C(X, Ω(∆ • ) ⊗ Y ). We will denote the simplicial categories obtained in this way from the cofibrant objects in the model categories above by Proof. The nth matching object for Ω(∆ • ) is Ω(∂∆ n ), and since Ω is a right Quillen functor, the map Ω(∆ n ) → Ω(∂∆ n ) is a fibration and hence Ω(∆ • ) is Reedy fibrant. We can moreover identify the matching object for Ω(∆ • ) ⊗ X with Ω(∂∆ n ) ⊗ X -this boils down to the fact that over a field the tensor product preserves finite limits in each variable. The levelwise surjectivity of Ω(∆ n ) → Ω(∂∆ n ) gives levelwise surjectivity of Ω(∆ n ) ⊗ X → Ω(∂∆ n ) ⊗ X, so this is again a fibration, as required. The second point follows from Lemma 2.1.6. The Quillen adjunctions induced by maps of algebras and operads of Proposition 2.1.7 and Proposition 2.1.8(iii-iv) induce adjunctions of ∞-categories (as proved in [MG16] for not necessarily simplicial model categories such as these). However, since tensor products are not strictly associative, the left adjoints are only pseudofunctorial in the commutative algebra variable. Since these functors, unlike their right adjoints, are compatible with the simplicial categories we have just described, we quickly point out how to obtain a functor of ∞-categories: Lemma 2.2.6. Let R be a commutative algebra over k and O an operad in Mod(R). There is a functor Proof. The proof follows that of [GHN15, Lemma A.24], and we will freely use notation and ideas from there in the proof here (but nowhere else in this paper). We have a normal pseudofunctor from commutative algebras over k to (fibrant) simplicial categories taking A to Alg O (R ⊗ A). Using the Duskin nerve [Dus02] of 2-categories as in [GHN15,§A] this gives a functor of quasicategories NComm(k) → N (2,1) CAT ∆ . If we restrict to cofibrant commutative algebras, then this functor takes quasi-isomorphisms of commutative algebras to weak equivalences of simplicial categories by Proposition 2.1.8(iv)(c); it thus induces a functor from the localization of NComm(k) cof at the quasi-isomorphisms, which is Comm(k), to the localization of N (2,1) CAT ∆ at the weak equivalences of simplicial categories, which is Cat ∞ since by [Lur14, Theorem 1.3.4.20] it is equivalent to the localization of the 1-category of simplicial categories at the weak equivalences.
We also note a useful technical result: 2.3. Some Operads. In this section, we introduce the operads relevant to our construction: Lie n , P 0 , and E 0 , which live in cochain complexes over k, and BD, which lives in cochain complexes over the algebra k[ ], where has degree zero.
Before defining these operads, we need to review some material, for which we use [LV12] as a convenient reference.
where the permuation group Σ n acts diagonally on the tensor product, and the composition of operations is in O and P independently. For instance, composition • i in the ith input is given by We also need to discuss shifts of operations, particularly shifted Lie brackets, for which we follow the treatment of [LV12] (notably Section 7.2). In the setting of cochain complexes, it is convenient to view shifting a complex as tensoring with the complex k[1], the one-dimensional vector space placed in degree −1. Similarly, shifting an operad amounts to tensoring with a distinguished, simple operad, namely the the endomorphism operad End Note that for any cochain complex V , its suspension Remark 2.3.5. A Lie n -algebra g has an n-shifted Lie bracket Λ 2 g → g [n]. Giving a Lie n -algebra structure on V is equivalent to giving a Lie algebra structure on V [−n].
As we will primarily be interested in Lie 1 , we will call an algebra over Lie 1 a shifted Lie algebra, only mentioning the level of shifting when it is not 1. Note in particular that the binary operations Lie 1 (2) consist of the trivial S 2 -representation in degree 1.
Definition 2.3.6. The operad P 0 is generated by two binary operations: • in degree 0 called "multiplication" and {} in degree 1 called "bracket." The operation • satisfies the relations for a commutative algebra, and the operation {} satisfies the relations for a shifted Lie algebra. The remaining ternary relation is that the bracket acts as a biderivation for multiplication. (It is also known as the Poisson 0 or Gerstenhaber or -1-braid operad. For a description of the operad using generators and relations, see Section 13.3.4 of [LV12] .) Note that each space of n-ary operations P 0 (n) has zero differential. As remarked in Section 13.3.4 of [LV12], P 0 is an "extension" of the shifted Lie operad Lie 1 by the commutative operad Comm, and hence there are canonical operad maps Comm → P 0 → Lie 1 . There is also a natural operad map Lie 1 → P 0 . Definition 2.3.7. The operad E 0 is the operad with just a single nullary operation. Its algebras in a symmetric monoidal category C are therefore just objects of C equipped with a map from the monoidal unit.
We construct now an operad quasi-isomorphic to E 0 as a variant of the operad P 0 : Definition 2.3.8. The operad E 0 is a modification of P 0 by changing the differentials. Let the binary operations E 0 (2) be P 0 (2) with differential d(•) = {}. Let E 0 (n) denote P 0 (n) equipped with the differential induced by the differential on binary operations.
By construction, the cohomology operad H * E 0 has trivial n-ary operations for n > 1. As E 0 is cochain homotopic to H * E 0 , this operad provides a model for the E 0 -operad.
Note that there is a map of operads Lie 1 → E 0 , induced by the map Lie 1 → P 0 . In contrast, the map Comm → P 0 does not lift to a map Comm → E 0 as such a map would not respect the differential on E 0 .
Finally, we introduce an operad interpolating between P 0 and E 0 ; it is a kind of "Rees operad." Definition 2.3.9. The operad BD is a modification of P 0 ⊗ k[ ] by changing the differentials. Let the binary operations BD(2) be P 0 (2) ⊗ k[ ] with differential d(•) = {}. Let BD(n) denote P 0 (n) ⊗ k[ ] equipped with the differential induced by the differential on binary operations.
This definition implies that for a BD-algebra A, so that d is a second-order differential operator on the underlying graded algebra A ♯ . Thus, modulo , the differential d is a derivation, so that the bracket measures the failure of A to be a commutative algebra in cochain complexes.
Observe that P 0 is isomorphic to and that E 0 is isomorphic to Thus lifting a P 0 -algebra to a BD-algebra produces an E 0 -algebra by setting = 1 in the algebra.
In this sense, a BD-algebra "quantizes" a P 0 -algebra to an E 0 -algebra.
Remark 2.3.10. The operad P 0 is a cocommutative Hopf operad, just as the Poisson operad is. The coproduct ∆ : P 0 → P 0 ⊗ H P 0 is given by which is the direct analogue for the Poisson operad. One simply checks directly that this choice works. The same coproduct works for the operads BD and E 0 , which are thus also Hopf.
2.4. Enveloping Algebras on the Model Category Level. We now wish to analyze the relationship between algebras over the three operads P 0 , BD, and E 0 . As we remarked above, we have a map of k[ ]-operads Lie 1 [ ] → BD that induces both the standard inclusion Lie 1 → P 0 when we set = 0 and also a map Lie 1 → E 0 when we set = 1. Combining these with the right Quillen functors induced by the algebra maps we get a commutative diagram of right Quillen functors: We will give explicit descriptions of the corresponding left adjoints to the horizontal morphisms, which can be thought of as "enveloping algebras": • the P 0 -enveloping functor U P0 is left adjoint to the "forgetful" functor from Alg P0 (A) to Lie 1 (A), • the BD-enveloping functor U BD is left adjoint to the "forgetful" functor from Alg • the E 0 -enveloping functor U E0 is left adjoint to the "forgetful" functor from Alg E0 (A) to Lie 1 (A). From the explicit descriptions it will be clear that these enveloping functors interact well with the natural symmetric monoidal structures on these categories of algebras. Note that P 0 , BD, and E 0 are all Hopf and so the natural monoidal structures amount, on the level of the underlying modules, to just tensor product ⊗ (over A for P 0 and E 0 or A[ ] for BD). By contrast, we equip Lie algebras with the monoidal structure given by the Cartesian product, which is the direct sum ⊕ on the level of underlying modules.
In this terminology, we have shown that is a functorial BD quantization of U P0 (g) for any shifted Lie algebra g in Mod(A).
Remark 2.4.2. In the setting of deformation quantization, people require that a quantization is flat over or topologically free. Since the functors involved are left Quillen, our construction always produces a module that is nicely behaved with respect to provided the input is cofibrant.
The P 0 -enveloping functor is explicitly provided by the following construction, which should seem obvious: if we have a shifted Lie algebra and we want a P 0 -algebra, all we need to do is freely construct the commutative algebra structure.
with the commutative multiplication of the symmetric algebra and with the bracket where x, y ∈ g. Thus U P0 is a strong symmetric monoidal functor: for any shifted Lie algebras g and g ′ .
Proof. Let g be a shifted Lie algebra, and let us write U P0 (g) for the explicit P 0 -algebra above; we will then show that this gives a left adjoint to the forgetful functor. Observe that the inclusion g ֒→ Sym A (g) of the commutative algebra generators is a map of Lie algebras, if we equip Sym A (g) with the bracket that defines U P0 (g). We want to show that composing with this map induces for every P 0 -algebra R an isomorphism (where we have not explicitly denoted the forgetful functor to Lie algebras).
To see this, observe that a Lie algebra map g → R induces a unique map of commutative algebras Sym A (g) → R, and this respects the Lie bracket giving U P0 (g) its Poisson structure, i.e. it is a map of P 0 -algebras. By inspection, this construction provides the desired inverse.
The fact that the functor is strong symmetric monoidal is then an immediate consequence of the fact that Sym is.
By a completely parallel argument, we obtain an analogous description of the E 0 -enveloping functor, except that the construction of the enveloping algebra looks slightly more complicated than in the P 0 case, since we need to describe the differential explicitly. Recall that for a commutative algebra A, we use A ♯ to denote the underlying commutative graded algebra.
Lemma 2.4.4. For a Lie 1 -algebra g in Mod(A), the E 0 -enveloping algebra U E0 (g) has underlying A ♯ -module Sym A ♯ (g) with the commutative multiplication of the symmetric algebra, with the bracket where x, y ∈ g, and with differential d E0 determined by Thus U E0 is a strong symmetric monoidal functor: for any shifted Lie algebras g and g ′ .
The fact that an E 0 -algebra satisfies for any elements a and b means that we can inductively define the differential on higher symmetric powers in U E0 (g), as we have specified it on its Sym ≤2 summand.
The situation with BD is parallel, after adjoining everywhere. for x ∈ g and d BD (x · y) = (d g x) · y + (−1) x x · (d g y) + {x, y} for x, y ∈ g. Thus U BD is a strong symmetric monoidal functor: for any shifted Lie algebras g and g ′ .
2.5. Relationship with Lie Algebra Homology. The enveloping algebra constructions described above may seem reminiscent of the Chevalley-Eilenberg chains of a Lie algebra, since the differential is determined by the Lie bracket in a similar way. We now pin down a precise relationship.
Let Cocomm(A) denote the category of cocommutative coalgebras in Mod(A). Let Sym c A (V ) denote the symmetric coalgebra on the A-module V , whose underlying A-module is n≥0 Sym n A (V ) and whose coproduct satisfies ∆(x) = 1 ⊗ x + x ⊗ 1 for every x ∈ Sym 1 A (V ). Definition 2.5.1. Let C L : Lie 1 (A) → Cocomm(A) denote the functor sending g to the cocommutative coalgebra Sym c A ♯ (g ♯ ) over A ♯ equipped with the differential d C L , which is the degree 1 coderivation such that for x ∈ Sym 1 and for xy ∈ Sym 1 (In other words, this functor agrees with the Chevalley-Eilenberg chains functor after shifting g to an unshifted Lie algebra g[−1].) Proposition 2.5.2. For a Lie 1 -algebra g in Mod(A), the underlying cochain complex of the E 0enveloping algebra U E0 (g) is naturally isomorphic to the underlying cochain complex of C L (g).
In other words, if θ denotes the forgetful functor from Alg E0 (A) to Mod(A), then there is a natural isomorphism θ • C L ⇒ θ • U E0 .
Remark 2.5.3. This relationship should not seem implausible. Consider the underived setting of Lie algebras in vector spaces. The inclusion of Vect into Lie as abelian Lie algebras is right adjoint to the functor g → g/[g, g] that "abelianizes" a Lie algebra (or takes its coinvariants). Hence the functor g → C Lie * (g, g) (whose cohomology is the Lie algebra cohomology groups H Lie * (g, g)) should provide a model for the derived left adjoint of the abelian Lie algebra functor. Now let us turn to our situation of shifted Lie algebras. An E 0 -algebra is simply a "pointed" module A → M , so we see that the functor g → A ⊕ C L (g, g) -where the first summand is the "pointing" -provides a derived left adjoint to the functor (A → M ) → M/A, with M/A an abelian shifted Lie algebra. But the composite θ • C L (g) is isomorphic to A ⊕ C L (g, g). In short, θ • C L should be a derived left adjoint to the "forgetful" functor (i.e. inclusion functor) from Alg E0 (A) to Lie 1 (A).
Remark 2.5.4. The result also fits nicely with the perspective of derived deformation theory: if we view a differential graded Lie algebra g as presenting a formal moduli space, then C L (g) describes the coalgebra of distributions on this space. As distributions are a natural home for "things that integrate," it is not surprising that this derived version exhibits the formal, algebraic properties axiomatized by physicists in BD-algebras when they sought to formalize properties of the putative path integral.
Proof. Both U E0 and C L assign to g the same underlying A ♯ -module Sym A ♯ (g ♯ ). Moreover, the differentials on both modules respect the filtration by symmetric powers: By definition, the differentials agree on Sym ≤2 The key difference is that • d E0 is extended to higher symmetric powers as a second-order differential operator on the symmetric algebra whereas • d C L is extended to higher symmetric powers as a coderivation on the symmetric coalgebra.
Hence we must show these conditions coincide, which follows immediately from Lemma 2.5.5, as the differential of the BD-enveloping algebra is a linear-coefficient second-order differential operator.
Lemma 2.5.5. Let R be a graded commutative algebra. For V in Mod(R), a coderivation on the symmetric coalgebra Sym c R (V ) provides a linear-coefficient, arbitrary order differential operator on the symmetric algebra Sym R (V ).
Proof. This lemma is the coalgebraic twin of the fact that a derivation is a first-order differential operator with no constraints on the coefficients. First, consider the multiplication map m x : p → xp given by multiplying in Sym R (V ) by a linear element x ∈ Sym 1 R (V ). This map m x is a coderivation: Second, consider a constant-coefficient derivation ∂ of the form f → ι λ f , where λ ∈ Hom R (V, R) and ι λ denotes contraction with λ. Thus ∂ is the derivation on Sym R (V ) obtained by extending λ from Sym 1 R (V ) ∼ = V by the Leibniz rule. This map ∂ is a comodule map, as we show by direct computation. Let x 1 · · · x n be a pure product in Sym n R (V ), and compute and then compute Comodule maps are closed under composition, so any constant-coefficient differential operator D = ∂ 1 · · · ∂ n is a comodule map. Thus the composition m x D is a coderivation.
Remark 2.5.6. As noted in [BV14], this lemma implies that the Chevalley-Eilenberg chains of an L ∞ -algebra g is the specialization to = 1 of a kind of BD ∞ -algebra U BD ∞ (g[1]). Here one weights the kth Taylor coefficient of the Chevalley-Eilenberg differential by k−1 .
2.6. Enveloping Algebras on the ∞-Category Level. As a special case of Lemma 2.2.6, we know that the enveloping algebra functors described above give functors of ∞-categories, compatible with base change in the commutative algebra variable. What is a bit less straightforward is showing that these functors are symmetric monoidal at the ∞-category level. The issue is that our model categories are not monoidal model categories: in particular, the tensor products do not preserve cofibrant objects, so our simplicial categories will not be symmetric monoidal. We therefore have to do a bit more work to see we have symmetric monoidal structures on the ∞-categories at all; we will proceed analogously to the proof of [Lur14, Proposition 4.1.3.10]: we enhance our simplicial category to a simplicial operad and check that the associated ∞-operad is actually a symmetric monoidal ∞-category. Once this is done, it is straightforward to see that our enveloping functors give maps between these simplicial operads, (pseudofunctorially) compatible with base change, and these induce symmetric monoidal functors on the ∞-category level. We focus on the case of Lie algebras; the same idea works for the other operads.
Definition 2.6.1. We define a simplicial (coloured) operad structure on the simplicial category Lie(A) op by defining the multimorphism spaces as with composition induced from that in Lie(A). This is compatible with the simplicial enrichment, given by tensoring with Ω(∆ • ), since tensoring commutes with direct sums.
To get from this simplicial operad to an ∞-operad we need to pass through its simplicial category of operators. Recall that any simplicial operad O has a simplicial category of operators O ⊗ . This has objects pairs ( n , (X 1 , . . . , X n )), where n is an object of op -the category of finite pointed sets -and the X i are objects of O. A morphism ( n , (X 1 , . . . , X n )) → ( m , (Y 1 , . . . , Y m )) is given by a morphism φ : n → m in op and for each i ∈ m a multimorphism ( Recall that a symmetric monoidal ∞-category can be defined as an ∞-operad O such that the projection O → op is a coCartesian fibration. This holds for our ∞-operad N(Lie(A) op,⊕ ): Proposition 2.6.2. The projection π : N(Lie(A) op,⊕ ) → op is a coCartesian fibration. That is, π is a symmetric monoidal ∞-category.
Proof. It suffices to show that for every object (X 1 , . . . , X n ) of Lie(A) op,⊕ and every map φ : n → m in op , there exists a morphism (X 1 , . . . , X n ) → (X ′ 1 , . . . , X ′ m ) over φ such that for any (Y 1 , . . . , Y k ), the square Hom( m , k ) Hom( n , k ) is homotopy Cartesian. Choose a weak equivalence X ′ i → j,φ(j)=i X j in Lie n (A) with X ′ i cofibrant. We claim the resulting map (X 1 , . . . , X n ) → (X ′ 1 , . . . , X ′ m ) in Lie(A) op,⊕ has this property. To see this it suffices to show that we have a weak equivalence on fibres over each ψ : m → k , since the objects in the bottom row are discrete. These fibres decompose as products, so it is enough to show that is a weak equivalence for all j. We can identify this map with Since Y j is cofibrant, to see that this map is a weak equivalence of simplicial sets, it suffices to show that i X ′ i → j X j is a weak equivalence in Lie algebras. But this map is the product over i of the maps X ′ i → j,φ(j)=i X j , which are weak equivalences, and since weak equivalences are detected in Mod(k), it is clear that direct sums of weak equivalences are again weak equivalences.
We thus have a symmetric monoidal ∞-category with underlying ∞-category Lie n (A) op . This induces a symmetric monoidal structure on Lie n (A), i.e. we have: Corollary 2.6.3. The Cartesian product ⊕ of Lie algebras induces a symmetric monoidal structure on the ∞-category Lie n (A).
Proposition 2.6.4. Proof. (i) follows from the same argument as for Lie algebras. We only need to check that if X is a cofibrant algebra then X ⊗ A -preserves quasi-isomorphisms. (In the case of BD-algebras, we use X ⊗ A[ ] -instead.) This claim follows from Lemma 2.1.4, since by Proposition 2.1.8(ii), the underlying A-module of a cofibrant algebra is cofibrant. For (ii), if U is either U E0 or U P0 , we must show that if L and L ′ are cofibrant Lie algebras and L ′′ → L ⊕ L ′ is a cofibrant replacement, then U (L ′′ ) → U (L ⊕ L ′ ) is a weak equivalence. It suffices to check weak equivalences at the level of the underlying modules, and there we have a natural filtration U (g) = colim n U ≤n (g) for any Lie algebra g, with U ≤n (g) being the subcomplex whose underlying graded module is Sym ≤n (g). It is manifest that U ≤0 (L ′′ ) = A = U ≤0 (L ⊕ L ′ ). Now consider the map of cofiber sequences By Lemma 2.1.5 the functor Sym n A preserves quasi-isomorphisms for cofibrant modules. Since the underlying A-module of a cofibrant Lie algebra is cofibrant by Proposition 2.1.8(ii), the rightmost vertical map in the diagram is a quasi-isomorphism. Inducting on n, it follows that U ≤n (L ′′ ) → U ≤n (L⊕L ′ ) is a quasi-isomorphism for all n. As quasi-isomorphisms commute with filtered colimits, we conclude that U (L ′′ ) → U (L ⊕ L ′ ) is a quasi-isomorphism. The proof for U BD is the same, except with some 's.
For the functors induced by the two maps A[ ] → A, it again suffices to show that we get a quasi-isomorphism of underlying modules, which is true since A ⊗ A[ ] -is a left Quillen functor, the underlying module of a cofibrant algebra is cofibrant, and the tensor product of cofibrant modules is again cofibrant.
Taking the base change (pseudo)functors into account, we have: Comm(k) to the ∞-category Comm( Cat ∞ ) of (large) symmetric monoidal ∞-categories taking A to the symmetric monoidal ∞-categories constructed above. The enveloping algebra functors U BD , U E0 and U P0 , as well as the functors ev =0 and ev =1 induce natural symmetric monoidal functors between these.
Proof. As Lemma 2.2.6, just replacing simplicial categories with simplicial operads (and passing to opposite ∞-categories).

2.7.
Aside: An E n -Enveloping Algebra Functor. In this section, we will describe an "enveloping algebra" adjunction where E n is the "little n-discs" ∞-operad. (This section is something of a digression from our main objective, although it is relevant as motivation for Conjecture 1.3.1.) We expect that this construction agrees with the enveloping functor for the map of operads constructed by Fresse [Fre14] using Koszul duality as well as that recently constructed (in greater generality) by Knudsen [Knu16] using factorization algebras. However, although we will show that our functor satisfies some of the same formal properties as Knudsen's, we will not attempt to compare them here. By our work in the preceding sections, we have a symmetric monoidal left adjoint functor where the second equivalence follows from the quasi-isomorphism E 0 ≃ E 0 of operads. By [Lur14, Corollary 7.3.2.7], the right adjoint to the E 0 -enveloping functor is lax monoidal, and so the resulting relative adjunction over op induces an adjunction for every ∞-operad O. Taking O to be the ∞-operad E n , we have since the Boardman-Vogt tensor E n ⊗ E 0 is equivalent to E n ; thus we get an adjunction Alg En (Lie 1 (A)) ⇄ Alg En (A).
To get the enveloping functor we want, we combine this construction with a result we learned from Nick Rozenblyum. Before we state it, we must recall the bar/cobar adjunction, as set up for ∞categories by Lurie in [Lur14, §5.2.2]. If C is a monoidal ∞-category with simplicial colimits and cosimplicial limits, there is an adjunction Bar : Alg aug Ass (C) ⇄ CoAlg coaug Ass (C) : Cobar between augmented associative algebras and coaugmented coassociative coalgebras. If C has a zero object and the monoidal structure is the Cartesian product, then this simplifies to an adjunction Bar : Alg Ass (C) ⇄ C : Cobar.
Proposition 2.7.1. Suppose C is a presentable stable ∞-category, D is a presentable ∞-category, and U : D → C is a functor that detects equivalences and preserves limits and sifted colimits. Then, regarding D as a monoidal ∞-category via the Cartesian product, the bar/cobar adjunction Proof. Since C is stable, the Cartesian product in C is also the coproduct, and hence it commutes with sifted colimits and cosifted limits in each variable. As U detects equivalences and preserves limits and sifted colimits, we find that the Cartesian product in D also preserves sifted colimits and cosifted limits in each variable. Thus using [Lur14, Example 5.2.2.3], for any X ∈ Alg Ass (D), we can identify U Bar(X) with the suspension ΣU X. Dually, if U ′ denotes the forgetful functor then for Y ∈ D, we can identify U ′ Cobar(Y ) with the loop object ΩU (Y ).
To show that the bar/cobar functors are an adjoint equivalence, it suffices to show that the unit and counit transformations are natural equivalences. But since the functors U and U ′ detect equivalences, we are finished because suspension/loops is an adjoint equivalence on the stable ∞category C.

The Heisenberg Functor
The usual Heisenberg Lie algebra of a symplectic vector space (V, ω : Λ 2 V → k) is the vector space V ⊕ kc equipped with the Lie bracket [x + αc, y + βc] = ω(x, y)c.
In other words, it is a central extension of the abelian Lie algebra V by the one-dimensional abelian Lie algebra kc. Specializing c to i , one recovers Heisenberg's celebrated relation [x, p] = i . Note that the pairing ω need not be non-degenerate, so the construction works even for "presymplectic" vector spaces.
Our goal in this section is to articulate a version of this construction where the input is a quadratic module of degree 1 -a module V over a commutative algebra A equipped with a shifted skew-symmetric pairing ω : Λ 2 A V → A[1] -and the output is a shifted Lie algebra given by centrally extending the abelian Lie algebra V by Ac, with c in degree zero. This construction makes sense "on the nose" for objects of the natural category of quadratic modules: given a quadratic module (V, ω), we can define the 1-shifted Heisenberg Lie algebra Heis 1 (V, ω) as V ⊕ Ac, where c has degree zero, with shifted Lie bracket [x + αc, y + βc] = ω(x, y)c. If we have a map f : V → V ′ such that f * ω ′ = ω, then we get a Lie algebra map Heis 1 (f ) : . This definition, however, has two issues. Firstly, the Lie algebra Heis 1 (V ) is not cofibrant, and so it needs to be cofibrantly replaced in order to get a homotopically meaningful answer when we apply the enveloping functors described above. Secondly, a more subtle issue is that, as we will see in §3.1, the obvious way to make a simplicial category of quadratic modules does not define the correct ∞-category. It turns out that we can fix the second issue by taking the maps of quadratic modules to be maps that preserve the pairings only up to a specific cochain homotopy. Unfortunately, this notion of map does not give maps between Heisenberg Lie algebras: if F : (V, ω) → (V ′ , ω ′ ) is a map of quadratic modules in this sense, given by a map of A-modules f : V → V ′ and a homotopy η between ω and f * ω, meaning then we see that In other words, f produces a Lie algebra map only up to homotopy.
For this reason we take a technical detour through L ∞ -algebras, as the formalism of L ∞ -algebras provides a convenient tool for working with Lie algebras up to homotopy. The key advantage is that one works with the coalgebra of Chevalley-Eilenberg chains C L (g) of a Lie algebra g -its bar construction B(V ) -rather than directly with g. In particular, maps between bar constructions capture the notion of "maps of Lie algebras up to homotopy." Using the flexibility of L ∞ -algebras, we will see in §3.3 that the corrected maps of quadratic modules induce natural maps on the bar constructions BHeis 1 (V ). We can then apply the cobar construction è to get a functor to shifted Lie algebras; this approach also fixes the first issue mentioned above, since èBHeis 1 (V ) is a natural cofibrant replacement of Heis 1 (V ). Passing to ∞-categories, we produce a functor where the right vertical functor is the forgetful functor that takes a moprhism to A[n] to its domain.
Remark 3.1.1. The closely related situation of modules equipped with (shifted) symmetric pairings has been studied by Vezzosi [Vez13].
For our purposes, it will be convenient to have a simplicial category that models the ∞-category Quad n (A) of such quadratic modules, so that we can give explicit constructions that play nicely with the enveloping algebra functors. As a first attempt, let us try to mimic the pullback construction above in the setting of simplicial categories.
By as we did with the ∞-categories. This pullback gives a simplicial enrichment of the obvious strict category of quadratic modules, but it is not a homotopy pullback diagram of simplical categories: the right vertical functor is not a fibration in the model category of simplicial categories. We therefore need to replace it with a map that is a fibration, which we do as follows: with C cofibrant in Mod(A) and f a fibration, and in which a morphism : (C, f ) → (C ′ , f ′ ) is a map φ : C → C ′ together with a cochain homotopy η : C → Ω(∆ 1 ) ⊗ X from f to f ′ • φ. This category has an obvious simplicial enrichment Mod(A) ′ /X , defined as usual by tensoring with Ω(∆ • ). Note that a morphism between such objects respects the maps down to X only up to homotopy. Recall that a functor of simplicial categories is a fibration if and only if it is an isofibration on homotopy categories (i.e. every isomorphism in the target can be lifted to one in the source) and it is given by Kan fibrations on the mapping spaces. Let (Y, f : Y → X) and (Z, g : Z → X) be two objects of Mod(A) ′ /X ; for brevity we will refer to these objects as just f and g. Then the simplicial set of maps between them is given by the pullback square where the bottom horizontal map evaluates a map in Ω(∆ 1 ) ⊗ X at the two endpoints of the 1simplex.
Since Ω(∆ • ) is Reedy fibrant, the bottom horizontal map is a Kan fibration, and hence so is the top horizontal map. To see that p is an isofibration on homotopy categories, observe that since Mod(A) is a model category and Mod(A) contains only the fibrant-cofibrant objects, the isomorphisms in the homotopy category of Mod(A) are precisely the cochain homotopy equivalences. Given a cochain homotopy equivalence φ : Y → Y ′ and a map f : Y ′ → X, a trivial cochain homotopy η from f • φ to itself gives a mapη : This map induces a simplicial homotopy equivalence on all mapping spaces, and thus becomes an isomorphism in the homotopy category.
Since the simplicial categories Mod(A) /X and Mod(A) ′ /X have the same objects, the functor i is obviously essentially surjective on the homotopy categories. To see that it is a weak equivalence, it therefore only remains to show that for any two objects f : Y → X, g : Z → X, the map of simplicial sets Hom A/X (f, g) → Hom ′ A/X (f, g) is a weak equivalence. To prove this, we consider the commutative diagram of simplicial sets Note that by definition the bottom square, the upper right square, and the outer composite square in the top row are all pullback squares. Hence the top left square is also a pullback. The top right vertical arrow is a Kan fibration because g : Z → X is a fibration, and so as indicated in the diagram, it follows that all three top vertical arrows are Kan fibrations. The bottom right arrow ev 0 is a trivial Kan fibration and so the bottom left arrow is a trivial Kan fibration. By the 2-out-of-3 property, we thus deduce that the bottom map in the upper left square is a weak equivalence. Hence the top left horizontal map is also a weak equivalence, as required, as simplicial sets form a right proper model category.
We then define our simplicial category of quadratic modules by: Definition 3.1.4. Let Quad n (A) be the simplicial category defined by the pullback square which is also a homotopy pullback square. We also write Quad n (A) for the underlying category of Quad n (A), which sits in a pullback square We now turn to equipping these categories with a symmetric monoidal structure. On the underlying modules, we simply use ⊕, but we need to describe how the quadratic forms are combined. Given (X, ω) and (X ′ , ω ′ ), let ω + ω ′ on the direct sum X ⊕ X ′ be given by the composite map where π is projection. We then define (X, ω) ⊕ (X ′ , ω ′ ) to be (X ⊕ X ′ , ω + ω ′ ). Given two maps Proposition 3.1.5. This definition extends naturally to a symmetric monoidal structure on the simplicial category Quad n (A). Moreover, this symmetric monoidal structure is pseudonatural in A.
Proof. Since Quad n (A) is the simplicial category associated to Quad n (A ⊗ Ω(∆ • )) by the construction of Proposition A.1.1, it suffices to show that the categories Quad n (A) are symmetric monoidal, pseudonaturally in A. It is easy to see that our definition does indeed give such a pseudonatural symmetric monoidal structure.
Corollary 3.1.6. The ∞-category Quad n (A) has a natural symmetric monoidal structure.
3.2. L ∞ -Algebras. In this section we introduce a version of L ∞ algebras well-suited to our purposes. A crucial requirement is that our notion must work over any commutative algebra A over a field k of characteristic zero and must play nicely with base-change. We will not develop a general framework, but rather proceed in a somewhat ad hoc fashion that produces the limited results we need.
Recall that Cocomm(A) denotes the category of cocommutative coalgebras in Mod(A). A cocommutative coalgebra C is coaugmented if there is a retract of coalgebras A η − → C → A. Its reduced coalgebra C is the kernel of the counit map, so that C = A ⊕ C, and C inherits a coproduct∆ bȳ For us, a coaugmented cocommutative coalgebra A η − → C is conilpotent if for any element c in the reduced coalgebra C, there is some integer n such that∆ n (c) = 0. The key example is the symmetric coalgebra Sym c A (V ), whose reduced coalgebra Sym ≥1 A (V ) is manifestly conilpotent as ker(∆ n ) = n j=1 Sym i A (V ). We write Cocomm conil (A) for the category of conilpotent coaugmented cocommutative coalgebras in Mod(A), where we require maps to preserve the coaugmentations.
in parallel with our construction of simplicial categories of algebras over operads. The bar construction B is given by the functor C L of Definition 2.5.1. (Recall this is the usual Chevalley-Eilenberg chains, after shifting.) The cobar construction è assigns to C ∈ Cocomm conil (A), the semi-free Lie 1 -algebra (Free Lie1 (C), d è ) whose differential is the Lie algebra derivation of degree 1 determined by the shift of the coproduct on C. This adjunction is natural in A, so in particular it gives a simplicial adjunction between the associated simplicial categories.
Note that by working with shifted Lie algebras, we obviate the need to shift in constructing the Chevalley-Eilenberg chains.
Remark 3.2.2. For a field k of characteristic zero, Hinich [Hin01] constructs a model structure on Cocomm conil (k) where all objects are cofibrant and the fibrant objects are the semi-free coalgebras, i.e. those whose underlying graded coalgebra is Sym c (V ) for some graded vector space V . The cobar-bar adjunction is a Quillen equivalence between this model category and that of Lie algebras.
In particular, for any L ∈ Lie 1 (A), the adjunction counit èBL → L is a cofibrant replacement of L.
We do not know if an analogous model structure exists on Cocomm conil (A) when A is not a field, but it turns out that èBL is still often a cofibrant replacement, which is enough for our purposes: Lemma 3.2.3. Let L be a shifted Lie algebra over A whose underlying A-module is cofibrant. Then (i) the counit map èBL → L is a weak equivalence, and (ii) the shifted Lie algebra èBL is cofibrant.
Proof. The bar coalgebra is the colimit of a sequence of coalgebras since the coproduct on the symmetric coalgebra decreases symmetric powers and the differential preserves and lowers the symmetric powers. Note that the cokernel of the map B k−1 (g) → B k (g) is simply Sym k A (g). Consider the following pushout square in A-modules: where • the top horizontal map is the differential on B k (g), restricted to Sym k A (g)[−1], and viewed as a degree zero map, and • the bottom left corner C(id) denotes the cone of the identity map from Sym k A (g)[−1] to itself. This is a pushout square because it is a pushout on the underlying graded vector spaces and the maps are also compatible with the differentials. The left vertical map is a cofibration in A-modules, and hence the right vertical map is a cofibration of A-modules.
We can also view this square as a commutative diagram of cocommutative coalgebras, where the two coalgebras on the left side have zero coproduct. Apply the cobar functor to this square. On the left side, it reduces to the free Lie 1 -algebra functor, and hence we have a cofibration of Lie 1 -algebras in A-modules. As it is a pushout square of Lie 1 -algebras, the right vertical map is also a cofibration.
The base case B 1 (g) is a free Lie 1 -algebra on a cofibrant A-module and hence a cofibrant Lie 1algebra. Hence B k (g) is also cofibrant as a Lie 1 -algebra, since it is the k-iterated pushout along a cofibration.
Definition 3.2.4. Let L ∞ (A) denote the full subcategory of Cocomm conil (A) spanned by the objects BL where L is a Lie 1 -algebra over A whose underlying A-module is cofibrant. Let L ∞ (A) denote the analogous simplicial category.  Here the left diagonal map is an isomorphism, since the cobar-bar adjunction is simplicial, and the right diagonal map is a weak equivalence by Proposition A.2.5 since èBL is cofibrant and èBL ′ → L ′ is a weak equivalence by Lemma 3.2.3. Thus è is weakly fully faithful.
It remains to show that è is essentially surjective on the homotopy category. Given L ∈ Lie 1 (A), then by definition L is a cofibrant Lie 1 -algebra, so by Proposition 2.1.8(ii) its underlying A-module is also cofibrant. Lemma 3.2.3 therefore implies that the counit èBL → L is a weak equivalence of cofibrant Lie algebras, and hence an equivalence in the simplicial category Lie 1 (A).
We need to know that this equivalence respects symmetric monoidal structures coming from the Cartesian product on both sides. For Lie 1 (A), we constructed this product in §2.6. The case of L ∞ (A) is easy: The product of conilpotent cocommutative coalgebras is given by the tensor product over A, and thus L ∞ (A) is closed under products -B is a right adjoint and so BL⊗BL ′ ∼ = B(L⊕L ′ ). This construction is also compatible with base change, so the simplicial category L ∞ (A) inherits a symmetric monoidal structure, and hence so does L ∞ (A).
Since this right adjoint B preserves products, its left adjoint è is oplax monoidal. We thus have a lax monoidal functor on the opposite categories. This functor is compatible with the simplicial enrichments, so we get a map of simplicial operads from L ∞ (A) op (which is a symmetric monoidal simplicial category) to Lie 1 (A) op (with the simplicial operad structure described in §2.6). Taking coherent nerves, we get a lax symmetric monoidal functor of ∞-categories L ∞ (A) op → Lie 1 (A) op . Proof. We must show that for any objects BL and BL ′ in L ∞ (A), the oplax structure map is a weak equivalence of (cofibrant) Lie algebras. The counit transformation gives a commutative diagram where the diagonal maps are weak equivalences (for the right diagonal map, this holds since weak equivalences are closed under ⊕). By the 2-out-of-3 property the horizontal map is hence also a weak equivalence.
The symmetric monoidal functor L ∞ (A) op → Lie 1 (A) op then induces a symmetric monoidal functor on opposite ∞-categories, L ∞ (A) → Lie 1 (A). Moreover, it is easy to see (using the same argument as in Lemma 2.6.5) that this construction is natural in the commutative algebra variable.
3.3. The Heisenberg L ∞ -Algebra. We construct here a symmetric monoidal functor that produces a Heisenberg L ∞ -algebra from a quadratic module of degree 1. As earlier, we begin by constructing a 1-categorical functor, upgrade it to a functor of simplicial categories, and then take the coherent nerves.
We will construct a functor that sends (V, ω) to BHeis 1 (V, ω) = C L (Heis 1 (V, ω)). We then need to associate functorially to each map F : This map H ∞ (F ) is easy to describe once we have some elementary results about coalgebras in hand.
Lemma 3.3.1. Let D be a degree zero coderivation of a conilpotent graded coalgebra C (i.e. with trivial differential). Then exp(D) is a coalgebra automorphism of C.
Proof. We compute The inverse is clearly exp(−D).
If (C, d C ) is a differential graded coalgebra and δ is a Maurer-Cartan element in the Lie algebra of coderivations Coder(C), i.e. a degree one element such that [d C , δ] + δ 2 = 0, then (C, d C + δ) defines another coalgebra (with the same coproduct). then exp(D) provides a coalgebra automorphism from (C, d C + δ 1 ) to (C, d C + δ 2 ).
Proof. Recall that the differential on C L (Heis 1 (V, ω)) is a sum of degree 1 coderivations d V + δ ω where d V denotes the differential on V extended to Sym c A (V ) as a coderivation and δ ω likewise denotes ω -viewed as a degree 1 map from Sym 2 A (V ) to A -extended as a coderivation. (The obvious analogues hold for the other coalgebras, such as C L (Heis 1 (V ′ , ω ′ )).) Thus, Sym c A (f ) is a coalgebra map from C L (Heis 1 (V, f * ω ′ )) to C L (Heis 1 (V ′ , ω ′ )) since f naturally provides a map of shifted Lie algebras from Heis 1 (V, f * ω) to Heis 1 (V ′ , ω ′ ). It remains to show that exp(D η ) is a map of coalgebras.
This claim follows from Lemma 3.3.2 once we verify that [D η , δ ω ] = 0 = [D η , δ f * ω ′ ]. Without loss of generality, we simply verify that the commutator with δ ω vanishes. Note that it suffices to compute the commutator [D η , δ ω ] just on the second stage of the filtration since any coderivation preserves the filtration by symmetric powers and a coderivation is determined by its behavior on cogenerators. But on this stage of the filtration, both η and ω map into Ac, and they both vanish on Ac, so their commutator vanishes.
We need to show that this construction respects composition.
is the homotopy from ω to f * g * ω ′′ obtained by composing η and f * γ is the natural way.
, essentially by the definition of D f * γ . Next observe that D η and D f * γ commute, by the argument in the preceding lemma: they are determined by their behavior on cogenerators and that is defined on the second stage of the filtration, but both have image in Ac and vanish on Ac. Hence Putting these results together, we can make the following definition: Proposition 3.3.6. The functor H ∞ is lax symmetric monoidal, sending ⊕ to ⊗ A .
The laxness is a consequence of the fact that each Heisenberg algebra contributes a central element. Thus, H ∞ (V, ω) ⊗ A H ∞ (V ′ , ω ′ ) has two central elements c and c ′ . By contrast, if we take a direct sum before constructing the Heisenberg algebra, we only have one central element c. We identify these two central elements with one another to produce a map This construction provides the natural transformation making H ∞ lax symmetric monoidal.
The functor H ∞ is natural in A, and we now want to use this naturality, applied to A ⊗ Ω(∆ • ), to get a functor of simplicial categories that is again natural in A. However, the naturality in A is not strict: since the tensor product is not strictly associative, but only associative up to isomorphism, it is only pseudonatural. The following is therefore not entirely obvious, but uses some pseudofunctorial observations we have delegated to the appendix.  via the coherent nerve. Moreover, this functor is natural in A ∈ Comm(k).
Proof. We saw above that H ∞ determines a pseudofunctor Comm(k)×[1] → CAT ∆ . If we restrict to cofibrant commutative algebras, then this functor takes weak equivalences of commutative algebras to weak equivalences of simplicial categories. Proceeding as in the proof of Lemma 2.2.6 we get from this the desired functor of ∞-categories Comm(k) × ∆ 1 → Cat ∞ . A similar argument with simplicial operads gives the lax monoidal structure.
The composite è • H ∞ is then a lax symmetric monoidal functor H : Quad 1 (A) → Lie 1 (A). It takes the unit 0 in Quad 1 (A) to a 1-dimensional abelian Lie algebra we'll denote by Ac, and therefore factors through a lax monoidal functor H : Quad 1 (A) → Mod Ac (Lie 1 (A)).
Lemma 3.3.9. The lax symmetric monoidal functor H is symmetric monoidal.
Proof. Let us write X ⊗ Ac Y for the tensor product in Mod Ac (Lie 1 (A)), which is by definition the geometric realization |X ⊕ Ac ⊕ · · · ⊕ Y | in the ∞-category Lie n (A). Then we must show that the natural map H(V, ω) ⊗ Ac H(V ′ , ω ′ ) → H(V ⊕ V ′ , ω + ω ′ ) is an equivalence. Since the forgetful functor to Mod(k) detects equivalences and preserves sifted colimits by Proposition 2.2.7, it suffices to check that the underlying map in Mod(k) is an equivalence. But in Mod(k) we can identify the image of H(V, ω) ⊗ Ac H(V ′ , ω ′ ) with the pushout H(V, ω) ∐ Ac H(V ′ , ω ′ ). It therefore suffices to show that H(V ⊕ V ′ , ω + ω ′ ) is correspondingly a homotopy pushout in Mod(k). To see this we can replace H(V ⊕ V ′ , ω + ω ′ ) with the quasi-isomorphic cochain complex V ⊕ V ′ ⊕ Ac, which is clearly the pushout of V ⊕ Ac and V ′ ⊕ Ac over Ac. Since the inclusions Ac → V ⊕ Ac, V ′ ⊕ Ac are cofibrations (as V and V ′ are cofibrant, and cofibrations are closed under pushouts) this is a homotopy pushout in Mod(k), which completes the proof.

Linear BV Quantization
Combining the constructions of the previous sections, we get a symmetric monoidal functor of ∞-categories that we call linear BV quantization. (As explained in Section 1.4, setting c = recovers the construction typically seen in the literature.) In this section we will explore some properties of this functor and its close cousin which we call simply linear quantization. In §4.1 we show that there is a natural extension from modules to quasicoherent sheaves on derived stacks, so that linear BV quantization is a well-posed construction in derived geometry. Then in §4.3 and §4.4 we show that this functor behaves like a determinant when restricted either to symplectic modules on a formal moduli problem or to symplectic vector bundles on a derived stack, after dealing with the base case of symplectic k-modules in §4.2. Finally, in §4.5 we combine our functors with the higher Morita category construction of [Hau14] to get symmetric monoidal functors of (∞, n)-categories.

4.1.
Linear BV quantization as a Map of Derived Stacks. We will show here that our BV quantization functor extends for formal reasons from commutative algebras to derived stacks.
Recall that a C-valuedétale sheaf is a presheaf F : Comm(k) → C that satisfiesétale descent : it preserves finite products and takes derivedétale covers (which we will not define here, cf. [TV08, Definition 2.2.2.12] or [Lur09b, Definition 4.3.13]) to cosimplicial limits. A derived stack (in the most general sense) is an S-valuedétale sheaf, where S is the ∞-category of large spaces. We use dSt k to denote the full subcategory of Fun(Comm(k), S) spanned by the derived stacks. It is then a formal consequence of the definition (cf. [Lur09b, Proposition 5.7]) that for any (very large) presentable ∞-category C, the ∞-category Fun R (dSt op k , C) of limit-preserving functors is equivalent to the ∞-category of functors Comm(k) → C that areétale sheaves, via restricting along the Yoneda embedding Comm(k) op → dSt k . The inverse functor is given simply by taking right Kan extensions.
We have constructed natural transformations Quad 1 (-) → Lie 1 (-), Lie 1 (-) → Alg BD (-), etc., of functors Comm(k) → Cat ∞ . To see that these extend to natural transformations of functors on derived stacks, it suffices to show that the functors in question areétale sheaves. This claim will follow quite straightforwardly from Lurie's descent theorem for modules: Remark 4.1.2. In fact, Lurie's result is substantially more general: he shows that Mod(-) is a hypercomplete sheaf in the flat topology, and that this holds for modules over commutative ring spectra.
As a trivial consequence we have: Lemma 4.1.3. The functor Quad n (-) satisfiesétale descent, and so has a natural extension to a limit-preserving functor Quad n : dSt op k → Cat ∞ . Proof. Immediate from Lurie's descent theorem and the description of Quad n (A) as a pullback of ∞-categories.
For any commutative algebra R ∈ Comm(k) and operad O in Mod(R), we have a functor Alg O (R ⊗ -) : Comm(k) → Cat ∞ (cf. Lemma 2.2.6). We now prove that this functor also satisfies descent; this is no doubt well-known to the experts -in particular, in the case of Lie algebras it is stated by Hennion as [Hen13, Proposition 2.1.3].
Proposition 4.1.4. Let R be a commutative algebra over k and let O be an operad in Mod(R). Then the functor Alg O (R ⊗ -) satisfiesétale descent, and so determines a limit-preserving functor where Pr L is the ∞-category of presentable ∞-categories and left adjoint functors.
Remark 4.1.5. It is easy to enhance this to get, for instance, a functor Opd(k)×dSt op k → Pr L , where Opd(k) is the ∞-category of operads in Mod(k), e.g. obtained by inverting the weak equivalences between (cofibrant) operads in Mod(k). To see this, observe that such a functor is equivalent to a functor Comm(k) → Fun(Opd(k), Pr L ) that satisfiesétale descent, which it does if and only if it does so when evaluated at each operad (since limits in functor ∞-categories are computed objectwise). The simplicial categories Alg (-) (-) are naturally pseudofunctorial in both variables, so they determine a pseudofunctor Opd(k) × Comm(k) → CAT ∆ . By Proposition 2.1.8(iii-iv), if we restrict to cofibrant objects of Comm(k) then this functor takes quasi-isomorphisms in both variables to weak equivalences of simplicial categories. Localizing, we obtained the required functor Opd(k) × Comm(k) → Pr L .
Remark 4.1.6. If we had a good theory of enriched ∞-operads, we would be able to formally identify Alg O (X), for X a derived stack, with the ∞-category of O-algebras in the ∞-category QCoh(X) of quasicoherent sheaves on X, regarded as enriched over k-modules.
For any derived stack X, we thus obtain natural functors and Q(X) : Quad 1 (X) → Mod OX (Alg E0 (X)) from our earlier work.
For the proof of this proposition, we need a technical result: Proposition 4.1.7. Let p : E → C ⊳ and q : F → C ⊳ be Cartesian and coCartesian fibrations. Suppose the functor φ : C ⊳ → Cat ∞ associated to q is a limit diagram. If a functor F : E → F over C ⊳ satisfies (a) the functor F preserves both Cartesian and coCartesian morphisms, (b) for every x ∈ C ⊳ , the functor F x : E x → F x detects equivalences and preserves limits, (c) the ∞-categories E x are complete for all x ∈ C ⊳ , then the functor ǫ : C ⊳ → Cat ∞ associated to p is also a limit diagram.
Proof. For c ∈ C, let e c denote the unique map −∞ → c in C ⊳ . By [Lur11, Lemma 5.17], we know that ǫ is a limit diagram if and only if the following conditions hold: (1) The functors e c,! : E −∞ → E c are jointly conservative, i.e. if f is a morphism in E −∞ such that e c,! f is an equivalence in E c for all c ∈ C, then f is an equivalence.
(2) If G : C → E is a coCartesian section of p over C, then G can be extended to a p-limit diagram G : C ⊳ → E, such thatḠ carries e c to a coCartesian morphism for all c ∈ C. Let us first prove (1). Suppose f is a morphism in E −∞ such that e c,! f is an equivalence in E c for all c ∈ C. Since F −∞ detects equivalences, to show that f is an equivalence it suffices to prove that F −∞ f is an equivalence. But as φ is a limit diagram, F −∞ f is an equivalence if and only if e c,! F −∞ f is an equivalence for all c ∈ C. And since F preserves coCartesian morphisms, we have natural equivalences e c,! F −∞ f ≃ F c e c,! f , hence these maps are indeed equivalences. Now we prove (2). The functor p is a Cartesian fibration, its fibres are complete, and the Cartesian pullback functors preserve limits since they are right adjoints. Therefore the p-limit of any diagram G exists by [Lur09a,Corollary 4.3.1.11]. Moreover, by [Lur09a, Proposition 4.3.1.10] we know that the limit is given by the limit in E −∞ of the Cartesian pullback of the diagram G to this fibre. Given the p-limit diagramḠ we are left with proving that the maps G(e c ) :Ḡ(−∞) → G(c) are all coCartesian, i.e. that the induced maps e c,!Ḡ (−∞) → G(c) are equivalences in E c . Since the functors F c detect equivalences, it suffices to show that the maps e c,! F −∞Ḡ (−∞) ≃ F c e c,!Ḡ (−∞) → F c G(c) are equivalences in F c , i.e. that the maps FḠ(e c ) are q-coCartesian. But since F −∞ preserves limits and F preserves Cartesian morphisms, we know that F −∞Ḡ (−∞) is the limit of the Cartesian pullback of F G to F −∞ , which is the q-limit of F • G. Since φ is a limit diagram, we know that (2) holds for q, i.e. that FḠ(e c ) is coCartesian for all c 4.2. Quantization over k. Over the field k, the linear quantization functor Q is particularly wellbehaved on quadratic k-modules that are non-degenerate and have finite-dimensional cohomology: the quantization has one-dimensional cohomology. In fact, we will see that for a cohomologically finite W , there is a numerical factor d W depending on the Betti numbers of W such that where T * [1]W denotes W ⊕ W * [1] with symplectic pairing given by the skew-symmetrization of the evaluation pairing. In other words, Q is a determinant-type functor, which illuminates one sense in which it provides a homological approach to integration, as the determinant of a vector space is the natural home for volume forms on it.
Every cochain complex is quasi-isomorphic to its cohomology, so it suffices to verify this invertibility property on such modules.
Lemma 4.2.1. Let V be a cochain complex with zero differential such that dim k V d < ∞ for all d and it vanishes for d ≫ 0 and d ≪ 0. Suppose V is equipped with a non-degenerate pairing ω : In short, where m = n (2n+1) dim k V 2n+1 is the index from the lemma. In particular, when V = T * [1]W = W ⊕ W * [1] with the natural pairing ω ev , we find that where d W = n (2n + 1)(dim k W 2n+1 − dim k W 2n ).
Proof. The vector space V is a direct sum of atomic components of the following form: Let V n denote the graded vector space with a copy of k in degree n and a copy of k in degree −1 − n, and the obvious pairing ω n . Denote the degree n generator by x and the dual generator in degree −1 − n by ξ. Then ω n (x, ξ) = 1. Observe that by unraveling the definitions. Without loss of generality, assume that ξ has odd degree. (Otherwise, just swap the labels on x and ξ.) Then compute that △(x a+1 ξ) = ±(a+1)x a , and so every monomial x a is exact and only the monomial ξ is closed. Hence when n is even. If x 1 , . . . , x N is a set of even degree basis elements for V and ξ 1 , . . . , ξ N the dual set of odd degree basis elements, then where m = i |ξ i |. In short, H * Q(V, ω) is isomorphic to the top exterior power (or determinant) of the odd-degree components of V , but placed in degree m.

Quantization over Formal Moduli Stacks.
The main result of this section is that this determinant-type behavior extends to formal moduli problems. We carefully state the result here and spend the rest of the section working through the proof.
To do this, we need to introduce the notion of a symplectic module. Following Lurie, a commutative algebra A is small if it sits in a finite sequence of algebras where each map A i → A i+1 is an elementary extension, i.e. sits in a pullback square f with n ≥ 0. Note that A is naturally augmented. The ∞-category Alg sm (k) of small algebras is the full subcategory of augmented connective commutative algebras Comm(k) ≤0 /k . (Lurie uses "small" where many people use "Artinian.") Definition 4.3.2. [Lur16, Chapter 13] A formal moduli problem is a functor X from Alg sm (k) to S such that (1) X(k) ≃ * and (2) given a pullback square σ of small algebras where φ is elementary, then its image X(σ) is a pullback square in S. Let Moduli denote the ∞-category of formal moduli problems over the field k. In other words, Q is a determinant-type functor on the symplectic modules over any formal moduli problem. To prove this theorem, it suffices to verify it on all small algebras.
By Lemma 4.2.1, we know the theorem holds on k, so now we need to extend to an arbitrary small commutative algebra. In the usual style of arguments in formal geometry, we will show that the relevant property -here, invertibility -plays nicely with an elementary extension. . We will show that an A-module X is invertible if and only if f ! X and g ! X are invertible; this will then imply the result by induction. The "only if" direction is obvious, since any strong symmetric monoidal functor preserves invertible objects.
To prove the other direction, we start with the pullback square of A-modules If we write HOM A for the internal Hom, then any A-module X yields a pullback square We have natural isomorphisms HOM A (X, f * B) ∼ = f * HOM B (f ! X, B), so writing D A X for the dual HOM A (X, A), and so on, we have a pullback square of A-modules Tensoring with X and applying the projection formula (i.e. the natural equivalences M ⊗ f * M ′ ≃ f * (f ! M ⊗ M ′ ), etc.), we finally get a pullback square Using the evaluation maps, we get a map of pullback squares from this square to the original pullback square from above (with A in the upper left corner). Hence the evaluation map X ⊗ A D A X → A is an equivalence if the evaluation maps for f ! X, g ! X and h ! X are equivalences, i.e. X is invertible if f ! X, g ! X and h ! X are. But h ! X is the image of f ! X (and of g ! X) under a strong symmetric monoidal functor, hence it is invertible if f ! X (or g ! X) is. This means that X is invertible if f ! X and g ! X are invertible, as required.
In consequence, we obtain the desired claim: Proof. If k ⊗ A Q(V, ω) is invertible over k, then Q(V, ω) is invertible over A. But the construction of quantization commutes with base-change, so which is invertible since the base-change of (V, ω) is a small symplectic module over k.

Symplectic Vector Bundles on Derived Stacks.
In this section, we examine how quantization behaves on derived Artin stacks. As the conditions we are considering are checked locally, our work in the preceding sections implies: Corollary 4.4.1. For V a 1-shifted symplectic module on a derived Artin stack X, the fiber of Q(V ) at any k-point p : Spec(k) → X is invertible. More generally, the pullback of Q(V ) to the formal moduli stack X ∧ p is invertible. Thus, we know that very locally -in the formal neighborhood of any k-point -quantization is well-behaved, but its behavior as the point varies is complicated. In particular, the quantization of a perfect symplectic module need not be invertible. Here is a particularly simple example: , where t has degree 0, and the two-term k[t]-module concentrated in degrees 0 and −1, which is perfect, and equipped with the natural shifted pairing from evaluation. Then the quantization is given by the complex (k[t, x, ξ], △ = tx∂/∂ξ + ∂ 2 /∂x∂ξ), which is concentrated in degree −1 after specializing to t = 0 and is concentrated in degree 0 after specializing t to any nonzero value. In other words, the quantization jumps at the origin. This result is not so strange as it may appear at first: this quadratic module presents the skyscraper sheaf at the origin and hence already jumps at the origin. By contrast, the determinant of this complex is just k[t] concentrated in degree 0.
Remark 4.4.3. This example underscores a key difference between the usual determinant functor and linear BV quantization: the determinant ignores the differential on the cochain complex, whereas BV quantization depends on the differential. (Note that the Euler characteristic also has the remarkable property that it does not depend on the differential, and the determinant is a kind of categorification.) Hence, we would like to restrict our attention to some collection of symplectic modules on which quantization does produce invertible modules.
Recall that the notion of being locally free of rank d is well-behaved in derived algebraic geometry in theétale topology. (See Section 2.9 of [Lur16].) Let VB denote the derived stack of vector bundles, i.e. modules that are locally free of some finite rank, which clearly admits a natural map to Mod. Let GVB denote the derived stack of graded vector bundles, which consists of modules that are finite direct sums of shifts of vector bundles. We now restrict to a collection of 1-shifted symplectic vector bundles on which Q is well-behaved.  Proof. This can be checked locally on affines. By [Lur16, Proposition 2.9.2.3], we can reduce to the case where the graded vector bundle is a direct sum of shifted free modules. In this free case, working over a commutative algebra A, the cotangent quantization is equivalent to the module where ∆ is determined by the pairing. The argument from Lemma 4.2.1 applies here, suitably interpreted. (In fact, one can view this complex as base-changed from a complex over k.) Hence, there is a natural A-linear quasi-isomorphism with b m the number of degree m generators of the graded vector bundle V .
Remark 4.4.7. This construction applies to any 1-shifted symplectic module V for which, sufficiently locally, the module is free and the pairing has the standard form. In other words, it applies to any 1-shifted symplectic module that is locally a shifted cotangent space. It seems plausible that all 1-shifted symplectic modules have this form, but we do not pursue a classification of shifted symplectic vector bundles here.

4.5.
Higher BV Quantization. The paper [Hau14] constructs for any nice symmetric monoidal ∞-category C a symmetric monoidal (∞, n)-category Alg n (C), whose objects are E n -algebras in C and whose i-morphisms are i-fold iterated bimodules in E n−i -algebras in C. Here the precise meaning of "nice" holds in particular if C has sifted colimits and the tensor product preserves these in each variable. Moreover, the construction is natural in C with respect to symmetric monoidal functors that preserve sifted colimits. We will now show that these assumptions hold for the ∞-categories and functors involved in our linear BV quantization, giving: • for each object X ∈ C, a natural isomorphism u X : F (id X ) ⇒ id F (X) • for each pair of composable morphisms f : X → Y , g : Y → Z in C, a natural isomorphism η f,g : such that • for every morphism f : X → Y , the triangles both commute. • for composable triples of morphisms f : X → Y , g : Y → Z, h : Z → W , the square commutes.
Proposition A.1.1. Suppose C : op → CAT is a pseudofunctor (where we write C n for the image of [n] and φ * : C n → C m for the image of φ : [m] → [n] in ). Then C determines a simplicial category C ∆ as follows: • the objects of C ∆ are the objects of C 0 , • for x, y in C 0 the n-simplices in C ∆ are given by C n (σ * n x, σ * n y), where σ n denotes the unique map [n] → [0] in . • for φ : [m] → [n] in , the corresponding simplicial structure map in C ∆ (x, y), which we denotẽ φ * takes f : σ * n x → σ * n y to the composite −−−→ (σ n • φ)y = σ * m y. Proof. To see that this does indeed define a simplicial set, we must check that these maps respect composition and identities. For composition, take ψ : [k] → [m] and φ : [m] → [n] and consider for f : σ * n x → σ * n y the commutative diagram where the unlabelled maps come from the natural isomorphisms η. Here the top horizontal composite is ( ψφ) * f , the bottom horizontal composite is ψ * φ * f , and the composite from the top left to the top right along the bottom row isψ * φ * f . For identities, consider for f as above the commutative diagram σ * n x (id [n] ) * σ * n x (id [n] ) * σ * n y σ * n y σ * n x σ * n y. Here the top horizontal composite is id [n] * f and the composite along the bottom is f . It is then clear that composition in the categories C n induces composition maps for C ∆ .
A natural transformation λ from F to G of pseudofunctors from C to CAT consists of the data of: • for every X ∈ C a functor λ X : F (X) → G(X), • for every morphism f : satisfying the obvious pentagon and triangle identities.
A modification ǫ from λ to µ of natural transformations from F to G of pseudofunctors from C to CAT consists of the data of: • for every object X ∈ C, a natural transformation ǫ X : λ X ⇒ µ X , These give, respectively, the 1-and 2-morphisms in a 2-category Fun Ps (C, CAT) of pseudofunctors. It is easy to see that the construction taking a pseudofunctor C : op → CAT to the simplicial category C ∆ is natural with respect to these morphisms and 2-morphisms, giving: Corollary A.1.2. The construction (C : op → CAT) → C ∆ extends naturally to a functor of 2-categories Fun Ps ( op , CAT) → CAT ∆ , where CAT ∆ is the 2-category of simplicial categories.
We leave the (straightforward) details of the proof to the reader.
A.2. Simplicial Enrichment of Model Categories. The ∞-categories we work with in this paper mostly arise from model categories, and in order to carry out our construction we want to show that these ∞-categories can also be described using natural simplicial enrichments of these model categories. For simplicial model categories this is a standard result, originally due to Dwyer and Kan [DK80b]. However, in our case the model categories are not quite simplicial in the usual sense, so we will need a slight variant of the comparison of Dwyer-Kan; this result is no doubt well-known to the experts, but we have included a proof in this appendix as we were not able to find a reference in the literature.
We begin by recalling how we construct the ∞-category associated to a model category, or more generally to a relative category. A relative category (C, W ) is a category C equipped with a collection W of "weak equivalences", i.e. a collection of morphisms in C that contains all the isomorphisms and satisies the 2-out-of-3 property: if f and g are composable morphisms such that two out of f , g, and gf are in W , then so is the third. Relative categories are the most basic form of homotopical data on a category, and have been studied as a model for ∞-categories by Barwick and Kan [BK12]. If (C, W ) is a relative category, we can invert the weak equivalences W to obtain an ∞-category C[W −1 ]: Definition A.2.1. Let -: Cat ∞ → S be the left adjoint to the inclusion S ֒→ Cat ∞ of the ∞category S of spaces into that of ∞-categories. Thus if C is an ∞-category, C is the space or ∞-groupoid obtained by inverting all the morphisms in C. If (C, W ) is a relative category, let W denote the subcategory of C containing only the morphisms in W . Then the localization C[W −1 ] is defined by the pushout square of ∞-categories where W is the subcategory of C containing only the morphisms in W .
We are interested in simplicial enrichments that arise in a particularly pleasant way, as follows: Suppose T • is a simplicial monad on a category C, i.e. a simplicial object in the category of monads, or more explicitly a collection of functors T n : C → C together with natural transformations µ n : T n • T n → T n (the multiplication) and η n : id → T n (the unit) satisfying the usual identities, as well as natural transformations φ * : T n → T m for every map φ : [m] → [n] in , compatible with the multiplication and unit transformations. Then we can define a simplicial category C ∆ with the same objects as C, where the mapping space Map C∆ (X, Y ) is given by Hom C (X, T • Y ). The identity map of X corresponds to the unit X → T 0 X, and the composition Map C∆ (X, Y ) × Map C∆ (Y, Z) → Map C∆ (X, Z) is given on simplices by taking f : X → T n Y and g : Y → T n Z to the composite In other words, if we let C n be the Kleisli category of the monad T n , then C • is a simplicial object in categories with a constant set of objects, and C ∆ is the associated simplicial category. Note that there is a canonical functor C → C ∆ that is the identity on objects and sends a map Remark A.2.3. Our notation is slightly abusive, in that the underlying category of the simplicial category C ∆ is not in general C -a morphism X → Y in (C ∆ ) 0 corresponds to a morphism X → T 0 Y in C. However, the monad T 0 is typically the identity in examples.
We'll now use results of Hovey to give conditions for a simplicial monad to interact well with the weak equivalences in a model category: Definition A.2.4. Let C be a model category. A coherent right framing on C is a simplicial monad T • on C such that for every object X of C the unit maps X → T n X are weak equivalences for all n, and if X is fibrant then T • X is a Reedy fibrant simplicial object of C.
Proposition A.2.5 (Hovey). Suppose C is a model category equipped with a coherent right framing T • . Then: (i) If Y is a fibrant object of C, then the functor Hom C (-, T • Y ) is a right Quillen functor from C op to Set ∆ , i.e. it takes cofibrations and trivial cofibrations in C to Kan fibrations and trivial Kan fibrations. In particular, if X is cofibrant, then Hom C (X, T • Y ) is a Kan complex, and Hom C (-, T • Y ) preserves weak equivalences between cofibrant objects. (ii) If X is a cofibrant object of C, then Hom C (X, T • -) preserves weak equivalences between fibrant objects in C.
Proof. From the definition of a coherent right framing it is evident that for Y fibrant the simplicial object T • Y is a simplicial framing of Y in the sense of [ Corollary A.2.6. Suppose C is a model category equipped with a coherent right framing T • . Let C cf ∆ be the simplicial category of fibrant-cofibrant objects in C with mapping spaces Hom C (-, T • -). Then C cf ∆ is a fibrant simplicial category. Proof. By Proposition A.2.5(i) the mapping spaces in C cf ∆ are all Kan complexes, so it is fibrant as a simplicial category.