Braces

Let ${\mathcal{O}}$ be a dg operad and ${\mathcal{C}}$ its Koszul dual cooperad which is assumed to be Hopf. Following Tamarkin’s work [Reference TamarkinTam00] on the deformation complex of a $\mathbb{P}_{n}$ -algebra, Calaque and Willwacher [Reference Calaque and WillwacherCW15] introduced an operad $\text{Br}_{{\mathcal{C}}}$ of brace algebras which acts on the deformation complex of any homotopy ${\mathcal{O}}$ -algebra. Moreover, they have remarked that the brace construction is an analog of the Boardman–Vogt tensor product $\mathbb{E}_{1}\otimes ~\unicode[STIX]{x1D6FA}{\mathcal{C}}$ of operads. This is suggested by the following examples:

1. – if $\mathbf{1}$ is the trivial cooperad, $\text{Br}_{\mathbf{1}}\cong \mathbb{E}_{1}$ ;

2. – if $\text{coAss}$ is the cooperad of coassociative coalgebras, $\text{Br}_{\text{coAss}}\{1\}\cong \mathbb{E}_{2}$ ;

3. – if $\text{coComm}$ is the cooperad of cocommutative coalgebras, $\text{Br}_{\text{coComm}}\cong \text{Lie}$ ;

4. – if $\text{co}\mathbb{P}_{n}$ is the cooperad of $\mathbb{P}_{n}$ -coalgebras, $\text{Br}_{\text{co}\mathbb{P}_{n}}\{n\}\cong \mathbb{P}_{n+1}$ .

In this paper we explain to what extent this is true. Namely, suppose ${\mathcal{C}}$ is a Hopf cooperad satisfying a minor technical assumption. We construct a functor of $\infty$ -categories

from the $\infty$ -category of $\text{Br}_{{\mathcal{C}}}$ -algebras to the $\infty$ -category of associative algebras in the $\infty$ -category of ${\mathcal{O}}$ -algebras. Let us note that we did not assume that ${\mathcal{O}}$ is a Hopf operad and the symmetric monoidal structure on ${\mathcal{A}}\text{lg}_{{\mathcal{O}}}$ comes from the Koszul dual side.

Unfortunately, we do not know if (1) is an equivalence in general, but we do show that it is an equivalence in two examples of interest: namely, Lie algebras and Poisson algebras.

Suppose ${\mathcal{C}}=\text{coComm}$ . As we have mentioned, $\text{Br}_{\text{coComm}}\cong \text{Lie}$ and so we get a functor

We show that it is an equivalence and in fact coincides with the functor which sends a Lie algebra $\mathfrak{g}$ to the associative algebra object in the category of Lie algebras $0\times _{\mathfrak{g}}0$ (see Proposition 2.13). We also show that the same functor can be constructed as follows. Given a Lie algebra $\mathfrak{g}$ , the universal enveloping algebra $\text{U}(\mathfrak{g})$ is a cocommutative bialgebra, i.e. an associative algebra object in cocommutative coalgebras. Identifying cocommutative coalgebras with Lie algebras using Koszul duality we obtain the same functor (see Proposition 2.11). Let us mention that the underlying Lie algebra structure on $\text{add}(\mathfrak{g})$ is canonically trivial by Proposition 2.15.

Note that $\text{Br}_{\text{coComm}}$ is an important operad in itself and appears for instance in the description of the Atiyah bracket of vector fields, see § 2.1.1.

Poisson additivity

The additivity functor is more interesting in the case of $\mathbb{P}_{n+1}$ -algebras. So, take ${\mathcal{C}}=\text{co}\mathbb{P}_{n}$ . Since $\text{Br}_{\text{co}\mathbb{P}_{n}}\{n\}\cong \mathbb{P}_{n+1}$ , we obtain a functor

The following statement combines Propositions 2.19–2.21 and Theorem 2.22.

Rozenblyum has given an independent proof of this result in the language of factorization algebras. This statement is a Poisson version of the additivity theorem [Reference LurieLur17, Theorem 5.1.2.2] for $\mathbb{E}_{n}$ -algebras proved by Dunn and Lurie: one has an equivalence

of symmetric monoidal $\infty$ -categories, where $\mathbb{E}_{n}$ is the operad of little $n$ -disks.

One has the following explicit description of the additivity functor for Poisson algebras which uses some ideas of Tamarkin (see [Reference TamarkinTam00] and [Reference TamarkinTam07]). For simplicity, we describe the construction in the case of non-unital $\mathbb{P}_{n+1}$ -algebras. In the case of unital $\mathbb{P}_{n+1}$ -algebras one has to take care of the natural curving appearing on the Koszul dual side, but otherwise the construction is identical (see § 2.5). If $A$ is a commutative algebra, we can consider its Harrison complex $\text{coLie}(A[1])$ which is a Lie coalgebra. If $A$ is moreover a $\mathbb{P}_{n+1}$ -algebra, then the Harrison complex  $\text{coLie}(A[1])[n-1]$ has a natural structure of an $(n-1)$ -shifted Lie bialgebra (Definition 2.16) which defines a functor

Given a Lie algebra $\mathfrak{g}$ , its universal enveloping algebra $\text{U}(\mathfrak{g})$ is a cocommutative bialgebra. If $\mathfrak{g}$ is moreover an $(n-1)$ -shifted Lie bialgebra, then $\text{U}(\mathfrak{g})$ acquires a natural cobracket making it into an associative algebra object in $\mathbb{P}_{n}$ -coalgebras. Thus, we get a functor

Applying Koszul duality we identify $\text{CoAlg}_{\text{co}\mathbb{P}_{n}}$ with the category of $\mathbb{P}_{n}$ -algebras thus giving the required additivity functor.

An important point we have neglected in this discussion is that at the very end one has to pass from the localization of the category of associative algebras in $\mathbb{P}_{n}$ -coalgebras to the $\infty$ -category of (homotopy) associative algebras in the localization of the category of $\mathbb{P}_{n}$ -coalgebras. That is, we have a natural functor

where $W_{\text{Kos}}$ is a certain natural class of weak equivalences we define in the paper, $\text{Alg}$ is the 1-category of associative algebras and ${\mathcal{A}}\text{lg}$ is the $\infty$ -category of (homotopy) associative algebras. The fact that this functor is an equivalence is not automatic: the corresponding rectification statement was proved in [Reference LurieLur17, Theorem 4.1.8.4] under the assumption that the model category in question is a monoidal model category while the monoidal structure on $\text{CoAlg}_{\text{co}\mathbb{P}_{n}}$ does not even preserve colimits. Furthermore, if we do not pass to the Koszul dual side, the localization functor (where $W_{\text{qis}}$ is the class of quasi-isomorphisms)

is not an equivalence. Indeed, $\text{Alg}(\text{Alg}_{\mathbb{P}_{n}})$ is equivalent to the category of commutative algebras while we prove that ${\mathcal{A}}\text{lg}(\text{Alg}_{\mathbb{P}_{n}}[W_{\text{qis}}^{-1}])$ is equivalent to the $\infty$ -category of $\mathbb{P}_{n+1}$ -algebras.

Let us note that the underlying commutative structure on $\text{add}(A)$ for a $\mathbb{P}_{n+1}$ -algebra $A$ coincides with the commutative structure on $A$ and the underlying Lie structure on $\text{add}(A)$ is trivial. However, the underlying $\mathbb{P}_{n}$ -structure on $\text{add}(A)$ is not necessarily commutative.

One motivation for developing Poisson additivity is the recent work of Costello and Gwilliam [Reference Costello and GwilliamCG16] that formalizes algebras of observables in quantum field theories. In that work a topological quantum field theory is described by a locally constant factorization algebra on the spacetime manifold valued in $\mathbb{E}_{0}$ -algebras. Since locally constant factorization algebras on $\mathbb{R}^{n}$ are the same as $\mathbb{E}_{n}$ -algebras, we see that observables in an $n$ -dimensional topological quantum field theory are described by $\mathbb{E}_{n}\otimes \mathbb{E}_{0}=\mathbb{E}_{n}$ -algebras. Similarly, classical topological field theories are described by locally constant factorization algebras valued in $\mathbb{P}_{0}$ -algebras, which in the case of $\mathbb{R}^{n}$ are the same as $\mathbb{E}_{n}$ -algebras in $\mathbb{P}_{0}$ -algebras. Our result thus shows that observables in an $n$ -dimensional classical topological field theory are described by a $\mathbb{P}_{n}$ -algebra (a natural result one expects by extrapolating from the case of topological quantum mechanics which is $n=1$ ).

Coisotropic structures

Another motivation is given by the theory of shifted Poisson geometry developed by Calaque–Pantev–Toën–Vaquié–Vezzosi and, more precisely, derived coisotropic structures. Recall that an $n$ -shifted Poisson structure on an affine scheme $\operatorname{Spec}A$ for $A$ a commutative dg algebra is described by a $\mathbb{P}_{n+1}$ -algebra structure on $A$ . Now suppose $f:\operatorname{Spec}B\rightarrow \operatorname{Spec}A$ is a morphism of affine schemes. In [Reference Calaque, Pantev, Toën, Vaquié and VezzosiCPTVV17] the following notion of derived coisotropic structures was introduced. Assume the statement of Poisson additivity. Then one can realize $A$ as an associative algebra in $\mathbb{P}_{n}$ -algebras and a coisotropic structure on $f$ is a lift of the natural action of $A$ on $B$ in commutative algebras to $\mathbb{P}_{n}$ -algebras. Let us denote by $\text{Cois}^{\text{CPTVV}}(f,n)$ the space of such coisotropic structures.

A more explicit definition of derived coisotropic structures was given in [Reference SafronovSaf17] and [Reference Melani and SafronovMS16] which does not rely on Poisson additivity. An action of a $\mathbb{P}_{n+1}$ -algebra $A$ on a $\mathbb{P}_{n}$ -algebra $B$ was modeled by a certain colored operad $\mathbb{P}_{[n+1,n]}$ and a derived coisotropic structure was defined to be the lift of the natural action of $A$ on $B$ in commutative algebras to an algebra over the operad $\mathbb{P}_{[n+1,n]}$ . Let us denote by $\text{Cois}^{\text{MS}}(f,n)$ the space of such coisotropic structures.

In this paper we show that these two notions coincide. The following statement is Corollary 3.9.

This statement is proved by developing a relative analog of the Poisson additivity functor. Namely, Theorem 3.8 asserts that the $\infty$ -category of $\mathbb{P}_{[n+1,n]}$ -algebras is equivalent to the $\infty$ -category of pairs $(A,M)$ , where $A$ is an associative algebra and $M$ is an $A$ -module in the $\infty$ -category of $\mathbb{P}_{n}$ -algebras.

Notations

1. – Given a relative category $({\mathcal{C}},W)$ we denote by ${\mathcal{C}}[W^{-1}]$ the underlying $\infty$ -category.

2. – We work over a field $k$ of characteristic zero; $\text{Ch}$ denotes the category of chain complexes of $k$ -modules and ${\mathcal{C}}\text{h}$ the underlying $\infty$ -category.

3. – Given a topological operad ${\mathcal{O}}$ , we denote by $\text{Alg}_{{\mathcal{O}}}({\mathcal{C}})$ the category of ${\mathcal{O}}$ -algebras in a symmetric monoidal category ${\mathcal{C}}$ and by ${\mathcal{A}}\text{lg}_{{\mathcal{O}}}({\mathcal{C}})$ the $\infty$ -category of ${\mathcal{O}}$ -algebras in a symmetric monoidal $\infty$ -category ${\mathcal{C}}$ . If ${\mathcal{O}}$ is a dg operad, the category of ${\mathcal{O}}$ -algebras in complexes is simply denoted by $\text{Alg}_{{\mathcal{O}}}$ .

4. – All operads are non-unital unless specified otherwise. We denote by ${\mathcal{O}}^{\text{un}}$ the operad of unital ${\mathcal{O}}$ -algebras.

5. – All non-counital coalgebras are conilpotent.

1.1 Relative categories

In the paper we will extensively use relations between relative categories and $\infty$ -categories, so let us recall the necessary facts.

We will call morphisms belonging to $W$ weak equivalences. A functor of relative categories $({\mathcal{C}},W_{{\mathcal{C}}})\rightarrow ({\mathcal{D}},W_{{\mathcal{D}}})$ is a functor that preserves weak equivalences.

Recall that given a category ${\mathcal{C}}$ its nerve $\text{N}({\mathcal{C}})$ is an $\infty$ -category. Similarly, if ${\mathcal{C}}$ is a relative category, the nerve $\text{N}({\mathcal{C}})$ is an $\infty$ -category equipped with a system of morphisms $W$ and we introduce the notation

where the localization functor on the right is defined in [Reference LurieLur17, Proposition 4.1.7.2]. In particular, ${\mathcal{C}}[W^{-1}]$ is an $\infty$ -category which we call the underlying $\infty$ -category of the relative category $({\mathcal{C}},W)$ .

We will also need a construction of symmetric monoidal $\infty$ -categories from ordinary symmetric monoidal categories. We say that ${\mathcal{C}}$ is a relative symmetric monoidal category if the functor $x\otimes -:{\mathcal{C}}\rightarrow {\mathcal{C}}$ preserves weak equivalences for every object $x\in {\mathcal{C}}$ .

For instance, let $\text{Ch}$ be the symmetric monoidal category of chain complexes of $k$ -vector spaces. Let $W_{\text{qis}}\subset \text{Ch}$ be the class of quasi-isomorphisms. Since $M\otimes -$ preserves quasi-isomorphisms for any $M\in \text{Ch}$ , we obtain a natural symmetric monoidal structure on the $\infty$ -category

of chain complexes.

We will repeatedly use the following method to prove that a functor between $\infty$ -categories is an equivalence. Consider a commutative diagram of $\infty$ -categories.

Assuming $G_{1}$ and $G_{2}$ have left adjoints $G_{1}^{L}$ and $G_{2}^{L}$ respectively, we obtain a natural transformation $G_{2}^{L}\rightarrow FG_{1}^{L}$ of functors ${\mathcal{D}}\rightarrow {\mathcal{C}}_{2}$ . We say that the original diagram satisfies the left Beck–Chevalley condition if this natural transformation is an equivalence. The following is a corollary of the $\infty$ -categorical version of the Barr–Beck theorem proved in [Reference LurieLur17, Corollary 4.7.3.16].

1.2 Operads

Our definitions and notations for operads follows those of Loday and Vallette [Reference Loday and ValletteLV12]. Unless specified otherwise, by an operad we mean an operad in chain complexes.

Given a symmetric sequence $V$ , we define the shift $V[n]$ to be the symmetric sequence with

Let $\text{sgn}_{m}$ be the one-dimensional sign representation of $S_{m}$ . We will also use the notation $V\{n\}$ to denote the symmetric sequence with

If $V$ is an operad or a cooperad, so is $V\{n\}$ .

Let $\mathbf{1}$ be the trivial operad. Recall that an augmentation on an operad ${\mathcal{O}}$ is a morphism of operads

In particular, one obtains a splitting of symmetric sequences

Similarly, one has a notion of a coaugmentation on a cooperad ${\mathcal{C}}$ .

Given an augmented operad ${\mathcal{O}}$ , its bar construction $\text{B}{\mathcal{O}}$ is defined to be the cofree cooperad on $\overline{{\mathcal{O}}}[1]$ equipped with the bar differential which consists of two terms: one coming from the differential on ${\mathcal{O}}$ and one coming from the product on ${\mathcal{O}}$ . Similarly, given a coaugmented cooperad ${\mathcal{C}}$ we have its cobar construction $\unicode[STIX]{x1D6FA}{\mathcal{C}}$ . We refer to [Reference Loday and ValletteLV12, § 6.5] for details. In particular, one has a quasi-isomorphism of operads

Most of the operads of interest that control non-unital algebras satisfy ${\mathcal{O}}(0)=0$ and ${\mathcal{O}}(1)\cong k$ and hence possess a unique augmentation. However, operads controlling unital algebras tend not to have an augmentation, so, following Hirsh and Millès [Reference Hirsh and MillèsHM12], we relax the condition a bit.

Given a semi-augmented operad ${\mathcal{O}}$ , one can still consider the bar construction $\text{B}{\mathcal{O}}$ , but the corresponding differential no longer squares to zero. Instead, we obtain a curved cooperad equipped with a curving $\unicode[STIX]{x1D703}:{\mathcal{C}}(1)\rightarrow k[2]$ (see [Reference Hirsh and MillèsHM12, Definition 3.2.1] for a complete definition).

We refer to [Reference Hirsh and MillèsHM12, § 3.3] for explicit formulas for the differential and the curving on the bar construction of a semi-augmented operad. Moreover, it is also shown there that the cobar construction $\unicode[STIX]{x1D6FA}{\mathcal{C}}$ on a coaugmented curved cooperad ${\mathcal{C}}$ is a dg operad equipped with a natural semi-augmentation.

Finally, we refer to [Reference Loday and ValletteLV12, § 7] for Koszul duality for augmented operads and to [Reference Hirsh and MillèsHM12, § 4] for Koszul duality for semi-augmented operads. The important point that we will use in the paper is that the Koszul dual cooperad ${\mathcal{C}}$ of ${\mathcal{O}}$ is naturally equipped with a quasi-isomorphism

which gives a semi-free resolution of the operad ${\mathcal{O}}$ . Such a quasi-isomorphism is equivalently given by a degree 1 (curved) Koszul twisting morphism ${\mathcal{C}}\rightarrow {\mathcal{O}}$ .

1.3 Operadic algebras

Given an operad ${\mathcal{O}}$ we denote by $\text{Alg}_{{\mathcal{O}}}$ the category of ${\mathcal{O}}$ -algebras in chain complexes. Similarly, for a cooperad ${\mathcal{C}}$ we denote by $\text{CoAlg}_{{\mathcal{C}}}$ the category of conilpotent ${\mathcal{C}}$ -coalgebras. To simplify the notation, we let

be the category of unital associative algebras.

If ${\mathcal{C}}$ is a curved cooperad, we denote by $\text{CoAlg}_{{\mathcal{C}}}$ the category of curved conilpotent ${\mathcal{C}}$ -coalgebras (see [Reference Hirsh and MillèsHM12, Definition 5.2.1]) which are cofibrant. Note that morphisms strictly preserve the differential.

If $A$ is an ${\mathcal{O}}$ -algebra, then $A[-n]$ is an ${\mathcal{O}}\{n\}$ -algebra; similarly, if $C$ is a ${\mathcal{C}}$ -coalgebra, then $C[-n]$ is a ${\mathcal{C}}\{n\}$ -coalgebra.

Now consider a (curved) cooperad ${\mathcal{C}}$ equipped with a (curved) Koszul twisting morphism ${\mathcal{C}}\rightarrow {\mathcal{O}}$ . Given an ${\mathcal{O}}$ -algebra $A$ we define its bar construction to be

equipped with the bar differential (see [Reference Hirsh and MillèsHM12, § 5.2.3]). Given a curved ${\mathcal{C}}$ -coalgebra $C$ we define its cobar construction to be

equipped with the cobar differential (see [Reference Hirsh and MillèsHM12, § 5.2.5]). Note that the cobar differential squares to zero. In particular, we get a bar-cobar adjunction

such that for any ${\mathcal{O}}$ -algebra $A$ the natural projection

is a quasi-isomorphism.

Let us denote by $W_{\text{qis}}\subset \text{Alg}_{{\mathcal{O}}}$ the class of morphisms of ${\mathcal{O}}$ -algebras which are quasi-isomorphisms of the underlying complexes. Let us also denote by $W_{\text{Kos}}\subset \text{CoAlg}_{{\mathcal{C}}}$ the class of morphisms of (curved) ${\mathcal{C}}$ -coalgebras which become quasi-isomorphisms after applying the cobar functor $\unicode[STIX]{x1D6FA}$ . The class of weak equivalences $W_{\text{Kos}}$ is independent of the choice of the operad ${\mathcal{O}}$ as shown by the following statement. Let us denote by $\unicode[STIX]{x1D6FA}_{{\mathcal{O}}}:\text{CoAlg}_{{\mathcal{C}}}\rightarrow \text{Alg}_{{\mathcal{O}}}$ the cobar construction associated to the operad ${\mathcal{O}}$ .

We introduce the notation

for the $\infty$ -category of ${\mathcal{O}}$ -algebras and by the previous proposition it can also be modeled by the relative category $(\text{CoAlg}_{{\mathcal{C}}},W_{\text{Kos}})$ .

Suppose ${\mathcal{O}}_{1}\rightarrow {\mathcal{O}}_{2}$ is a morphism of operads which is a quasi-isomorphism in each arity. The forgetful functor

automatically preserves quasi-isomorphisms and hence induces a functor on the level of $\infty$ -categories.

Finally, recall that the forgetful functor $\text{Alg}_{{\mathcal{O}}}\rightarrow \text{Ch}$ creates sifted colimits since the category $\text{Alg}_{{\mathcal{O}}}$ can be written as the category of algebras over a monad which preserves sifted colimits. The same statement is true on the level of $\infty$ -categories.

1.4 Examples

Let us show how the bar-cobar duality works for unital algebras over the associative and Poisson operads to compare it to the classical bar-cobar duality.

Figure 1.

Generating operations of $\text{Ass}^{\text{un}}$ .

We begin with the case of the associative operad considered in [Reference Hirsh and MillèsHM12, § 6]. Let ${\mathcal{O}}=\text{Ass}^{\text{un}}$ be the operad governing unital associative algebras. The operad $\text{Ass}^{\text{un}}$ is quadratic and is generated by the symmetric sequence $V$ with $V(0)=k$ and $V(2)=k[S_{2}]$ whose elements are shown on Figure 1.

Figure 2.

Relations in $\text{Ass}^{\text{un}}$ .

The relations in $\text{Ass}^{\text{un}}$ have the form shown on Figure 2. This gives an inhomogeneous quadratic-linear-constant presentation of the operad $\text{Ass}^{\text{un}}$ . Given such an operad ${\mathcal{O}}$ , we denote by $q{\mathcal{O}}$ the operad with the same generators and where we only keep the quadratic part of the relations. Recall that the underlying graded cooperad of the Koszul dual is defined from the quadratic part of the relations, the differential uses the linear part and the curving comes from the constant part. In the relations we have there are no linear terms, so the Koszul dual cooperad coincides with the Koszul dual cooperad of the quadratic operad $q\text{Ass}^{\text{un}}$ equipped with a curving. From the relations we see that

where $\mathbb{E}_{0}$ is the operad governing complexes together with a distinguished vector and where $\oplus$ refers to the product in the category of operads.

Figure 3.

Curving on (Ass^un)^¡.

By [Reference Hirsh and MillèsHM12, Proposition 6.1.4] the Koszul dual of $q\text{Ass}^{\text{un}}$ is given by

where $\text{co}\mathbb{E}_{0}\cong \mathbb{E}_{0}$ is the cooperad of complexes together with a functional and $\oplus$ now denotes the product in the category of conilpotent cooperads, i.e. the conilpotent cooperad cogenerated by Ass^¡ and $\text{co}\mathbb{E}_{0}$ . We define

as graded cooperads. The degree $-2$ part of (Ass^un)^¡(1) is the two-dimensional vector space spanned by trees shown in Figure 3 and we define the curving 𝜃: (Ass^un)^¡(1)→k[2] to take value $-1$ on both of these. Let us denote

The cooperad $\text{coAss}^{\unicode[STIX]{x1D703}}$ governs coassociative coalgebras $C$ together with a coderivation $d:C\rightarrow C$ of degree $1$ and a curving $\unicode[STIX]{x1D703}:C\rightarrow k[2]$ satisfying the equations

where

Given a unital associative dg algebra $A$ , its bar complex is

equipped with the following differential. Let us denote elements of the bar complex by $[x_{1}|\cdots |x_{n}]$ with elements of $k[2]$ denoted by $\ast$ . Then

with $d\ast =1\in A$ and $\ast \cdot x=x\cdot \ast =0$ . The curving is given by $\unicode[STIX]{x1D703}([\ast ])=1$ .

Similarly, given a curved coalgebra $C$ , its cobar complex is

whose elements we denote by $[x^{1}|\cdots |x^{n}]$ for $x^{i}\in C$ such that the differential is

Let us similarly work out the Koszul dual of the operad of unital $\mathbb{P}_{n}$ -algebras. Recall that a $\mathbb{P}_{n}$ -algebra is a dg Poisson algebra whose Poisson bracket has degree $1-n$ . We denote by ${\mathcal{O}}=\mathbb{P}_{n}^{\text{un}}$ the operad controlling such algebras. It is generated by the symmetric sequence $V$ with $V(0)=k\cdot u$ and $V(2)=k\cdot m\oplus \text{sgn}_{2}^{\otimes n}\cdot \{\}$ with the following relations:

In particular, we see again that

Note that the Koszul dual of the operad of non-unital $\mathbb{P}_{n}$ -algebras is  Therefore, by [Reference Hirsh and MillèsHM12, Proposition 6.1.4] the Koszul dual of $\mathbb{P}_{n}^{\text{un}}$ is

equipped with the curving 𝜃: (ℙ _n ^un)^¡(1)→k[2] which sends the tree

to $-1$ , where $\unicode[STIX]{x1D6FF}$ is the cobracket in $\text{co}\mathbb{P}_{n}\{n\}$ . We denote

A curved coalgebra $C$ over the cooperad $\text{co}\mathbb{P}_{n}^{\unicode[STIX]{x1D703}}$ is given by the following data:

1. – a cocommutative comultiplication on $C$ ;

2. – a cobracket $\unicode[STIX]{x1D6FF}:C\rightarrow C\otimes C[1-n]$ for which we use Sweedler’s notation

   $$\begin{eqnarray}\unicode[STIX]{x1D6FF}(x)=x_{(1)}^{\unicode[STIX]{x1D6FF}}\otimes x_{(2)}^{\unicode[STIX]{x1D6FF}},\end{eqnarray}$$
   which satisfies the coalgebraic version of the Jacobi identity;

3. – a coderivation $d:C\rightarrow C$ of degree $1$ ;

4. – a curving $\unicode[STIX]{x1D703}:C\rightarrow k[1+n]$ .

Together these satisfy the relations

1.5 Hopf operads

Recall that a Hopf operad is an operad in counital cocommutative coalgebras. Dually, a Hopf cooperad is a cooperad in unital commutative algebras. Alternatively, recall that the category of symmetric sequences has two tensor structures: the composition product which is merely monoidal and which we use to define operads and cooperads and the Hadamard product which is symmetric monoidal. The Hadamard tensor product defines a symmetric monoidal structure on the category of cooperads and one can define a Hopf cooperad to be a unital commutative algebra in the category of cooperads.

One can similarly define a notion of a curved Hopf cooperad to be a unital commutative algebra in the category of curved cooperads.

Given a Hopf operad ${\mathcal{O}}$ and two ${\mathcal{O}}$ -algebras $A_{1},A_{2}$ the tensor product of the underlying complexes is also an ${\mathcal{O}}$ -algebra using

Dually, one defines the tensor product of two (curved or dg) ${\mathcal{C}}$ -coalgebras for a Hopf cooperad ${\mathcal{C}}$ to be the tensor product of the underlying chain complexes.

The operads $\text{Ass}^{\text{un}}$ and $\mathbb{P}_{n}^{\text{un}}$ we are interested in are Hopf operads. For instance, for $\mathbb{P}_{n}^{\text{un}}$ we have

By duality we get Hopf cooperad structures on $\text{coAss}^{\text{cu}}=(\text{Ass}^{\text{un}})^{\ast }$ and $\text{co}\mathbb{P}_{n}^{\text{cu}}=(\mathbb{P}_{n}^{\text{un}})^{\ast }$ , the cooperads of counital coassociative coalgebras and counital $\mathbb{P}_{n}$ -coalgebras respectively.

Note, however, that the curved cooperad $\text{coAss}^{\unicode[STIX]{x1D703}}$ admits no Hopf structure. Indeed, the degree zero part of $\text{coAss}^{\unicode[STIX]{x1D703}}(0)$ is trivial and hence one cannot define a unit. To remedy this problem, we introduce the following modification. Given an operad ${\mathcal{O}}$ we denote by ${\mathcal{O}}^{\text{un}}={\mathcal{O}}\oplus k$ the symmetric sequence which coincides with ${\mathcal{O}}$ in arities at least 1 and which is ${\mathcal{O}}(0)\oplus k$ in arity zero. Similarly, we define the symmetric sequence ${\mathcal{C}}^{\text{cu}}={\mathcal{C}}\oplus k$ for a cooperad ${\mathcal{C}}$ .

For instance, $\text{Ass}$ and $\mathbb{P}_{n}$ have Hopf unital structures given by the Hopf operads $\text{Ass}^{\text{un}}$ and $\mathbb{P}_{n}^{\text{un}}$ . Similarly, the curved cooperads $\text{coAss}^{\unicode[STIX]{x1D703}}$ and $\text{co}\mathbb{P}_{n}^{\unicode[STIX]{x1D703}}$ have Hopf counital structures given by the cooperads $\text{coAss}^{\unicode[STIX]{x1D703},\text{cu}}$ and $\text{co}\mathbb{P}_{n}^{\unicode[STIX]{x1D703},\text{cu}}$ respectively, where, for instance, $\text{coAss}^{\unicode[STIX]{x1D703},\text{cu}}$ governs curved counital coassociative coalgebras.

Given a Hopf operad ${\mathcal{O}}^{\text{un}}$ , the counits assemble to give a map of operads ${\mathcal{O}}^{\text{un}}\rightarrow \text{Comm}^{\text{un}}$ . We can equivalently define a Hopf unital structure on an operad ${\mathcal{O}}$ to be a Hopf operad ${\mathcal{O}}^{\text{un}}$ together with an isomorphism of operads

Using the morphism ${\mathcal{O}}^{\text{un}}\rightarrow \text{Comm}^{\text{un}}$ we see that the unital commutative algebra $k$ admits a natural ${\mathcal{O}}^{\text{un}}$ -algebra structure.

Dually, for a Hopf cooperad ${\mathcal{C}}^{\text{cu}}$ we define the notion of a coaugmented ${\mathcal{C}}^{\text{cu}}$ -coalgebra. We denote by $\text{Alg}_{{\mathcal{O}}^{\text{un}}}^{\text{aug}}$ the category of augmented ${\mathcal{O}}^{\text{un}}$ -algebras and by $\text{CoAlg}_{{\mathcal{C}}^{\text{cu}}}^{\text{coaug}}$ the category of coaugmented ${\mathcal{C}}^{\text{cu}}$ -coalgebras. Note that a coaugmented ${\mathcal{C}}^{\text{cu}}$ -coalgebra $C\rightarrow k$ is automatically assumed to be conilpotent in the sense that the non-counital coalgebra $\overline{C}$ is conilpotent.

The same construction works for a curved cooperad ${\mathcal{C}}$ . Note that since ${\mathcal{C}}^{\text{cu}}$ is a Hopf cooperad, the category $\text{CoAlg}_{{\mathcal{C}}}$ inherits a natural symmetric monoidal structure. Explicitly, given two ${\mathcal{C}}$ -coalgebras $C_{1},C_{2}$ , the underlying graded vector space of their tensor product is defined to be $C_{1}\otimes C_{2}\oplus C_{1}\oplus C_{2}$ .

We will need a certain compatibility between the Hopf structure on ${\mathcal{C}}^{\text{cu}}$ and weak equivalences.

We are now going to show that some standard examples of Hopf cooperads are admissible.

The statement has the following important corollaries.

Consider the symmetric monoidal structure on the category of non-unital algebras $\text{Alg}_{\text{Ass}}$ where the tensor product of $A_{1}$ and $A_{2}$ is

One also has a symmetric monoidal structure on the category of non-counital coalgebras $\text{CoAlg}_{\text{coAss}}$ where the tensor product of $C_{1}$ and $C_{2}$ is

Similarly, one introduces the symmetric monoidal structures on the categories $\text{Alg}_{\mathbb{P}_{n}}$ and $\text{CoAlg}_{\text{co}\mathbb{P}_{n}}$ . Finally, consider the Cartesian symmetric monoidal structure on the category of Lie algebras $\text{Alg}_{\text{Lie}}$ .

Therefore, we can use the relative symmetric monoidal category $(\text{CoAlg}_{\text{co}\mathbb{P}_{n}},W_{\text{Kos}})$ as a model for the symmetric monoidal $\infty$ -category ${\mathcal{A}}\text{lg}_{\mathbb{P}_{n}}$ .

2.1 Brace construction

Suppose ${\mathcal{C}}$ is a (curved) cooperad equipped with a Hopf counital structure (see Definition 1.12). Let us briefly recall from [Reference Calaque and WillwacherCW15] the definition of the associated operad of braces $\text{Br}_{{\mathcal{C}}}$ . Its operations are parametrized by rooted trees with ‘external’ vertices that are colored white in our pictures and ‘internal’ vertices that are colored black. An external vertex with $r$ children is labeled by elements of ${\mathcal{C}}^{\text{cu}}(r)$ while an internal vertex with $r$ children is labeled by elements of $\overline{{\mathcal{C}}}(r)[-1]$ .

The coproduct on ${\mathcal{C}}^{\text{cu}}$ defines a morphism

where in the second morphism we use ${\mathcal{C}}^{\text{cu}}(1)\rightarrow k$ given by the counit on the cooperad ${\mathcal{C}}^{\text{cu}}$ and ${\mathcal{C}}^{\text{cu}}(0)\cong {\mathcal{C}}(0)\oplus k\rightarrow k$ given by projection on the second summand. The operadic composition is given by grafting rooted trees with labels obtained by applying the coproduct on ${\mathcal{C}}^{cu}$ and combining different labels using the Hopf structure. The differential on internal vertices coincides with the cobar differential (in particular, if ${\mathcal{C}}$ is curved, it contains an extra curving term); the differential of an external vertex splits it into an internal and an external vertex. We refer to [Reference Dolgushev and WillwacherDW15, Formula (8.14)] for an explicit description of the differential.

Figure 4.

Generating operations of $\text{Br}_{{\mathcal{C}}}$ .

The operad $\text{Br}_{{\mathcal{C}}}$ is generated by trees shown in Figure 4, where the leaves are labeled by the unit element of ${\mathcal{C}}^{\text{cu}}(0)$ . For a $\text{Br}_{{\mathcal{C}}}$ -algebra $A$ we denote the operation given by the corolla with an internal vertex by $m(c|x_{1},\ldots ,x_{r})$ where $c\in \overline{{\mathcal{C}}}(r)[-1]$ and $x_{i}\in A$ . The operation given by the corolla with an external vertex is denoted by $x\{c|y_{1},\ldots ,y_{r}\}$ , where $c\in {\mathcal{C}}^{\text{cu}}(r)$ . We denote by $\unicode[STIX]{x1D6FA}{\mathcal{C}}\rightarrow \text{Br}_{{\mathcal{C}}}$ the natural morphism which sends a generator in $\overline{{\mathcal{C}}}[-1]$ to the corresponding corolla with an internal vertex.

The generating operations satisfy the following three relations.

1. (i) (Associativity).

   (6)

   

2. (ii) (Higher homotopies).

   (7)

   

3. (iii) (Distributivity).

   (8)

   

Let us give some examples of the brace construction that we will use. Note that the second corolla in Figure 4 with $k=2$ gives rise to a pre-Lie structure on any $\text{Br}_{{\mathcal{C}}}$ -algebra. However, in general the pre-Lie operation is not compatible with the differential.

The simplest example is the case ${\mathcal{C}}=\mathbf{1}$ , the trivial cooperad. In this case $\overline{{\mathcal{C}}}=0$ and hence the only operations are given by braces. However, since ${\mathcal{C}}(n)=0$ for $n\geqslant 2$ , we can only have vertices of valence 1 and 0, i.e. the operations of $\text{Br}_{\mathbf{1}}$ are parametrized by linear chains and hence $\text{Br}_{\mathbf{1}}(n)\cong k[S_{n}]$ . It is immediate to see that the pre-Lie operation in this case gives rise to an associative multiplication.

For ${\mathcal{C}}=\text{coAss}$ we obtain an $A_{\infty }$ structure of degree 1 together with degree $0$ brace operations $x\{y_{1},\ldots ,y_{n}\}$ . Recall the brace operad $\text{Br}$ introduced by Gerstenhaber and Voronov [Reference Gerstenhaber and VoronovGV95] and let $A_{\infty }$ be the operad controlling $A_{\infty }$ algebras.

Now consider ${\mathcal{C}}=\text{co}\mathbb{P}_{n}$ . Calaque and Willwacher [Reference Calaque and WillwacherCW15] following some ideas of Tamarkin [Reference TamarkinTam00] introduced a morphism of operads

on generators by the following rule.

1. – The generators

   $$\begin{eqnarray}\text{}\underline{x_{1}\cdots x_{k}}\in \text{coLie}\{1-n\}(k)\subset \text{co}\mathbb{P}_{n+1}\{1\}(k)\end{eqnarray}$$
   are sent to the tree drawn in Figure 5 with the root labeled by the element
   $$\begin{eqnarray}\text{}\underline{x_{1}\cdots x_{k}}\in \text{coLie}\{1-n\}(k)\subset \text{co}\mathbb{P}_{n}(k).\end{eqnarray}$$

   Here $\text{}\underline{x_{1}\cdots x_{k}}$ is the image of the $k$ -ary comultiplication under the projection

   $$\begin{eqnarray}\text{coAss}\{1-n\}\rightarrow \text{coLie}\{1-n\}.\end{eqnarray}$$

2. – The generator

   $$\begin{eqnarray}x_{1}\wedge x_{2}\in \text{coComm}\{1\}(2)\subset \text{co}\mathbb{P}_{n+1}\{1\}(2)\end{eqnarray}$$
   is sent to the linear combination of trees shown in Figure 6.

3. – The generators

   $$\begin{eqnarray}x_{1}\wedge \text{}\underline{x_{2}\cdots x_{k}}\in \text{co}\mathbb{P}_{n+1}\{1\}(k)\end{eqnarray}$$
   for $k>2$ are sent to the tree shown in Figure 7 with the root labeled by the element
   $$\begin{eqnarray}\text{}\underline{x_{2}\cdots x_{k}}\in \text{coLie}\{1-n\}(k-1)\subset \text{co}\mathbb{P}_{n}(k-1).\end{eqnarray}$$

4. – The rest of the generators are sent to zero.

Figure 5.

Image of $\text{}\underline{x_{1}\cdots x_{k}}$ .

Figure 6.

Image of $x_{1}\wedge x_{2}$ .

Figure 7.

Image of $x_{1}\wedge \text{}\underline{x_{2}\cdots x_{k}}$ .

Note that we have a quasi-isomorphism of operads

and hence we get a zig–zag of quasi-isomorphisms between the operads $\text{Br}_{\text{co}\mathbb{P}_{n}}\{n\}$ and $\mathbb{P}_{n+1}$ .

One can similarly define a morphism of operads

in the following way.

1. – The generators $y\in \text{coLie}^{\unicode[STIX]{x1D703}}\{1-n\}\subset \text{co}\mathbb{P}_{n+1}^{\unicode[STIX]{x1D703}}\{1\}$ are sent to the tree drawn in Figure 5 with the root labeled by $x\in \text{coLie}^{\unicode[STIX]{x1D703}}\{1-n\}\subset \text{co}\mathbb{P}_{n}^{\unicode[STIX]{x1D703}}$ .

2. – The generator

   $$\begin{eqnarray}x_{1}\wedge x_{2}\in \text{coComm}\{1\}(2)\subset \text{co}\mathbb{P}_{n+1}^{\unicode[STIX]{x1D703}}\{1\}(2)\end{eqnarray}$$
   is sent to the linear combination of trees shown in Figure 6.

3. – Suppose $y\in \text{coLie}^{\unicode[STIX]{x1D703}}\{1-n\}(k-1)\subset \text{co}\mathbb{P}_{n+1}^{\unicode[STIX]{x1D703}}\{1\}(k-1)$ for $k>2$ and let $x\wedge y$ be the image of $y$ under $\text{co}\mathbb{P}_{n+1}^{\unicode[STIX]{x1D703}}\{1\}(k-1)\rightarrow \text{co}\mathbb{P}_{n+1}^{\unicode[STIX]{x1D703}}\{1\}(k)$ .

   The generators $x\wedge y\in \text{co}\mathbb{P}_{n+1}^{\unicode[STIX]{x1D703}}\{1\}(k)$ are sent to the tree shown in Figure 7 with the root labeled by the element $y\in \text{coLie}^{\unicode[STIX]{x1D703}}\{1-n\}(k-1)\subset \text{co}\mathbb{P}_{n}^{\unicode[STIX]{x1D703}}(k-1)$ .

4. – The rest of the generators are sent to zero.

The following statement is a version of Proposition 2.3.

This proposition implies that we have a zig–zag of weak equivalences between the operads $\text{Br}_{\text{co}\mathbb{P}_{n}^{\unicode[STIX]{x1D703}}}\{n\}$ and $\mathbb{P}_{n+1}^{\text{un}}$ .

Finally, consider the case ${\mathcal{C}}=\text{coComm}$ . By construction we have a morphism of operads

given by sending the Lie bracket to the following combination.

2.2 Additivity for brace algebras

Recall that we have a morphism of operads $\unicode[STIX]{x1D6FA}{\mathcal{C}}\rightarrow \text{Br}_{{\mathcal{C}}}$ . In particular, a brace algebra has a bar complex which is a ${\mathcal{C}}$ -coalgebra. Now we are going to introduce an associative multiplication on the bar complex making the diagram

commute.

Let $A$ be a $\text{Br}_{{\mathcal{C}}}$ -algebra and consider ${\mathcal{C}}^{\text{cu}}(A)$ , the cofree conilpotent ${\mathcal{C}}^{\text{cu}}$ -coalgebra, equipped with the bar differential. It is naturally coaugmented using the decomposition

We introduce the unit on ${\mathcal{C}}^{\text{cu}}(A)$ to be given by the coaugmentation $k\rightarrow {\mathcal{C}}^{\text{cu}}(A)$ . The multiplication

is uniquely specified on the cogenerators by a morphism

In turn, it is defined via the composite

where the first morphism is induced by the counit on ${\mathcal{C}}^{\text{cu}}$ and the second morphism is given by braces, i.e. the morphism

is given by applying the second corolla in Figure 4 with the root labeled by the element of ${\mathcal{C}}^{\text{cu}}(n)$ and the leaves labeled by the unit $k\rightarrow {\mathcal{C}}^{\text{cu}}(0)$ . The following statement is shown in [Reference Melani and SafronovMS16, Proposition 3.4].

For a $\text{Br}_{{\mathcal{C}}}$ -algebra $A$ we denote by $\text{B}A={\mathcal{C}}^{\text{cu}}(A)$ the bar complex equipped with the bar differential and the above associative multiplication. This defines a functor

Now suppose the Hopf unital structure on ${\mathcal{C}}$ is admissible. Then the symmetric monoidal structure on $\text{CoAlg}_{{\mathcal{C}}^{\text{cu}}}^{\text{coaug}}$ preserves weak equivalences and hence $\text{CoAlg}_{{\mathcal{C}}^{\text{cu}}}^{\text{coaug}}[W_{\text{Kos}}^{-1}]$ is a symmetric monoidal $\infty$ -category.

Since the bar functor preserves weak equivalences, the composite functor

gives rise to the additivity functor

We do not know if the functor (13) is an equivalence in general, but we prove that it is an equivalence for Lie and Poisson algebras. Note that in the case ${\mathcal{C}}=\text{coAss}$ we obtain a functor

which we expect coincides with the Dunn–Lurie equivalence [Reference LurieLur17, Theorem 5.1.2.2].

2.3 Additivity for Lie algebras

In this section we work out how the additivity functor (13) looks like for Lie algebras and prove that it is an equivalence. We consider the cooperad ${\mathcal{C}}=\text{coComm}$ .

We have a functor

which sends a Lie algebra to its universal enveloping algebra. The following is proved e.g. in [Reference CartierCar07, Theorem 3.8.1].

We get two functors

where one is given by the brace bar construction $\text{B}$ and the other one is given by the composite

where the forgetful functor is given by Proposition 2.5. In fact, these are equivalent.

The $\infty$ -category of Lie algebras is pointed, i.e. the initial and final objects coincide. Therefore, we can consider the loop functor

given by sending a Lie algebra $\mathfrak{g}$ to its loop object $0\times _{\mathfrak{g}}0$ . More explicitly, since the monoidal structure on ${\mathcal{A}}\text{lg}_{\text{Lie}}$ is Cartesian, by [Reference LurieLur17, Proposition 4.1.2.10] we can identify ${\mathcal{A}}\text{lg}({\mathcal{A}}\text{lg}_{\text{Lie}})$ with the $\infty$ -category of Segal monoids, i.e. simplicial objects $M_{\bullet }$ of ${\mathcal{A}}\text{lg}_{\text{Lie}}$ such that $M_{0}$ is contractible and the natural maps $M_{n}\rightarrow M_{1}\times _{M_{0}}\times \cdots \times _{M_{0}}M_{1}$ are equivalences. Under this identification the loop object of $\mathfrak{g}$ is defined to be the simplicial object underlying the Cech nerve of $0\rightarrow \mathfrak{g}$ :

Its left adjoint is the classifying space functor

which sends a Segal monoid in Lie algebras to its geometric realization. The following is proved in [Reference ToënToë13, Lemma 5.3] and [Reference Gwilliam and HaugsengGH16, Corollary 2.7.2].

Since the operads $\text{Br}_{\text{coComm}}$ and $\text{Lie}$ are quasi-isomorphic, the additivity functor (13) becomes

Observe that now we have constructed two functors ${\mathcal{A}}\text{lg}_{\text{Lie}}\longrightarrow {\mathcal{A}}\text{lg}({\mathcal{A}}\text{lg}_{\text{Lie}})$ : the loop functor $\unicode[STIX]{x1D6FA}$ and the additivity functor $\text{add}$ .

Combining the previous proposition with Proposition 2.12 we obtain the following corollary.

Note that the underlying Lie algebra of ${\mathcal{A}}\text{lg}({\mathcal{A}}\text{lg}_{\text{Lie}})$ is canonically trivial. Indeed, let ${\mathcal{C}}\text{h}\longrightarrow {\mathcal{A}}\text{lg}_{\text{Lie}}$ be the functor which sends a complex $\mathfrak{g}$ to the trivial Lie algebra $\mathfrak{g}[-1]$ .

2.4 Additivity for Poisson algebras

This section is devoted to an explicit description of the additivity functor (13) in the case of $\mathbb{P}_{n}$ -algebras and to showing that it is a symmetric monoidal equivalence of $\infty$ -categories.

Consider the cooperad ${\mathcal{C}}=\text{co}\mathbb{P}_{n}$ . Since we have a zig–zag of quasi-isomorphisms between the operads $\mathbb{P}_{n+1}$ and $\text{Br}_{\text{co}\mathbb{P}_{n}}\{n\}$ , the additivity functor (13) becomes

Now we are going to give a different perspective on this functor closer to Tamarkin’s papers [Reference TamarkinTam00] and [Reference TamarkinTam07]. This new perspective will elucidate the formulas for the morphism (9) and allow us to show that the additivity functor is a symmetric monoidal equivalence.

We are going to introduce yet another version of the bar-cobar duality for Poisson algebras, this time the dual object will be a Lie bialgebra.

We will say an $n$ -shifted Lie bialgebra is conilpotent if the underlying Lie coalgebra is so and we denote the category of $n$ -shifted conilpotent Lie bialgebras by $\text{BiAlg}_{\text{Lie}_{n}}$ . Weak equivalences in $\text{BiAlg}_{\text{Lie}_{n}}$ are created by the forgetful functor

The commutative bar-cobar adjunction

extends to a bar-cobar adjunction

so that the diagram

commutes where the vertical functors are the forgetful functors.

Explicitly, if $A$ is a $\mathbb{P}_{n+1}$ -algebra, consider $\mathfrak{g}=\text{coLie}(A[1])[n-1]$ , the cofree conilpotent shifted Lie coalgebra equipped with the bar (i.e. Harrison) differential. By the cocycle equation a Lie bracket on $\mathfrak{g}$ is uniquely determined by its projection to cogenerators and the map

is defined to be the Lie bracket

The Jacobi identity is obvious. Compatibility of the bracket on $\mathfrak{g}$ with the bar differential follows from the Leibniz rule for $A$ .

Conversely, if $\mathfrak{g}$ is an $(n-1)$ -shifted Lie bialgebra, consider $A=\operatorname{Sym}(\mathfrak{g}[-n])$ equipped with the cobar differential using the Lie coalgebra structure on $\mathfrak{g}$ . The Lie bracket on $A$ by the Leibniz rule is defined on generators to be the Lie bracket on $\mathfrak{g}$ . The compatibility of the cobar differential on $A$ with the Lie structure can be checked on generators where it coincides with the cocycle (16).

By the definition of weak equivalences it is clear that the adjunction (17) induces an adjoint equivalence on $\infty$ -categories since the unit and counit of the adjunction are weak equivalences after forgetting down to commutative algebras and Lie coalgebras.

By the Cartier–Milnor–Moore theorem (Theorem 2.9) the universal enveloping algebra functor induces an equivalence of categories

If $\mathfrak{g}$ is an $(n-1)$ -shifted Lie bialgebra, we can define a $\mathbb{P}_{n}$ -coalgebra structure on $\text{U}(\mathfrak{g})$ as follows. By construction $\text{U}(\mathfrak{g})$ is a cocommutative bialgebra and we define the cobracket on the generators to be the cobracket on $\mathfrak{g}$ . The coproduct on $\text{U}(\mathfrak{g})$ is conilpotent and the cobracket on $\text{U}(\mathfrak{g})$ is conilpotent if the cobracket on $\mathfrak{g}$ is so. In this way we construct the following commutative diagram.

Note that if $A\in \text{Alg}(\text{CoAlg}_{\text{co}\mathbb{P}_{n}^{\text{cu}}}^{\text{coaug}})$ has an associative multiplication and a compatible $\mathbb{P}_{n}$ -coalgebra structure, then the space of primitive elements is closed under the cobracket by the Leibniz rule for the $\mathbb{P}_{n}$ -coalgebra structure. Thus, the inverse functor in both cases is simply given by the functor of primitive elements.

Now we are going to show that the brace bar construction for ${\mathcal{C}}=\text{co}\mathbb{P}_{n}$ is compatible with the bar-cobar duality between $\mathbb{P}_{n+1}$ -algebras and $(n-1)$ -shifted Lie bialgebras in the following sense.

By the previous proposition the two functors of $\infty$ -categories

given either by localizing the brace bar functor for ${\mathcal{C}}=\text{co}\mathbb{P}_{n}$ or by localizing the bar-cobar duality between $\mathbb{P}_{n+1}$ -algebras and shifted Lie bialgebras coincide.

As a corollary, we get a sequence of functors

But ${\mathcal{A}}\text{lg}({\mathcal{A}}\text{lg}({\mathcal{C}}))\cong {\mathcal{A}}\text{lg}_{\mathbb{E}_{2}}({\mathcal{C}})$ for any symmetric monoidal $\infty$ -category ${\mathcal{C}}$ by the Dunn–Lurie additivity theorem [Reference LurieLur17, Theorem 5.1.2.2]. Iterating this construction, we get a symmetric monoidal functor

The additivity functor (15) interacts in the obvious way with the commutative and the Lie structures on a $\mathbb{P}_{n}$ -algebra as shown by the next three propositions.

Let us denote by

the functor which sends a Lie algebra $\mathfrak{g}$ to the non-unital $\mathbb{P}_{n}$ -algebra $\overline{\operatorname{Sym}}(\mathfrak{g}[1-n])$ , the reduced symmetric algebra on $\mathfrak{g}[1-n]$ with the Poisson bracket induced by the Leibniz rule from the bracket on $\mathfrak{g}$ . Recall that the symmetric monoidal structure on $\text{Alg}_{\mathbb{P}_{n}}$ we consider is the one transferred under the equivalence $\text{Alg}_{\mathbb{P}_{n}}\cong \text{Alg}_{\mathbb{P}_{n}^{\text{un}}}^{\text{aug}}$ from the usual tensor product of augmented $\mathbb{P}_{n}$ -algebras. In particular, under this equivalence the functor $\overline{\operatorname{Sym}}$ corresponds to $\mathfrak{g}\mapsto \operatorname{Sym}(\mathfrak{g}[1-n])$ which is symmetric monoidal. The functor $\overline{\operatorname{Sym}}$ preserves weak equivalences, so we obtain a symmetric monoidal functor of $\infty$ -categories

The functor $\overline{\operatorname{Sym}}:{\mathcal{A}}\text{lg}_{\text{Lie}}\rightarrow {\mathcal{A}}\text{lg}_{\mathbb{P}_{n+1}}$ has a right adjoint $\text{forget}:{\mathcal{A}}\text{lg}_{\mathbb{P}_{n+1}}\rightarrow {\mathcal{A}}\text{lg}_{\text{Lie}}$ given by forgetting the commutative algebra structure. Since $\overline{\operatorname{Sym}}$ is symmetric monoidal, the right adjoint $\text{forget}$ is lax symmetric monoidal and hence sends associative algebras to associative algebras. Therefore, the commutativity data of Proposition 2.20 gives rise to a diagram of right adjoints

which commutes up to a natural transformation.

We will end this section by proving that the additivity functor for $\mathbb{P}_{n}$ -algebras is an equivalence.

A natural question is whether the Dunn–Lurie additivity functor (see [Reference LurieLur17, Theorem 5.1.2.2])

is compatible with the Poisson additivity functor in the following sense. Suppose that $n\geqslant 2$ . Then we have an equivalence of Hopf operads $\mathbb{P}_{n}\cong \mathbb{E}_{n}$ provided by the formality of the operad $\mathbb{E}_{n}$ which gives an equivalence of symmetric monoidal $\infty$ -categories ${\mathcal{A}}\text{lg}_{\mathbb{E}_{n}}\cong {\mathcal{A}}\text{lg}_{\mathbb{P}_{n}}$ .

2.5 Additivity for unital Poisson algebras

Let us explain the necessary modifications one performs to construct the additivity functor for unital Poisson algebras. Recall from § 1.4 the bar-cobar adjunction for unital Poisson algebras

where $\text{co}\mathbb{P}_{n}^{\unicode[STIX]{x1D703}}$ is the cooperad of curved $\mathbb{P}_{n}$ -coalgebras. One similarly has a bar-cobar adjunction

where $\text{coLie}^{\unicode[STIX]{x1D703}}$ is the cooperad of curved Lie coalgebras, that is, graded Lie coalgebras $\mathfrak{g}$ together with a coderivation $d$ of degree 1 satisfying

where $\unicode[STIX]{x1D6FF}(x)=x_{(1)}^{\unicode[STIX]{x1D6FF}}\otimes x_{(2)}^{\unicode[STIX]{x1D6FF}}$ .

The cobar construction is given by the same formula as in the non-curved case except that we add the curving to the differential. The bar construction of a unital commutative algebra $A$ is given by

with a bar differential and the curving which is given by projecting to $k[2]$ .

We denote the category of $n$ -shifted conilpotent curved Lie bialgebras by $\text{BiAlg}_{\text{Lie}_{n}}^{\unicode[STIX]{x1D703}}$ with weak equivalences created by the forgetful functor

The commutative bar-cobar adjunction

extends to a bar-cobar adjunction

The only modification from the case of non-unital Poisson algebras is the formula for the Lie bracket. Suppose $A$ is a unital $\mathbb{P}_{n+1}$ -algebra. Then as a graded vector space

By the cocycle (16), the Lie bracket on $\text{coLie}(A[1]\oplus k[2])$ is uniquely determined after projection to cogenerators and the morphism

has the zero component in $k[n+1]$ and its $A[n]$ component is defined to be the bracket

The universal enveloping algebra construction gives a functor

Therefore, we can define the additivity functor

to be given by the composite

The following statement is proved as for non-unital Poisson algebras by analyzing the forgetful functor to Lie algebras.

3.1 Two definitions

Let us introduce the following notations. Given a category ${\mathcal{C}}$ we denote by $\text{Arr}({\mathcal{C}})$ the category of morphisms in ${\mathcal{C}}$ , i.e. the functor category $\text{Fun}(\unicode[STIX]{x1D6E5}^{1},{\mathcal{C}})$ . Given a symmetric monoidal category ${\mathcal{C}}$ we denote by $\text{LMod}({\mathcal{C}})$ the category of pairs $(A,M)$ , where $A$ is a unital associative algebra in ${\mathcal{C}}$ and $M$ is a left $A$ -module. Let us denote by ${\mathcal{L}}\text{Mod}({\mathcal{C}})$ the same construction for a symmetric monoidal $\infty$ -category ${\mathcal{C}}$ .

Introduce the notation

We have a forgetful functor ${\mathcal{A}}\text{lg}_{\mathbb{P}_{(n+1,n)}}\rightarrow \text{Arr}({\mathcal{A}}\text{lg}_{\text{Comm}})$ which sends a pair $(A,B)$ with the action map $A\otimes B\rightarrow B$ to the morphism of commutative algebras $A\rightarrow B$ given by the composite $A\stackrel{\operatorname{id}\otimes 1}{\rightarrow }A\otimes B\rightarrow B$ .

The following definition of coisotropic structures is due to Calaque et al. [Reference Calaque, Pantev, Toën, Vaquié and VezzosiCPTVV17, Definition 3.4.3].

To relate this space of $n$ -shifted coisotropic structures to the space of $n$ -shifted Poisson structures on $A$ , one has to use the Poisson additivity functor

A more explicit definition of the space of coisotropic structures was given in [Reference Melani and SafronovMS16] and [Reference SafronovSaf17] as follows. Let $B$ be a $\mathbb{P}_{n}$ -algebra. We define its strict center to be the $\mathbb{P}_{n+1}$ -algebra

with the differential twisted by $[\unicode[STIX]{x1D70B}_{B},-]$ . We refer to [Reference SafronovSaf17, § 1.1] for explicit formulas for the $\mathbb{P}_{n+1}$ -structure. One can also consider its center

with the differential twisted by the $\mathbb{P}_{n}$ -structure on $B$ . By results of [Reference Calaque and WillwacherCW15], $\text{Z}(B)$ is a $\text{Br}_{\text{co}\mathbb{P}_{n}}\{n\}$ -algebra and hence in particular a homotopy $\mathbb{P}_{n+1}$ -algebra. Moreover, the natural inclusion

is compatible with the homotopy $\mathbb{P}_{n+1}$ structures on both sides and is a quasi-isomorphism if $B$ is cofibrant as a commutative algebra.

Let $\mathbb{P}_{[n+1,n]}$ be the colored operad whose algebras consist of a pair $(A,B,f)$ where $A$ is a $\mathbb{P}_{n+1}$ -algebra, $B$ is a $\mathbb{P}_{n}$ -algebra and $f:A\rightarrow \text{Z}^{\text{str}}(B)$ is a $\mathbb{P}_{n+1}$ -morphism. The projection $\text{Z}^{\text{str}}(B)\rightarrow B$ is a morphism of commutative algebras, so we get a natural forgetful functor

Our goal will be to construct an equivalence of $\infty$ -categories ${\mathcal{A}}\text{lg}_{\mathbb{P}_{[n+1,n]}}\rightarrow {\mathcal{A}}\text{lg}_{\mathbb{P}_{(n+1,n)}}$ which is compatible with the forgetful functor to $\text{Arr}({\mathcal{A}}\text{lg}_{\text{Comm}})$ which will show that the spaces $\text{Cois}^{\text{MS}}(f,n)$ and $\text{Cois}^{\text{CPTVV}}(f,n)$ are equivalent. We will construct the equivalence as a relative version of the additivity functor (13).

3.2 Relative additivity for Lie algebras

We begin with the relative analog of the additivity functor for Lie algebras.

Let us introduce a Swiss-cheese operad of Lie algebras. Given a dg Lie algebra $\mathfrak{h}$ the Chevalley–Eilenberg complex $\text{C}^{\bullet }(\mathfrak{h},\mathfrak{h})[1]$ carries a convolution Lie bracket. We let $\text{Lie}_{[1,0]}$ be the colored operad whose algebras consist of a pair of dg Lie algebras $(\mathfrak{g},\mathfrak{h})$ together with a map of dg Lie algebras $\mathfrak{g}\rightarrow \text{C}^{\bullet }(\mathfrak{h},\mathfrak{h})[1]$ .

Consider the morphism $\mathfrak{h}\rightarrow \text{Der}(\mathfrak{h})$ given by the adjoint action and denote by $\widetilde{\text{Der}}(\mathfrak{h})$ its cone. Note that we have an obvious inclusion $\widetilde{\text{Der}}(\mathfrak{h})\subset \text{C}^{\bullet }(\mathfrak{h},\mathfrak{h})[1]$ . We denote by $\text{Lie}_{[1,0]}^{\text{str}}$ the quotient operad of $\text{Lie}_{[1,0]}$ where we set all components of the map $\mathfrak{g}\rightarrow \text{C}^{\bullet }(\mathfrak{h},\mathfrak{h})[1]$ having $\mathfrak{h}$ -arity at least $2$ to be zero. Thus, a $\text{Lie}_{[1,0]}^{\text{str}}$ -algebra is a pair $(\mathfrak{g},\mathfrak{h})$ of dg Lie algebras together with a map of Lie algebras $\mathfrak{g}\rightarrow \widetilde{\text{Der}}(\mathfrak{h})$ .

Given a pair $(\mathfrak{g},\mathfrak{h})\in \text{Alg}_{\text{Lie}_{[1,0]}^{\text{str}}}$ we can construct a dg Lie algebra $\mathfrak{k}$ as follows. As a graded vector space we define $\mathfrak{k}=\mathfrak{g}\oplus \mathfrak{h}$ . The differential on $\mathfrak{k}$ is the sum of the differentials on $\mathfrak{g}$ and $\mathfrak{h}$ and the differential $\mathfrak{g}\rightarrow \mathfrak{h}$ coming from the composite

The Lie bracket on $\mathfrak{k}$ is the sum of Lie brackets on $\mathfrak{g}$ and $\mathfrak{h}$ and the action map of $\mathfrak{g}$ on $\mathfrak{h}$ given by the map $\mathfrak{g}\rightarrow \text{Der}(\mathfrak{h})$ . In this way we see that a pair $(\mathfrak{g},\mathfrak{h})\in \text{Alg}_{\text{Lie}_{[1,0]}^{\text{str}}}$ is the same as a dg Lie algebra extension

One can similarly construct an $L_{\infty }$ extension from the data of a general object of $\text{Alg}_{\text{Lie}_{[1,0]}}$ . Since every $L_{\infty }$ algebra is quasi-isomorphic to a dg Lie algebra, the following statement should be obvious.

Now we are going to introduce the bar construction

Consider a pair $(\mathfrak{g},\mathfrak{h})\in \text{Alg}_{\text{Lie}_{[1,0]}}$ . We send it to the pair $(\text{U}(\mathfrak{g}),\text{C}_{\bullet }(\mathfrak{h}))$ of a cocommutative bialgebra and a cocommutative coalgebra. The action map

is constructed as follows. Since $\text{U}(\mathfrak{g})$ is generated by $\mathfrak{g}$ , it is enough to specify the action $\mathfrak{g}\otimes \text{C}_{\bullet }(\mathfrak{h})\rightarrow \text{C}_{\bullet }(\mathfrak{h})$ that we denote by $x.c$ for $x\in \mathfrak{g}$ and $c\in \text{C}_{\bullet }(\mathfrak{h})$ satisfying the equations

Since $\text{C}_{\bullet }(\mathfrak{h})$ is cofree, by the first equation it is enough to specify the map $\mathfrak{g}\otimes \text{C}_{\bullet }(\mathfrak{h})\rightarrow \mathfrak{h}[1]$ which we define to be adjoint to the given map $\mathfrak{g}\rightarrow \text{C}^{\bullet }(\mathfrak{h},\mathfrak{h})[1]$ . It is then easy to see that since the map $\mathfrak{g}\rightarrow \text{C}^{\bullet }(\mathfrak{h},\mathfrak{h})[1]$ is compatible with Lie brackets, the second equation is satisfied and since it is compatible with the differentials, so is the map $\mathfrak{g}\otimes \text{C}_{\bullet }(\mathfrak{h})\rightarrow \text{C}_{\bullet }(\mathfrak{h})$ . This defines the functor

We can also introduce the cobar construction

Consider an element $(A,C)\in \text{LMod}(\text{CoAlg}_{\text{coComm}})$ where $A$ is an algebra and $C$ is an $A$ -module. By Theorem 2.9 we can identify $A\cong \text{U}(\mathfrak{g})$ for the Lie algebra $\mathfrak{g}$ of primitive elements. We send $(\text{U}(\mathfrak{g}),C)$ to the pair $(\mathfrak{g},\mathfrak{h}=\unicode[STIX]{x1D6FA}C)$ , where $\unicode[STIX]{x1D6FA}C$ is the Harrison complex $\text{Lie}(\overline{C}[-1])$ . Let us denote the action map $\text{U}\mathfrak{g}\otimes C\rightarrow C$ by $x.c$ for $x\in \mathfrak{g}$ and $c\in C$ . The morphism $\mathfrak{g}\longrightarrow \mathfrak{h}[1]$ is defined by the composite

where the first map is given by the action map $x.1$ . Since $\mathfrak{h}$ is semi-free, the morphism $\mathfrak{g}\rightarrow \text{Der}(\mathfrak{h})$ is uniquely determined by the map

where the first map is the action of $\mathfrak{g}$ on $C$ . The fact that the thus constructed morphism $\mathfrak{g}\rightarrow \widetilde{\text{Der}}(\mathfrak{h})$ is a morphism of Lie algebras follows from the associativity of the action map $\text{U}\mathfrak{g}\otimes C\rightarrow C$ . The compatibility of the morphism $\mathfrak{g}\rightarrow \widetilde{\text{Der}}(\mathfrak{h})$ with the differential follows from the compatibility of the action map $\text{U}(\mathfrak{g})\otimes C\rightarrow C$ with coproducts. This defines the functor

Note that in this way we obtain an adjunction

Indeed, the counit and unit morphisms

are defined to be the identities in the first slot and the counit and unit of the usual bar-cobar adjunction in the second slot.

The composite functor

defines a relative additivity functor for Lie algebras: