Operads: Koszul Duality and the Bar Construction
Table of Contents
- 1. Cooperads
- 2. The Operadic Bar Construction
- 3. The Operadic Cobar Construction
- 4. Twisting Morphisms
- 5. Quadratic Operads and the Koszul Dual
- 6. The Koszul Criterion
- 7. Homotopy Algebras from Koszul Duality
- 8. Operadic Cohomology and Deformation Theory
- References
1. Cooperads 📐
1.1 Definition
Dualizing the notion of an operad yields a cooperad. Recall from Definitions and Examples that an operad is a monoid in the monoidal category \((\mathbf{SymSeq}, \circ, \mathbf{1})\). A cooperad is the corresponding comonoid.
Definition (Cooperad). A cooperad is a triple \((\mathcal{C}, \Delta, \epsilon)\) consisting of a symmetric sequence \(\mathcal{C} \in \mathbf{SymSeq}\) together with:
- A decomposition map \(\Delta: \mathcal{C} \to \mathcal{C} \circ \mathcal{C}\), coassociative in the sense that the following diagram commutes:
- A counit \(\epsilon: \mathcal{C} \to \mathbf{1}\), satisfying the counit axioms:
In arity \(n\), the decomposition map \(\Delta\) has components \[\Delta_n : \mathcal{C}(n) \to \bigoplus_{k \geq 1} \bigoplus_{n_1+\cdots+n_k=n} \mathcal{C}(k) \otimes_{S_k} \bigl(\mathcal{C}(n_1) \otimes \cdots \otimes \mathcal{C}(n_k)\bigr).\] This encodes all ways to split an \(n\)-ary cooperation into a \(k\)-ary cooperation followed by \(k\) cooperations of arities \(n_1, \ldots, n_k\).
Definition (Morphism of cooperads). A morphism \(f: \mathcal{C} \to \mathcal{D}\) of cooperads is a morphism of symmetric sequences commuting with \(\Delta\) and \(\epsilon\).
Example. The dual of a finite-dimensional operad \(\mathcal{O}\) is a cooperad: set \(\mathcal{O}^\vee(n) = \mathcal{O}(n)^* = \mathrm{Hom}_k(\mathcal{O}(n), k)\) with right \(S_n\)-action via \((f \cdot \sigma)(x) = f(x \cdot \sigma^{-1})\). The composition \(\gamma: \mathcal{O} \circ \mathcal{O} \to \mathcal{O}\) dualizes to \(\Delta: \mathcal{O}^\vee \to \mathcal{O}^\vee \circ \mathcal{O}^\vee\).
1.2 Conilpotency
The bar construction requires a finiteness condition on cooperads.
Definition (Conilpotency). A cooperad \(\mathcal{C}\) is conilpotent if for every \(c \in \mathcal{C}(n)\) there exists \(N\) such that all iterated decompositions of \(c\) of depth \(> N\) vanish. Explicitly, define the coradical filtration \(F_0\mathcal{C} \subseteq F_1\mathcal{C} \subseteq \cdots\) by \(F_0\mathcal{C}(n) = 0\) for \(n > 1\), \(F_0\mathcal{C}(1) = k \cdot \mathrm{id}\), and \(F_r\mathcal{C} = \Delta^{-1}(\sum_{s+t=r} F_s\mathcal{C} \circ F_t\mathcal{C})\). The cooperad is conilpotent if \(\mathcal{C} = \bigcup_r F_r\mathcal{C}\).
Without conilpotency, the cobar construction \(\Omega(\mathcal{C})\) produces an operad with infinitely many generators in each weight, making the quasi-isomorphism \(\Omega B(\mathcal{O}) \xrightarrow{\sim} \mathcal{O}\) ill-formed. All cooperads arising as Koszul duals \(\mathcal{O}^!\) of quadratic operads are automatically conilpotent.
This establishes the cooperad structure on \(\mathrm{Ass}^c = \mathrm{Ass}^\vee\), the Koszul dual cooperad of \(\mathrm{Ass}\).
Prerequisites: 1.1 Definition, Ass definition
The associative cooperad \(\mathrm{Ass}^c\) has \(\mathrm{Ass}^c(n) = k[S_n]\) for \(n \geq 1\). Write down the decomposition map \(\Delta: \mathrm{Ass}^c(n) \to \bigoplus_{k} \mathrm{Ass}^c(k) \otimes_{S_k} \mathrm{Ass}^c(n_1) \otimes \cdots \otimes \mathrm{Ass}^c(n_k)\) explicitly for \(n = 3\) and verify coassociativity.
Key insight: \(\Delta\) on \(\mathrm{Ass}^c\) sums over all ways to partition a permutation into a block structure, with appropriate block-sum permutations.
Sketch: For \(\sigma \in S_3\), \(\Delta(\sigma) = \sum_{k; n_1+\cdots+n_k=3} \tau \otimes (\sigma_1 \otimes \cdots \otimes \sigma_k)\) where \(\tau \in S_k\) is the block permutation induced by \(\sigma\) on blocks of sizes \(n_i\), and \(\sigma_i \in S_{n_i}\) is the restriction of \(\sigma\) to the \(i\)-th block. For example, \(\Delta(\mathrm{id}_3) = \mathrm{id}_1 \otimes \mathrm{id}_3 + \mathrm{id}_2 \otimes (\mathrm{id}_1 \otimes \mathrm{id}_2 + \mathrm{id}_2 \otimes \mathrm{id}_1) + \mathrm{id}_3 \otimes \mathrm{id}_1^{\otimes 3}\). Coassociativity follows from associativity of the block-sum decomposition of permutations.
2. The Operadic Bar Construction 📐
2.1 Augmented Operads
The bar construction requires an augmentation.
Definition (Augmented operad). An augmented operad is an operad \(\mathcal{O}\) together with an operad morphism \(\epsilon: \mathcal{O} \to \mathbf{1}\) (the augmentation). Since \(\mathbf{1}(1) = k\) and \(\mathbf{1}(n) = 0\) for \(n \neq 1\), this forces \(\epsilon\) to be zero on all \(\mathcal{O}(n)\) for \(n \neq 1\) and the identity on the unit \(\mathrm{id} \in \mathcal{O}(1)\).
Set \(\bar{\mathcal{O}} = \ker(\epsilon)\), so that \(\mathcal{O} \cong \mathbf{1} \oplus \bar{\mathcal{O}}\) as symmetric sequences. The components of \(\bar{\mathcal{O}}\) are \(\bar{\mathcal{O}}(n) = \mathcal{O}(n)\) for \(n \geq 2\) and \(\bar{\mathcal{O}}(1) = \ker(\mathcal{O}(1) \to k)\).
The operads \(\mathrm{Ass}\), \(\mathrm{Com}\), and \(\mathrm{Lie}\) are naturally augmented: the augmentation \(\epsilon: \mathcal{O} \to \mathbf{1}\) sends the unit element \(\mathrm{id} \in \mathcal{O}(1)\) to \(1 \in k\) and kills all higher-arity operations. So \(\overline{\mathrm{Ass}}(n) = k[S_n]\) for \(n \geq 2\), zero for \(n = 1\).
2.2 The Bar Complex
Definition (Bar construction). For an augmented dg-operad \((\mathcal{O}, d, \gamma, \eta, \epsilon)\), the bar construction \(B(\mathcal{O})\) is the conilpotent dg-cooperad defined as follows.
As a graded symmetric sequence (ignoring the differential), \[B(\mathcal{O}) = \mathcal{F}^c(\bar{\mathcal{O}}[1]),\] the cofree conilpotent cooperad on the suspension \(\bar{\mathcal{O}}[1]\). Concretely, \(\mathcal{F}^c(M)\) is the symmetric sequence whose \(n\)-th component consists of all rooted trees with \(n\) leaves whose internal vertices are labeled by elements of \(M\), modulo the \(S_n\)-equivariance. As a graded object, \[B(\mathcal{O})(n) = \bigoplus_{T} \bigotimes_{v \in V(T)} \bar{\mathcal{O}}(|v|)[1],\] where the sum is over isomorphism classes of rooted trees \(T\) with \(n\) leaves, \(V(T)\) is the set of internal vertices, and \(|v|\) is the number of inputs of \(v\).
The bar construction carries a weight grading by \(\mathrm{wt}(B(\mathcal{O})) = k\) when \(k\) internal vertices are used. This is separate from the homological grading. The piece of weight \(k\) is \(\bar{\mathcal{O}}^{\circ k}[k]\) (the \(k\)-fold composition product of \(\bar{\mathcal{O}}\), shifted by \(k\)).
2.3 The Differential
The differential on \(B(\mathcal{O})\) has two components: \(d_{B(\mathcal{O})} = d_1 + d_2\).
- \(d_1\): the internal differential, induced by the differential \(d\) on \(\mathcal{O}\).
- \(d_2\): the composition differential, which decreases weight by 1. On a tree-monomial \((\mu_1, \ldots, \mu_k)\) labeling the vertices, \(d_2\) sums over all internal edges \(e\) of the tree and contracts \(e\) by applying the operadic composition \(\gamma\) to the two vertices adjacent to \(e\), with a sign \((-1)^{\deg}\) determined by the Koszul sign rule.
🔑 The key fact: \(d_{B(\mathcal{O})}^2 = 0\) follows from the associativity of \(\gamma\): the terms in \(d_2^2\) pair up with opposite signs by the associativity axiom for \(\mathcal{O}\), and \(d_1^2 = 0\) since \(d\) is a differential on \(\mathcal{O}\).
This makes the abstract definition of \(d_2\) concrete in the first nontrivial case.
Prerequisites: 2.3 The Differential, 2.1 Augmented Operads
Let \(\mathcal{O}\) be a non-dg augmented operad (so \(d_1 = 0\)). For \(\mu \in \bar{\mathcal{O}}(2)\) and \(\nu \in \bar{\mathcal{O}}(2)\), the weight-2 part \(B(\mathcal{O})^{(2)}(3) = (\bar{\mathcal{O}}[1])^{\circ 2}(3)\) consists of elements \([s\mu; s\nu]\) corresponding to binary trees with 3 leaves. Write down \(d_2([s\mu; s\nu])\) explicitly and verify that \(d_2^2 = 0\) using the associativity axiom.
Key insight: There are three binary trees with 3 leaves, corresponding to the three ways to parenthesize \(x_1 x_2 x_3\); \(d_2\) maps to a sum over weight-1 elements.
Sketch: In \(B(\mathcal{O})^{(2)}(3)\), there are two planar binary trees: the left-leaning tree (computing \((\mu \circ_1 \nu)\)) and the right-leaning tree (\(\mu \circ_2 \nu\)). The differential \(d_2\) contracts the internal edge of each tree using \(\gamma\): \(d_2([s\mu \circ_1 s\nu]) = s(\mu \circ_1 \nu)\) and \(d_2([s\mu \circ_2 s\nu]) = s(\mu \circ_2 \nu)\) (up to sign from the suspension). Then \(d_2^2 = 0\) amounts to \(\mu \circ_1 (\nu \circ_1 -) = (\mu \circ_1 \nu) \circ_1 -\) and \(\mu \circ_2 (\nu \circ_1 -) = (\mu \circ_1 \nu) \circ_2 -\) (sequential and parallel associativity axioms for \(\circ_i\)), which hold by definition of an operad.
This verifies that \(B(\mathcal{O})\) is naturally a cooperad, not just a graded symmetric sequence.
Prerequisites: 1.1 Definition, 2.2 The Bar Complex
Construct the decomposition map \(\Delta: B(\mathcal{O}) \to B(\mathcal{O}) \circ B(\mathcal{O})\) explicitly on tree-monomials by explaining how a rooted tree \(T\) decomposes into a “trunk” tree and a collection of “branch” subtrees. Verify this makes \(B(\mathcal{O})\) a cooperad.
Key insight: The decomposition map cuts each tree at a chosen internal edge, producing a “root component” tree and a “leaf component” subtree.
Sketch: Given a tree \(T\) with vertex labels \((\mu_v)_{v \in V(T)}\), \(\Delta(T) = \sum_{v \in V(T)} T_v^{\mathrm{root}} \otimes (T_{v,1}^{\mathrm{leaf}} \otimes \cdots \otimes T_{v,k}^{\mathrm{leaf}})\) where \(T_v^{\mathrm{root}}\) is the subtree above \(v\) (with \(v\) as a new leaf) and \(T_{v,i}^{\mathrm{leaf}}\) is the subtree rooted at the \(i\)-th child of \(v\). Coassociativity follows because both \((\Delta \circ \mathrm{id}) \circ \Delta\) and \((\mathrm{id} \circ \Delta) \circ \Delta\) sum over all pairs of edges in \(T\), with the two orderings producing the same set of terms.
3. The Operadic Cobar Construction 📐
3.1 Definition
Definition (Cobar construction). For an augmented conilpotent dg-cooperad \((\mathcal{C}, d, \Delta, \epsilon)\), the cobar construction \(\Omega(\mathcal{C})\) is the augmented dg-operad defined as follows.
As a graded operad (ignoring the differential), \(\Omega(\mathcal{C}) = \mathcal{F}(\bar{\mathcal{C}}[-1])\), the free operad on the desuspension \(\bar{\mathcal{C}}[-1]\). Elements are linear combinations of rooted trees with internal vertices labeled by elements of \(\bar{\mathcal{C}}[-1]\).
The differential \(d_{\Omega(\mathcal{C})} = d_1 + d_2\) where: - \(d_1\): induced by the internal differential \(d\) on \(\mathcal{C}\). - \(d_2\): the cocomposition differential, which increases weight by 1. On a single vertex labeled by \(c \in \bar{\mathcal{C}}(n)\), \(d_2(s^{-1}c) = \sum s^{-1}c' \circ_i s^{-1}c''\) summing over the components of \(\Delta(c)\) (split into a tree with one interior edge).
The cooperad \(\mathrm{Com}^c\) (dual of \(\mathrm{Com}\)) has \(\mathrm{Com}^c(n) = k\) with trivial \(S_n\)-action. The cobar construction \(\Omega(\mathrm{Com}^c)\) is the operad encoding \(L_\infty\)-algebras, with generators \(\ell_n\) in each arity \(n \geq 2\) of degree \(2-n\). This is the content of the Koszul duality \(\mathrm{Com}^! = \mathrm{Lie}^c\).
3.2 The Bar-Cobar Adjunction
Theorem (Bar-cobar adjunction). There is an adjunction \[\Omega : \mathbf{dg\text{-}Coop}^{\mathrm{aug, conil}} \rightleftharpoons \mathbf{dg\text{-}Op}^{\mathrm{aug}} : B\] with unit \(\mathcal{C} \to B\Omega(\mathcal{C})\) and counit \(\Omega B(\mathcal{O}) \to \mathcal{O}\).
🔑 The counit is a quasi-isomorphism: For any augmented dg-operad \(\mathcal{O}\), the natural map \(\Omega B(\mathcal{O}) \xrightarrow{\sim} \mathcal{O}\) is a quasi-isomorphism. This makes \(\Omega B(\mathcal{O})\) a cofibrant resolution of \(\mathcal{O}\) — a free dg-operad with differential encoding all operadic compositions in \(\mathcal{O}\).
This unpacks the universal property of the bar-cobar adjunction in the simplest case.
Prerequisites: 3.2 The Bar-Cobar Adjunction
Let \(\mathcal{O}\) be a non-dg augmented operad. Describe the counit map \(\Omega B(\mathcal{O}) \to \mathcal{O}\) explicitly on generators: what does it send a weight-1 generator \(s^{-1}(s\mu) \in \bar{\mathcal{C}}[-1]\) (for \(\mu \in \bar{\mathcal{O}}(n)\)) to? What is the image of a weight-2 generator?
Key insight: The counit sends weight-1 generators to the corresponding operation in \(\mathcal{O}\), and kills all weight \(\geq 2\) generators — which is why \(\Omega B(\mathcal{O}) \to \mathcal{O}\) is a surjection onto \(\mathcal{O}\) with acyclic kernel.
Sketch: A weight-1 generator of \(\Omega B(\mathcal{O})\) is \(s^{-1}(s\mu) = \mu \in \bar{\mathcal{O}}(n)\) (the suspension and desuspension cancel). The counit sends this to \(\mu \in \mathcal{O}(n)\). A weight-2 generator is \(s^{-1}(s\mu) \circ_i s^{-1}(s\nu)\) for \(\mu \in \bar{\mathcal{O}}(m)\), \(\nu \in \bar{\mathcal{O}}(n)\); the counit sends this to \(\mu \circ_i \nu \in \mathcal{O}(m+n-1)\). But in \(\Omega B(\mathcal{O})\) there is also the differential term \(d_2(s^{-1}s(\mu \circ_i \nu)) = s^{-1}(s\mu) \circ_i s^{-1}(s\nu) - s^{-1}s(\mu \circ_i \nu)\), which shows these weight-2 elements are exact in the acyclic complex \(\ker(\Omega B(\mathcal{O}) \to \mathcal{O})\).
This is a concrete computation showing \(B(\mathrm{Com})\) has the right homology, illustrating the bar-cobar quasi-isomorphism.
Prerequisites: 2.2 The Bar Complex, 2.3 The Differential
Compute the homology of \(B(\mathrm{Com})(n)\) in weight 1 and weight 2 for \(n = 2, 3\). (Recall \(\bar{\mathrm{Com}}(n) = k\) for \(n \geq 2\).) Identify the pattern and state what it predicts for the Koszul criterion.
Key insight: \(H_\bullet(B(\mathrm{Com})(n))\) is concentrated in weight \(n-1\), the minimum possible weight, with \(H = k\) in that weight and zero elsewhere — exactly the Koszul condition.
Sketch: \(B(\mathrm{Com})(2)\): only weight 1, one generator \([s\mu_2]\) where \(\mu_2 \in \bar{\mathrm{Com}}(2) = k\). No differential. \(H = k\) in weight 1. For \(n=3\), weight 1: one generator \([s\mu_3]\). Weight 2: three generators \([s\mu_2 \circ_i s\mu_2]\) for \(i=1,2\) (two binary trees with 3 leaves), but \(\mathrm{Com}\) is symmetric so all \(S_3\)-orbits collapse to two basis elements. The differential \(d_2([s\mu_3]) = 0\) (weight 1 has no decomposition) and \(d_2([s\mu_2 \circ_1 s\mu_2]) = [s\mu_3]\), \(d_2([s\mu_2 \circ_2 s\mu_2]) = [s\mu_3]\) (up to sign). Thus \(d_2\) on weight 2 maps onto weight 1; the kernel in weight 2 is the difference \([s\mu_2 \circ_1 s\mu_2] - [s\mu_2 \circ_2 s\mu_2]\), which has no further boundary. So \(H\) is concentrated in weight 2 = \(n-1\) for \(n=3\), as required.
4. Twisting Morphisms 💡
4.1 The Maurer-Cartan Equation
The bar-cobar adjunction is best understood through twisting morphisms, which are the “structure constants” mediating between cooperads and operads.
Definition (Twisting morphism). Let \(\mathcal{C}\) be an augmented conilpotent dg-cooperad and \(\mathcal{O}\) an augmented dg-operad. A twisting morphism \(\alpha \in \mathrm{Hom}_{\mathbf{SymSeq}}(\bar{\mathcal{C}}, \bar{\mathcal{O}})\) of degree \(-1\) is a collection of \(S_n\)-equivariant maps \(\alpha(n): \bar{\mathcal{C}}(n) \to \bar{\mathcal{O}}(n)\) of degree \(-1\) satisfying the Maurer–Cartan equation: \[\partial(\alpha) + \gamma \circ (\alpha \circ \alpha) \circ \Delta = 0 \quad \in \mathrm{Hom}_{\mathbf{SymSeq}}(\bar{\mathcal{C}}, \bar{\mathcal{O}}).\]
Here \(\partial(\alpha) = d_\mathcal{O} \circ \alpha - (-1)^{|\alpha|} \alpha \circ d_\mathcal{C}\) is the chain-complex differential on \(\mathrm{Hom}\), and \(\gamma \circ (\alpha \circ \alpha) \circ \Delta\) encodes the sum over all binary decompositions of \(\mathcal{C}\) followed by applying \(\alpha\) twice and composing in \(\mathcal{O}\).
The set of twisting morphisms is denoted \(\mathrm{Tw}(\mathcal{C}, \mathcal{O})\).
The space \(\mathrm{Hom}_{\mathbf{SymSeq}}(\bar{\mathcal{C}}, \bar{\mathcal{O}})\) carries a convolution pre-Lie algebra structure: \((f \star g)(c) = \gamma(\mathrm{id} \circ (f \otimes g))(\Delta(c))\). The Maurer–Cartan equation is \(\partial(\alpha) + \alpha \star \alpha = 0\) in this algebra. This is the operadic analogue of the Maurer–Cartan equation \(dA + A \wedge A = 0\) in gauge theory.
4.2 The Universal Twisting Morphisms
Theorem (Representability). The functor \(\mathrm{Tw}(-, \mathcal{O})\) is representable by \(B(\mathcal{O})\), and \(\mathrm{Tw}(\mathcal{C}, -)\) is representable by \(\Omega(\mathcal{C})\): \[\mathrm{Tw}(\mathcal{C}, \mathcal{O}) \cong \mathrm{Hom}_{\mathbf{dg\text{-}Op}}(\Omega\mathcal{C}, \mathcal{O}) \cong \mathrm{Hom}_{\mathbf{dg\text{-}Coop}}(\mathcal{C}, B\mathcal{O}).\]
The universal twisting morphism from the bar construction is \(\pi: B(\mathcal{O}) \to \mathcal{O}\), the map projecting onto weight-1 trees. The universal twisting morphism into the cobar construction is \(\iota: \mathcal{C} \to \Omega(\mathcal{C})\), the inclusion as weight-1 generators.
🔑 A twisting morphism \(\alpha: \mathcal{C} \to \mathcal{O}\) is a quasi-isomorphism iff the twisted composite \(\mathcal{C} \circ_\alpha \mathcal{O}\) (with differential \(d_\alpha = d + \alpha\)) is acyclic. This is the key criterion underlying Koszul duality.
This shows that the identity morphism \(\mathrm{id}: \mathcal{O}^! \to \mathcal{O}^!\) does not in general solve the Maurer-Cartan equation, distinguishing the canonical twisting morphism \(\kappa\) from naive candidates.
Prerequisites: 4.1 The Maurer-Cartan Equation
Let \(\mathcal{O}\) be a non-dg quadratic operad with \(\mathcal{O}^!\) its Koszul dual. Explain why the degree-0 identity map \(\mathrm{id}: \bar{\mathcal{O}}^! \to \bar{\mathcal{O}}^!\) cannot be a twisting morphism \(\mathcal{O}^! \to \mathcal{O}^!\). Then explain why the canonical \(\kappa: \mathcal{O}^! \to \mathcal{O}\) (of degree \(-1\)) can satisfy the Maurer–Cartan equation.
Key insight: Twisting morphisms must have degree \(-1\); the identity has degree 0. The degree shift from the Koszul dual construction (\(sE^\vee\) instead of \(E^\vee\)) is exactly what gives \(\kappa\) the correct degree.
Sketch: The Maurer–Cartan equation requires \(\alpha\) to have degree \(-1\) (so that \(\alpha \star \alpha\) has degree \(-2\) and \(\partial(\alpha)\) has degree \(-1 - 1 = -2\), making the equation degree-homogeneous at \(-2\)). The identity has degree 0, so \(\partial(\mathrm{id}) = 0\) but \(\mathrm{id} \star \mathrm{id}\) has degree 0, and the equation \(0 + \mathrm{id} \star \mathrm{id} = 0\) says \(\gamma \circ \Delta = 0\), which is false for non-trivial cooperads. The canonical \(\kappa: \mathcal{O}^!(n) \to \mathcal{O}(n)\) is defined on generators \(sE^\vee \to E\) by the degree-\((-1)\) evaluation pairing, and satisfies MC because the quadratic relations of \(\mathcal{O}^!\) are the orthogonal complement of those of \(\mathcal{O}\).
5. Quadratic Operads and the Koszul Dual 📐
5.1 Quadratic Operads
Most operads encountered in practice are quadratic: generated by binary operations with relations of degree 2.
Definition (Quadratic operad). A quadratic operad is an operad of the form \(\mathcal{O} = \mathcal{F}(E)/(R)\), where: - \(E \in \mathbf{SymSeq}\) is a symmetric sequence concentrated in arity 2 (the generators), - \(R \subseteq \mathcal{F}(E)(3)\) is an \(S_3\)-submodule (the relations), and - \((R)\) is the operadic ideal generated by \(R\).
We write \(\mathcal{O} = \mathcal{F}(E, R)\) for the quadratic operad with generators \(E\) and relations \(R\).
- \(\mathrm{Ass} = \mathcal{F}(E_\mathrm{Ass}, R_\mathrm{Ass})\) with \(E_\mathrm{Ass}(2) = k[S_2]\) (two binary operations \(\mu\) and \(\mu \cdot (12)\)) and \(R_\mathrm{Ass} = \langle \mu \circ_1 \mu - \mu \circ_2 \mu \rangle\) (associativity). - \(\mathrm{Com} = \mathcal{F}(E_\mathrm{Com}, R_\mathrm{Com})\) with \(E_\mathrm{Com}(2) = k\) (one binary symmetric operation) and \(R_\mathrm{Com} = \langle \mu \circ_1 \mu - \mu \circ_2 \mu \rangle\) (associativity; commutativity is imposed by the trivial \(S_2\)-action). - \(\mathrm{Lie} = \mathcal{F}(E_\mathrm{Lie}, R_\mathrm{Lie})\) with \(E_\mathrm{Lie}(2) = k \cdot [-, -]\) (antisymmetric bracket, \(S_2\) acts by \(-1\)) and \(R_\mathrm{Lie}\) the Jacobi identity \([[a,b],c] + [[b,c],a] + [[c,a],b] = 0\).
5.2 The Koszul Dual Cooperad
Definition (Koszul dual cooperad). For a quadratic operad \(\mathcal{O} = \mathcal{F}(E, R)\), the Koszul dual cooperad is \[\mathcal{O}^! = \mathcal{F}^c(sE^\vee, s^2 R^\perp),\] the quadratic cooperad with cogenerators \(sE^\vee\) (the suspension of the linear dual of \(E\)) and corelations \(s^2 R^\perp\), where \(R^\perp \subseteq \mathcal{F}(E^\vee)(3)\) is the annihilator of \(R\) under the natural pairing \(\mathcal{F}(E^\vee)(3) \otimes \mathcal{F}(E)(3) \to k\).
The suspension shifts are: \((sE^\vee)(2) = E^\vee(2)[-1]\), so if \(E(2)\) is concentrated in degree 0, then \((sE^\vee)(2)\) is in degree \(+1\). This gives \(\mathcal{O}^!(n)\) a homological degree of \(n-1\) (for \(n\)-ary cooperations).
There is a related but distinct notion, the Koszul dual operad \(\mathcal{O}^* = \mathcal{F}(E^\vee, R^\perp)\) (without the suspension). The cooperad \(\mathcal{O}^!\) is related by \(\mathcal{O}^! = s\mathcal{O}^*\) (suspension of the dual operad, viewed as a cooperad). Loday–Vallette use \(\mathcal{O}^!\) for the cooperad; some sources write \(\mathcal{O}^*\) for the dual operad. Be careful not to confuse them.
5.3 The Canonical Twisting Morphism
There is a canonical degree-\((-1)\) map \(\kappa: \mathcal{O}^! \to \mathcal{O}\), defined on cogenerators \(sE^\vee \to E\) by the evaluation pairing \((se^\vee)(e) = e^\vee(e) \in k\), extended to all of \(\mathcal{O}^!\) by the universal property of the cofree cooperad, and zero on all weight \(\geq 2\).
Proposition. \(\kappa: \mathcal{O}^! \to \mathcal{O}\) is a twisting morphism: it satisfies the Maurer–Cartan equation \(\partial(\kappa) + \gamma \circ (\kappa \circ \kappa) \circ \Delta = 0\).
Proof sketch. The equation in weight 2 reduces to \(\gamma(e^\vee_1 \otimes e^\vee_2)(\Delta(c)) = 0\) for all \(c \in \mathcal{O}^!(3)\). Since \(c \in s^2 R^\perp\) by definition, and \(R^\perp\) annihilates \(R\), this is exactly the condition that the image of \(\gamma \circ (\kappa \circ \kappa)\) lies in the image of the relations of \(\mathcal{O}\), which is zero in \(\mathcal{O}\). □
5.4 Examples: Ass, Com, and Lie
| Operad | Generators | Relations | Koszul dual \(\mathcal{O}^!\) |
|---|---|---|---|
| \(\mathrm{Ass}\) | \(k[S_2]\) | Associativity | \(\mathrm{Ass}^c\) (self-dual) |
| \(\mathrm{Com}\) | \(k\) (trivial) | Associativity + commutativity | \(\mathrm{Lie}^c\) |
| \(\mathrm{Lie}\) | \(k \cdot [-,-]\) (sign rep) | Jacobi | \(\mathrm{Com}^c\) |
🔑 The Com–Lie Koszul duality \(\mathrm{Com}^! = \mathrm{Lie}^c\) and \(\mathrm{Lie}^! = \mathrm{Com}^c\) is the operadic shadow of the classical PBW theorem and the Chevalley–Eilenberg complex.
This establishes the self-duality of \(\mathrm{Ass}\) and clarifies why it is special among the classical operads.
Prerequisites: 5.2 The Koszul Dual Cooperad, 5.1 Quadratic Operads
Let \(\mathrm{Ass} = \mathcal{F}(k[S_2], R)\) with \(R = \langle \mu \circ_1 \mu - \mu \circ_2 \mu \rangle \subseteq \mathcal{F}(k[S_2])(3) \cong k[S_3]^{\oplus 2}\). Compute \(R^\perp \subseteq \mathcal{F}(k[S_2]^\vee)(3)\) and show that \(\mathrm{Ass}^! \cong \mathrm{Ass}^c\) as cooperads.
Key insight: \(R\) and \(R^\perp\) have the same dimension (both \(\frac{1}{2}\)-dimensional in each \(S_3\)-isotypic component), and the pairing swaps them, giving \(\mathrm{Ass}^* \cong \mathrm{Ass}\).
Sketch: \(\mathcal{F}(k[S_2])(3) \cong k[S_3]\) (the free associative operad in arity 3 has basis all parenthesizations of 3 inputs, i.e., \(S_3\) elements). The relation space \(R = \ker(\mathcal{F}(k[S_2])(3) \to \mathrm{Ass}(3))\): since \(\mathrm{Ass}(3) = k[S_3]\) and the map is an isomorphism (no relations kill things), we have \(R = 0\)! Wait — more carefully, \(\mathrm{Ass}(3) = k[S_3]\) and \(\mathcal{F}(k[S_2])(3) = k[S_3]^2\) (two planar binary trees), so \(R = \langle (\sigma, -\sigma) : \sigma \in S_3\rangle\), a rank-6 subspace of the rank-12 space \(k[S_3]^2\). Then \(R^\perp\) has rank 6 as well and one checks the natural pairing identifies it with a copy of \(k[S_3]\) — i.e., \(\mathrm{Ass}^*\) is again generated by \(k[S_2]^\vee \cong k[S_2]\) with the same relations. So \(\mathrm{Ass}^! \cong \mathrm{Ass}^c\).
This establishes the Com-Lie Koszul duality at the level of generators and relations.
Prerequisites: 5.2 The Koszul Dual Cooperad
Show that \(\mathrm{Com}^! \cong \mathrm{Lie}^c\): identify the cogenerators of \(\mathrm{Com}^!\) (as a cooperad) with the generators of \(\mathrm{Lie}\) (as an operad), and show that the corelations of \(\mathrm{Com}^!\) are the dual of the Jacobi identity.
Key insight: The annihilator of the associativity+commutativity relations in \(\mathcal{F}(k)(3) = k[S_3]\) is spanned by the antisymmetrizer and its cyclic permutations — which is exactly the Jacobi identity in antisymmetric form.
Sketch: \(E_\mathrm{Com} = k\) with trivial \(S_2\)-action, so \(E_\mathrm{Com}^\vee = k\) with trivial \(S_2\)-action, but \(sE_\mathrm{Com}^\vee = k[-1]\) with \(S_2\) acting by the sign representation (the suspension of a trivial representation under the Koszul sign rule becomes the sign representation). This is exactly \(E_\mathrm{Lie}\). The corelation space \(s^2 R_\mathrm{Com}^\perp \subseteq \mathcal{F}(E_\mathrm{Lie})(3)\): one computes \(R_\mathrm{Com}^\perp = \langle e_1 \otimes e_2 \circ_1 e_3 + e_2 \otimes e_3 \circ_1 e_1 + e_3 \otimes e_1 \circ_1 e_2 \rangle\), which is the Jacobi identity. So the quadratic operad with these generators and relations is \(\mathrm{Lie}\), confirming \(\mathrm{Com}^* \cong \mathrm{Lie}\) and \(\mathrm{Com}^! \cong \mathrm{Lie}^c\).
6. The Koszul Criterion 🔑
6.1 Statement
Definition (Koszul operad). A quadratic operad \(\mathcal{O}\) is Koszul if the canonical twisting morphism \(\kappa: \mathcal{O}^! \to \mathcal{O}\) is a quasi-isomorphism, i.e., if \(H_\bullet(\mathcal{O}^! \circ_\kappa \mathcal{O}) \cong \mathbf{1}\) (the unit symmetric sequence).
Theorem (Equivalent criteria). The following are equivalent for a quadratic operad \(\mathcal{O}\):
- \(\kappa: \mathcal{O}^! \to \mathcal{O}\) is a quasi-isomorphism.
- The twisted composite product \(\mathcal{O}^! \circ_\kappa \mathcal{O}\) (the Koszul complex) is acyclic.
- \(H_\bullet(B(\mathcal{O}))\) is concentrated in weight $= $ homological degree (i.e., pure weight).
- For each \(n\), the component \(\mathcal{O}(n)\) has a basis given by a PBW-type basis of decorated trees.
Criteria 1–3 are equivalent but not directly computable without knowing the full bar complex. Criterion 4 (the PBW/rewriting criterion, due to Dotsenko–Khoroshkin and Loday–Vallette) gives an algorithmic check: \(\mathcal{O}\) is Koszul iff its relations form a Gröbner basis for the free operad, equivalently iff a quadratic PBW basis exists. This is how one proves that \(\mathrm{Perm}\), \(\mathrm{PreLie}\), and \(\mathrm{Dend}\) are Koszul.
6.2 Koszulness of Ass and Com
Theorem. The operads \(\mathrm{Ass}\), \(\mathrm{Com}\), and \(\mathrm{Lie}\) are Koszul.
Proof for \(\mathrm{Ass}\). The Koszul complex \(\mathrm{Ass}^c \circ_\kappa \mathrm{Ass}\) has components \((\mathrm{Ass}^c \circ_\kappa \mathrm{Ass})(n)\). By direct computation using the fact that \(\mathrm{Ass}^c(k) = k[S_k]\) and \(\mathrm{Ass}(n) = k[S_n]\), the complex in arity \(n\) is isomorphic to the augmented chain complex of the \((n-2)\)-simplex \(\Delta^{n-2}\) (tensored with \(k[S_n]\)). Since \(\Delta^{n-2}\) is contractible, the complex is acyclic. □
Proof sketch for \(\mathrm{Com}\). The Koszul complex \(\mathrm{Lie}^c \circ_\kappa \mathrm{Com}\) in arity \(n\) is isomorphic (as a chain complex of \(S_n\)-modules) to the top-weight part of the homology of the partition lattice \(\Pi_n\). By a theorem of Sundaram, this is acyclic in all degrees except the top, where \(H_{n-2}(\Pi_n) = \mathrm{Lie}(n)\) (the representation of \(S_n\) on the free Lie algebra). This confirms \(H_\bullet(\mathrm{Lie}^c \circ_\kappa \mathrm{Com}) \cong \mathbf{1}\), so \(\mathrm{Com}\) is Koszul. □
This makes the proof of Koszulness of Ass concrete in the simplest nontrivial case.
Prerequisites: 6.1 Statement, 5.4 Examples: Ass, Com, and Lie
Write out the Koszul complex \((\mathrm{Ass}^c \circ_\kappa \mathrm{Ass})(3)\) explicitly: identify the chain groups in each weight, write the differential \(d_\kappa\), and verify it is acyclic. Identify it as the chain complex of the 1-simplex \(\Delta^1\).
Key insight: The complex has two chain groups (weight 1 and weight 2), and the differential is an isomorphism — the chain complex of an interval.
Sketch: \((\mathrm{Ass}^c \circ_\kappa \mathrm{Ass})(3)\): weight 1 part is \(\mathrm{Ass}^c(3) \otimes_{S_3} \mathrm{Ass}(1)^{\otimes 3} \oplus \cdots\), but the leading term is \(\mathrm{Ass}^c(1) \cdot \mathrm{Ass}(3) = k[S_3]\) (one vertex, all 3 inputs entering directly) and \(\mathrm{Ass}^c(3) \cdot \mathrm{Ass}(1)^3 = k[S_3]\) in weight 1. Actually weight 1 is \(\bar{\mathrm{Ass}}^c(3) \otimes 1 \cong k[S_3]\) and weight 2 is \(\bigoplus_{n_1+n_2=3} \bar{\mathrm{Ass}}^c(2) \otimes (\bar{\mathrm{Ass}}(n_1) \otimes \bar{\mathrm{Ass}}(n_2)) \cong k[S_3] \oplus k[S_3]\) (two splittings: \((2,1)\) and \((1,2)\)). The differential \(d_\kappa\) on the weight-2 generators is \(\kappa\) applied to the cooperadic generator: \(d_\kappa(\mu^* \otimes \nu_1 \otimes \nu_2) = \kappa(\mu^*) \circ (\nu_1 \otimes \nu_2) = \mu \circ (\nu_1 \otimes \nu_2)\). Computing the chain groups as \(k[S_3]\) in weight 1 and \(k[S_3]^2\) in weight 2, the differential is the map \((\mathrm{id}, -\mathrm{id}): k[S_3]^2 \to k[S_3]\), which is surjective with kernel \(k[S_3]\) (the diagonal). The kernel is the image of the weight-3 differential (which is zero since there is no weight 3 for \(n=3\)). So \(H_0 = 0\) and \(H_{-1} = 0\) except at weight = homological degree, confirming acyclicity. This is the chain complex \(k[S_3] \xrightarrow{\sim} k[S_3]\) of \(\Delta^1 \otimes k[S_3]\).
This gives an example where the Koszul criterion fails, illustrating that Koszulness is a non-trivial condition.
Prerequisites: 6.1 Statement
The pre-Lie operad \(\mathrm{PreLie}\) is generated by one binary operation \(a \rhd b\) with the single relation \((a \rhd b) \rhd c - a \rhd (b \rhd c) = (a \rhd c) \rhd b - a \rhd (c \rhd b)\). Show this is a quadratic operad. State (without proof) whether it is Koszul, and identify its Koszul dual.
Key insight: \(\mathrm{PreLie}\) is Koszul, with Koszul dual \(\mathrm{Perm}\) (the permutation operad), reflecting the fact that pre-Lie algebras are the correct framework for Rota–Baxter structures.
Sketch: The pre-Lie relation is a quadratic relation in \(\mathcal{F}(E)(3)\) where \(E(2) = k \cdot \mu\) (one non-symmetric binary operation). The Koszul dual has generators \(k \cdot \mu^\vee\) with the orthogonal complement relation, which turns out to be \(\mathrm{Perm}\): the operad encoding algebras with a bilinear product satisfying \((a \cdot b) \cdot c = a \cdot (b \cdot c)\) and \((a \cdot b) \cdot c = (a \cdot c) \cdot b\) (the permutation associativity). \(\mathrm{PreLie}\) is Koszul by the rewriting criterion (Dotsenko–Khoroshkin): its generators form a Gröbner basis for the free operad with the appropriate monomial order.
7. Homotopy Algebras from Koszul Duality 💡
7.1 The Resolution O-infinity
For a Koszul operad \(\mathcal{O}\), the bar-cobar adjunction produces a canonical cofibrant resolution.
Definition (\(\mathcal{O}_\infty\)). For a Koszul operad \(\mathcal{O}\), the homotopy \(\mathcal{O}\)-operad is \[\mathcal{O}_\infty := \Omega(\mathcal{O}^!),\] the cobar construction on the Koszul dual cooperad.
Theorem. The canonical map \(\mathcal{O}_\infty = \Omega(\mathcal{O}^!) \xrightarrow{\sim} \mathcal{O}\) induced by \(\kappa: \mathcal{O}^! \to \mathcal{O}\) is a quasi-isomorphism of dg-operads. Moreover, \(\mathcal{O}_\infty\) is a free graded operad (as a graded operad, ignoring the differential), hence cofibrant in the model structure on dg-operads.
Both \(\Omega B(\mathcal{O}) \xrightarrow{\sim} \mathcal{O}\) and \(\Omega(\mathcal{O}^!) \xrightarrow{\sim} \mathcal{O}\) are cofibrant resolutions. The bar-cobar resolution \(\Omega B(\mathcal{O})\) always exists but is enormous (exponential in the weight). The Koszul resolution \(\Omega(\mathcal{O}^!)\) is much smaller — it uses only \(\mathcal{O}^!\) rather than all of \(B(\mathcal{O})\). Koszulness is precisely the condition that these two resolutions are quasi-isomorphic.
7.2 A-infinity and L-infinity Algebras
The two most important instances:
\(A_\infty\)-algebras. Since \(\mathrm{Ass}\) is Koszul and \(\mathrm{Ass}^! = \mathrm{Ass}^c\), \[A_\infty = \Omega(\mathrm{Ass}^c).\] A graded module \(A\) is an \(A_\infty\)-algebra iff it carries operations \(m_n: A^{\otimes n} \to A\) of degree \(2-n\) (\(n \geq 1\)) satisfying the Stasheff relations: \[\sum_{r+s+t=n} (-1)^{rs+t} m_{r+1+t}(\mathrm{id}^{\otimes r} \otimes m_s \otimes \mathrm{id}^{\otimes t}) = 0 \quad \forall n \geq 1.\] The first few: \(m_1^2 = 0\) (so \(d = m_1\) is a differential), \(m_1 m_2 + m_2(m_1 \otimes \mathrm{id} + \mathrm{id} \otimes m_1) = 0\) (so \(m_2\) is a chain map), and \(m_2(\mathrm{id} \otimes m_2) - m_2(m_2 \otimes \mathrm{id}) = m_1 m_3 + m_3(\text{boundary}) + \cdots\) (so \(m_2\) is homotopy-associative with \(m_3\) as the homotopy).
See Algebras and Modules §7 for the full treatment.
\(L_\infty\)-algebras. Since \(\mathrm{Lie}\) is Koszul and \(\mathrm{Lie}^! = \mathrm{Com}^c\), \[L_\infty = \Omega(\mathrm{Com}^c).\] An \(L_\infty\)-algebra is a graded module with antisymmetric brackets \(\ell_n: \Lambda^n A \to A\) of degree \(2-n\) satisfying generalized Jacobi identities. An ordinary dg-Lie algebra is the special case \(\ell_n = 0\) for \(n \geq 3\). \(L_\infty\)-algebras are the correct notion of a “Lie algebra up to all higher homotopies” and govern formal deformation problems via the Maurer–Cartan equation \(\sum_n \frac{1}{n!}\ell_n(\gamma^{\otimes n}) = 0\).
7.3 The Homotopy Transfer Theorem
The key application of \(\mathcal{O}_\infty\)-algebras is that they transfer along deformation retracts.
Theorem (Homotopy Transfer). Let \(A\) be an \(\mathcal{O}_\infty\)-algebra and suppose there is a diagram \[\begin{tikzcd} H \arrow[r, shift left, "i"] & A \arrow[l, shift left, "p"] \end{tikzcd} \quad \text{with } pi = \mathrm{id}_H, \quad ip \simeq \mathrm{id}_A \text{ (chain homotopy } h: A \to A[-1]).\] Then \(H\) carries an induced \(\mathcal{O}_\infty\)-algebra structure, and \(i\) extends to an \(\infty\)-quasi-isomorphism of \(\mathcal{O}_\infty\)-algebras.
The transferred operations on \(H\) are given by explicit tree formulas: the \(n\)-ary operation on \(H\) is a sum over binary trees with \(n\) leaves, where each leaf is labeled by \(i\), each internal vertex by an operation of \(A\), and each internal edge by the homotopy \(h\).
If \(A\) is a dg-associative algebra and \(H = H^\bullet(A)\) with the zero differential, the homotopy transfer theorem gives \(H\) the structure of an \(A_\infty\)-algebra (the minimal model of \(A\)) with \(m_1 = 0\), $m_2 = $ induced product, and \(m_n\) (\(n \geq 3\)) the higher Massey products. The \(A_\infty\)-algebra \(H\) is quasi-isomorphic to \(A\) as an \(A_\infty\)-algebra. This is why cohomology “knows” the quasi-isomorphism type of \(A\).
This makes the homotopy transfer theorem concrete for the simplest nontrivial case.
Prerequisites: 7.3 The Homotopy Transfer Theorem, 7.2 A-infinity and L-infinity Algebras
Let \(A\) be a dg-associative algebra with a deformation retract \((i, p, h)\) onto \(H = H^\bullet(A)\). Write down the formula for the transferred operation \(m_3: H^{\otimes 3} \to H\) using the tree formula, and identify it as a Massey product.
Key insight: \(m_3(x,y,z) = p(h(i(x) \cdot i(y)) \cdot i(z)) \pm p(i(x) \cdot h(i(y) \cdot i(z)))\), which is exactly the triple Massey product \(\langle [x], [y], [z] \rangle\).
Sketch: The tree formula for \(m_3\) sums over the two binary trees with 3 leaves and one internal edge. Tree 1 (left-leaning): \(p \circ m_2^A \circ (h \circ m_2^A \circ (i \otimes i) \otimes i)\), giving \(p(h(i(x) \cdot i(y)) \cdot i(z))\). Tree 2 (right-leaning): \(p \circ m_2^A \circ (i \otimes h \circ m_2^A \circ (i \otimes i))\), giving \((-1)^\star p(i(x) \cdot h(i(y) \cdot i(z)))\) with sign from the Koszul rule. The sum is the Massey product \(\langle x, y, z \rangle\) (when \(xy = yz = 0\) in cohomology, so the Massey product is defined); more precisely, \(m_3\) computes the indeterminacy-free Massey product, which is always defined for an \(A_\infty\)-algebra.
This connects L-infinity algebras to the deformation theory of algebraic structures.
Prerequisites: 7.2 A-infinity and L-infinity Algebras
Let \(\mathfrak{g}\) be an \(L_\infty\)-algebra (concentrated in non-negative degrees). The Maurer–Cartan set is \(\mathrm{MC}(\mathfrak{g}) = \{\gamma \in \mathfrak{g}^1 : \sum_{n \geq 1} \frac{1}{n!} \ell_n(\gamma^{\otimes n}) = 0\}\). Show that when \(\mathfrak{g}\) is an ordinary dg-Lie algebra (i.e., \(\ell_n = 0\) for \(n \geq 3\)), this reduces to the classical Maurer–Cartan equation \(d\gamma + \frac{1}{2}[\gamma, \gamma] = 0\).
Key insight: For a dg-Lie algebra, only the \(n=1\) and \(n=2\) terms survive in the \(L_\infty\) MC equation, giving exactly \(\ell_1(\gamma) + \frac{1}{2}\ell_2(\gamma, \gamma) = d\gamma + \frac{1}{2}[\gamma, \gamma] = 0\).
Sketch: The \(L_\infty\) MC equation is \(\sum_{n=1}^\infty \frac{1}{n!} \ell_n(\gamma^{\otimes n}) = 0\). For \(n=1\): \(\ell_1 = d\) (the differential). For \(n=2\): \(\frac{1}{2}\ell_2(\gamma,\gamma) = \frac{1}{2}[\gamma,\gamma]\). For \(n \geq 3\): \(\ell_n = 0\) by assumption. So the equation is \(d\gamma + \frac{1}{2}[\gamma,\gamma] = 0\). The factor of \(\frac{1}{2}\) absorbs the antisymmetry \([\gamma,\gamma] = -[\gamma,\gamma]\): since \(\gamma \in \mathfrak{g}^1\) is degree 1, the Koszul sign gives \([\gamma,\gamma] \neq 0\) in general (it equals \(2\gamma^2\) in the universal enveloping algebra).
8. Operadic Cohomology and Deformation Theory 🔑
8.1 The Two-Sided Bar Resolution
The bar construction extends to a two-sided version that resolves an algebra \(A\) as a module over itself.
Definition (Two-sided bar resolution). For an \(\mathcal{O}\)-algebra \(A\) and \(A\)-modules \(M\), \(N\) (left and right respectively), the two-sided bar complex is \[B(\mathcal{O}, A, A) := \mathcal{O} \circ_\kappa A,\] with \(A\) playing the role of both left and right arguments. As a graded symmetric sequence, \[B(\mathcal{O}, A, A)(n) = \bigoplus_k \mathcal{O}^!(k) \otimes_{S_k} A^{\otimes k} \otimes \cdots\] with differential combining the internal differential on \(A\) with the twisted differential from \(\kappa\).
🔑 \(B(\mathcal{O}, A, A)\) is a free resolution of \(A\) as an \(\mathcal{O}\)-algebra (when \(\mathcal{O}\) is Koszul). This replaces the classical bar resolution \(A \otimes A^{\otimes \bullet} \otimes A \to A\) for associative algebras.
Definition (Operadic cohomology). The operadic cohomology of an \(\mathcal{O}\)-algebra \(A\) with coefficients in an \(A\)-module \(M\) is \[H^\bullet_\mathcal{O}(A, M) := H^\bullet\bigl(\mathrm{Hom}_{\mathcal{O}\text{-bimod}}(B(\mathcal{O}, A, A), M)\bigr).\]
8.2 Classical Cohomology Theories as Special Cases
| Operad \(\mathcal{O}\) | \(\mathcal{O}\)-algebra \(A\) | \(H^\bullet_\mathcal{O}(A, M)\) |
|---|---|---|
| \(\mathrm{Ass}\) | Associative algebra | Hochschild cohomology \(HH^\bullet(A, M)\) |
| \(\mathrm{Com}\) | Commutative algebra | Harrison cohomology \(\cong\) André–Quillen cohomology \(AQ^\bullet(A, M)\) |
| \(\mathrm{Lie}\) | Lie algebra \(\mathfrak{g}\) | Chevalley–Eilenberg cohomology \(H^\bullet_\mathrm{CE}(\mathfrak{g}, M)\) |
The operadic framework unifies all three under a single definition and makes transparent why they satisfy the same formal properties (long exact sequences, cup products when \(M = A\), Gerstenhaber algebra structure, etc.).
Harrison cohomology (defined via the anti-symmetrization of the Hochschild complex) and André–Quillen cohomology (the derived functor of derivations for commutative algebras) agree over \(\mathbb{Q}\) but differ in characteristic \(p\) (where Harrison lacks the higher-order terms). The operadic cohomology \(H^\bullet_\mathrm{Com}(A,M)\) recovers Harrison cohomology; André–Quillen cohomology arises from the full cotangent complex \(\mathbb{L}_{A/k}\).
8.3 The Deformation Complex of an Operad
The bar construction also computes deformations of the operad \(\mathcal{O}\) itself.
Definition (Deformation complex). The deformation complex of a Koszul operad \(\mathcal{O}\) is \[\mathrm{Def}(\mathcal{O}) := \mathrm{Hom}_{\mathbf{SymSeq}}(\mathcal{O}^!, \mathcal{O})\] with the convolution \(L_\infty\)-algebra structure: the bracket \([\alpha, \beta] = \alpha \star \beta - (-1)^{|\alpha||\beta|} \beta \star \alpha\) where \(\star\) is the convolution pre-Lie product from §4.1, and higher brackets from the \(L_\infty\) structure on the convolution algebra.
Theorem. The Maurer–Cartan elements of \(\mathrm{Def}(\mathcal{O})\) are in bijection with deformations of the operad structure on \(\mathcal{O}\): dg-operad structures on \(\mathcal{O}[t](/notes/t/)\) that reduce to \(\mathcal{O}\) modulo \(t\).
The cohomology groups of \(\mathrm{Def}(\mathcal{O})\) control: - \(H^0\): symmetries/automorphisms of \(\mathcal{O}\) (infinitesimal) - \(H^1\): infinitesimal deformations of \(\mathcal{O}\) as an operad - \(H^2\): obstructions to extending deformations
The Kontsevich formality theorem states that the dg-Lie algebra of polyvector fields on \(\mathbb{R}^n\) (the Hochschild complex \(C^\bullet(C^\infty, C^\infty)\) with Gerstenhaber bracket) is formal — quasi-isomorphic to its cohomology. In the operadic language, this is a quasi-isomorphism of \(L_\infty\)-algebras \(\mathrm{Def}(\mathcal{O}) \xrightarrow{\sim} H^\bullet(\mathrm{Def}(\mathcal{O}))\). The formality implies Kontsevich’s deformation quantization: every Poisson manifold has a canonical \(\star\)-product deformation of its function algebra.
This recovers the classical Hochschild complex as a special case of the operadic deformation complex.
Prerequisites: 8.3 The Deformation Complex of an Operad, 5.4 Examples: Ass, Com, and Lie
Show that \(\mathrm{Def}(\mathrm{Ass}) = \mathrm{Hom}_{\mathbf{SymSeq}}(\mathrm{Ass}^c, \mathrm{Ass})\) is isomorphic (as a graded vector space) to the Hochschild cochain complex \(C^\bullet(A, A) = \prod_{n \geq 0} \mathrm{Hom}(A^{\otimes n}, A)\) for the free associative algebra \(A = T(V)\). Identify which part of the Gerstenhaber bracket on \(HH^\bullet\) comes from the convolution pre-Lie product.
Key insight: \(\mathrm{Hom}_{\mathbf{SymSeq}}(\mathrm{Ass}^c, \mathrm{Ass}) = \prod_n \mathrm{Hom}_{S_n}(k[S_n], k[S_n]) = \prod_n \mathrm{End}(k[S_n])^{S_n}\), which for the free algebra \(T(V)\) specializes to \(\prod_n \mathrm{Hom}(V^{\otimes n}, V^{\otimes n})^{S_n} \supseteq \prod_n \mathrm{Hom}(V^{\otimes n}, V)\), recovering the Hochschild complex.
Sketch: An element \(f \in \mathrm{Hom}_{\mathbf{SymSeq}}(\mathrm{Ass}^c(n), \mathrm{Ass}(n)) = \mathrm{Hom}_{S_n}(k[S_n], k[S_n])\) is an \(S_n\)-equivariant endomorphism of \(k[S_n]\), i.e., an element of \(k[S_n]^{\mathrm{op}} = k[S_n]\) itself (by Schur’s lemma for the regular representation). For the free algebra \(T(V) = \bigoplus_n V^{\otimes n}\), a Hochschild \(n\)-cochain \(f: A^{\otimes n} \to A\) corresponds to an \(S_n\)-equivariant map \(k[S_n] \to k[S_n]\) (since \(\mathrm{Ass}(n) = k[S_n]\) acts on \(T(V)(n) = V^{\otimes n}\)). The convolution pre-Lie product \(f \star g(n) = \sum_{i} f \circ_i g\) (inserting \(g\) at the \(i\)-th input of \(f\)) recovers the Gerstenhaber pre-Lie product on Hochschild cochains, whose antisymmetrization is the Gerstenhaber bracket \([f,g] = f \star g - (-1)^{|f||g|} g \star f\).
This shows that deformations of the Com operad are classified by Harrison cohomology, connecting operad deformation theory to commutative algebra.
Prerequisites: 8.3 The Deformation Complex of an Operad, 8.2 Classical Cohomology Theories as Special Cases
Show that \(H^1(\mathrm{Def}(\mathrm{Com})) = H^1_\mathrm{Harrison}(k, k)\), where the Harrison cohomology is taken with coefficients in the ground field \(k\) viewed as a \(k\)-module. Interpret the Maurer–Cartan elements of \(\mathrm{Def}(\mathrm{Com})\) in terms of deformations of the commutative operad structure.
Key insight: \(H^1(\mathrm{Def}(\mathrm{Com}))\) classifies deformations of \(\mathrm{Com}\) that remain quadratic — i.e., deformations within the world of quadratic operads. These correspond to antisymmetric “Lie-type” corrections, linking to the Com–Lie Koszul duality.
Sketch: \(\mathrm{Def}(\mathrm{Com}) = \mathrm{Hom}_{\mathbf{SymSeq}}(\mathrm{Lie}^c, \mathrm{Com})\). In degree 1, this is \(\mathrm{Hom}_{\mathbf{SymSeq}}(\mathrm{Lie}^c(2), \mathrm{Com}(2)) = \mathrm{Hom}_{S_2}(k \cdot \mathrm{sgn}, k) = 0\) (since sgn and trivial representations don’t map nontrivially). This reflects the fact that \(\mathrm{Com}\) is rigid (no nontrivial first-order deformations), consistent with the classical result \(H^2_\mathrm{Harrison}(A,A) = 0\) for polynomial algebras. At degree 2, \(\mathrm{Def}(\mathrm{Com})^2 = \mathrm{Hom}_{S_3}(\mathrm{Lie}^c(3), \mathrm{Com}(3))\); a Maurer–Cartan element here is an antisymmetric ternary operation satisfying a Jacobi-like identity — a deformation toward \(\mathrm{Lie}\).
Mathematical Development Exercises (continued)
This shows that the bar construction of a free operad has trivial homology, consistent with the fact that free operads are “Koszul” in a degenerate sense.
Prerequisites: 2.2 The Bar Complex, 2.3 The Differential
Let \(\mathcal{O} = \mathcal{F}(E)\) be a free operad on a symmetric sequence \(E\) concentrated in arity 2. Show that \(B(\mathcal{F}(E))\) has homology \(H_\bullet(B(\mathcal{F}(E))) = \mathbf{1}\) (the unit cooperad). What does this say about the Koszul criterion for free operads?
Key insight: The bar complex of a free operad is contractible: the generating trees provide an explicit contracting homotopy via “expanding” each arity-2 composition back into a free tree.
Sketch: In \(B(\mathcal{F}(E))\), the weight-\(n\) component consists of trees-of-trees, which can be “straightened” into single trees using the free operad multiplication. The contracting homotopy \(h: B(\mathcal{F}(E)) \to B(\mathcal{F}(E))[-1]\) is defined by picking a splitting of the composition \(\gamma: \mathcal{F}(E) \circ \mathcal{F}(E) \to \mathcal{F}(E)\), i.e., a map \(s: \mathcal{F}(E) \to \mathcal{F}(E) \circ \mathcal{F}(E)\) sending each tree to its “top vertex” and the remaining forest. Then \([d_\gamma, h] = \mathrm{id}\) on the positive-weight part, showing acyclicity. This is consistent with the Koszul criterion: free operads satisfy the criterion trivially since all their relations are zero.
This connects the PBW basis theorem for universal enveloping algebras to Koszulness of the Lie operad.
Prerequisites: 6.1 Statement, 6.2 Koszulness of Ass and Com
The PBW theorem states that \(U(\mathfrak{g}) \cong S(\mathfrak{g})\) as filtered vector spaces (for a Lie algebra \(\mathfrak{g}\)). Show how this follows from the Koszulness of \(\mathrm{Lie}\), using the fact that \(\mathrm{Lie}(n) \cong k[S_n] \otimes_{S_n} \mathrm{sgn}\) (the free Lie algebra) as an \(S_n\)-module.
Key insight: The Koszul complex \(\mathrm{Com}^c \circ_\kappa \mathrm{Lie}\) being acyclic is exactly the PBW theorem: it says that the associated graded of \(U(\mathfrak{g})\) is \(S(\mathfrak{g})\).
Sketch: The Koszul complex for \(\mathrm{Lie}\) is \(\mathrm{Com}^c \circ_\kappa \mathrm{Lie}\). In arity \(n\), this is a complex of \(S_n\)-modules \(\bigoplus_k (\mathrm{Com}^c(k) \otimes \mathrm{Lie}(n_1) \otimes \cdots \otimes \mathrm{Lie}(n_k))_{n_1+\cdots+n_k=n}\) with differential from \(\kappa\). Acyclicity of this complex implies that \(\mathrm{Lie}(n)\) has a PBW basis indexed by the same combinatorial data as \(\mathrm{Sym}^n\) — i.e., \(\dim \mathrm{Lie}(n) = (n-1)!\) (the well-known dimension formula), and the associated graded of \(U(\mathfrak{g})\) is \(S(\mathfrak{g})\) for any \(\mathfrak{g}\).
Algorithmic Applications
This gives a concrete algorithm for computing \(B(\mathcal{O})(n)\) in low weights.
Prerequisites: 2.2 The Bar Complex, 2.3 The Differential
Write a Python pseudocode algorithm that, given a quadratic operad \(\mathcal{O}\) specified by its generators \(E\) and relations \(R\) (as matrices over \(k\)), computes the chain groups and differential of \(B(\mathcal{O})(n)\) for a given \(n\) and weight \(\leq W\).
Key insight: The weight-\(w\) part of \(B(\mathcal{O})(n)\) is a direct sum over rooted trees with \(w\) internal vertices and \(n\) leaves, with each vertex labeled by a basis element of \(\bar{\mathcal{O}}\).
Sketch:
def bar_construction(O_gens, O_rels, n, max_weight):
"""
O_gens: dict {arity: basis_list} for the generators of O-bar
O_rels: list of linear relations as dicts {monomial: coefficient}
Returns: chain_groups, differential_matrices
"""
from itertools import product
def rooted_trees(n_leaves, weight):
"""Enumerate rooted trees with n_leaves leaves and `weight` internal vertices."""
if weight == 1:
yield [n_leaves] # single vertex of arity n_leaves
else:
for arity in range(2, n_leaves):
for partition in integer_partitions(n_leaves, arity):
for subtrees in product(
*[rooted_trees(p, w)
for p, w in distribute_weight(partition, weight-1)]
):
yield (arity, subtrees)
chain_groups = {}
for w in range(1, max_weight + 1):
basis = []
for tree in rooted_trees(n, w):
for labels in label_tree(tree, O_gens):
basis.append((tree, labels))
chain_groups[w] = basis
# Differential: contract each internal edge using O_rels
differentials = {}
for w in range(2, max_weight + 1):
mat = {}
for idx, (tree, labels) in enumerate(chain_groups[w]):
for edge in internal_edges(tree):
contracted = contract_edge(tree, labels, edge, O_rels)
sign = koszul_sign(tree, labels, edge)
mat[idx] = mat.get(idx, {})
for jdx, coeff in contracted:
mat[idx][jdx] = mat[idx].get(jdx, 0) + sign * coeff
differentials[w] = mat
return chain_groups, differentialsThe key subroutines are rooted_trees (enumerate by weight), label_tree (assign operad generators to vertices), contract_edge (apply \(\gamma\) using the relations), and koszul_sign (track the degree-shift signs from suspension). The differential matrix has shape len(chain_groups[w]) × len(chain_groups[w-1]).
This turns the Koszul criterion into a linear algebra computation.
Prerequisites: 6.1 Statement, Exercise 17
Using the output of the algorithm in Exercise 17, describe how to check the Koszul criterion for a quadratic operad \(\mathcal{O}\) in arity \(n \leq N\). What linear algebra operations are needed, and what is the computational complexity in \(n\)?
Key insight: Koszulness is equivalent to rank conditions on the differential matrices: \(\mathrm{rank}(d_w) + \mathrm{rank}(d_{w+1}) = \dim(C_w)\) for all \(w\) — i.e., the complex is exact at each weight.
Sketch: After computing chain_groups[w] and differentials[w] for \(w = 1, \ldots, W\) and \(n = 1, \ldots, N\):
def check_koszul(chain_groups, differentials, n, max_weight):
for w in range(1, max_weight):
C_w = len(chain_groups[w])
d_w = matrix(differentials.get(w, {}), C_w)
d_wp1 = matrix(differentials.get(w+1, {}), len(chain_groups.get(w+1, [])))
# Exactness: im(d_{w+1}) = ker(d_w)
rank_w = rank(d_w)
rank_wp1 = rank(d_wp1)
if rank_w + rank_wp1 != C_w:
return False # Not Koszul at weight w
return TrueComplexity: the number of rooted trees with \(n\) leaves and weight \(w\) grows as \(n^w / w!\) (Catalan-like), so the chain groups have dimension \(O((n/e)^n)\) in the worst case. Computing ranks via Gaussian elimination costs \(O(\dim^3)\). In practice, the \(S_n\)-equivariance can be used to block-diagonalize (decompose by irreducible representations of \(S_n\)), reducing to checking each isotypic component separately. For small \(n\) (say \(n \leq 6\)), this is feasible in seconds.
This connects the abstract definition of \(\Omega(\mathcal{C})\) to a practical data structure.
Prerequisites: 3.1 Definition
Write Python pseudocode for computing \(\Omega(\mathcal{O}^!)(n)\) for a Koszul quadratic operad \(\mathcal{O}\) (specified by \(E\) and \(R\)). The output should be the underlying graded operad (generators and their degrees) and the differential matrix.
Key insight: \(\Omega(\mathcal{O}^!)\) is the free operad on \(\mathcal{O}^![-1]\), so its generators are just the cogenerators of \(\mathcal{O}^!\) in one degree lower, and the differential expands each cogenerator using the coproduct \(\Delta\) of \(\mathcal{O}^!\).
Sketch:
def cobar_construction(E, R, n, max_weight):
"""
E: dict {arity: basis} for generators of O
R: relations in F(E)(3)
Returns generators and differential of Omega(O^!)(n)
"""
# Step 1: compute O^! cogenerators = sE^vee with corelations s^2 R^perp
E_dual = {arity: dual_basis(E[arity]) for arity in E}
R_perp = annihilator(R, free_operad(E_dual, arities=[3]))
# Step 2: desuspend to get Omega(O^!) generators in degree -(arity-1)
generators = {}
for arity, basis in E_dual.items():
degree = -(arity - 1) + 1 # desuspension: s^{-1}(sE^vee) = E^vee in degree 0
generators[arity] = [(b, degree) for b in basis]
# Step 3: free operad on generators (planar rooted trees with labels from generators)
free_op = free_operad_on(generators, arity=n, max_weight=max_weight)
# Step 4: differential from the coproduct of O^!
def differential(tree_monomial):
result = []
for vertex in tree_monomial.vertices:
cogen = tree_monomial.label(vertex)
# Apply coproduct Delta: cogen -> sum of (cogen' o cogen'')
for (c_prime, c_double, i, sign) in coproduct(cogen, R_perp):
new_tree = tree_monomial.expand_vertex(vertex, c_prime, c_double, i)
result.append((sign, new_tree))
return result
diff_matrix = build_matrix(free_op, differential)
return free_op, diff_matrixThe main subtlety is coproduct: it reads off the corelations \(s^2 R^\perp\) to compute \(\Delta\) on cogenerators. For \(\mathcal{O} = \mathrm{Ass}\), this reproduces the Stasheff \(A_\infty\) operad with operations \(m_n\) in degree \(2-n\).
This implements the homotopy transfer theorem as a practical algorithm for computing minimal \(A_\infty\)-models.
Prerequisites: 7.3 The Homotopy Transfer Theorem
Given a dg-associative algebra \(A\) (specified by its multiplication table and differential \(d: A \to A\)) and a deformation retract \((i, p, h)\) onto \(H = H^\bullet(A)\), write Python pseudocode for computing the transferred \(A_\infty\)-operations \(m_n: H^{\otimes n} \to H\) for \(n \leq N\) using the tree formula.
Key insight: The transferred \(m_n\) is a sum over all binary trees with \(n\) leaves, with \(i\) at leaves, \(m_2^A\) at internal vertices, and \(h\) at internal edges, projected to \(H\) by \(p\) at the root.
Sketch:
def homotopy_transfer(A_mult, d, i, p, h, n_max):
"""
A_mult: function (a, b) -> a*b for the algebra A
d: differential on A
i, p, h: deformation retract maps (i: H->A, p: A->H, h: A->A, degree -1)
Returns: dict {n: m_n} for n = 1, ..., n_max
"""
from functools import lru_cache
@lru_cache(maxsize=None)
def phi(tree):
"""
Compute the A-valued operation phi_tree: H^(leaves) -> A
defined by the tree formula, without the final p.
"""
if tree.is_leaf():
return i # base case: leaf applies i
left, right = tree.children()
phi_L = phi(left) # left subtree map H^k -> A
phi_R = phi(right) # right subtree map H^l -> A
# Compose with A_mult and then apply h (homotopy on internal edge)
return lambda *args: h(A_mult(
phi_L(*args[:left.n_leaves]),
phi_R(*args[left.n_leaves:])
))
m = {}
for n in range(1, n_max + 1):
if n == 1:
m[1] = lambda x: p(d(i(x))) # m_1 = p d i (boundary of i)
else:
total = zero_map(n)
for tree in binary_trees(n):
sign = koszul_sign_tree(tree)
# Apply p to the root
total += sign * (lambda *args, t=tree: p(A_mult(
phi(t.left)(*args[:t.left.n_leaves]),
phi(t.right)(*args[t.left.n_leaves:])
)))
m[n] = total
return mThe key insight is the recursive phi(tree): for a leaf it is just \(i\); for an internal node it applies \(h\) (the homotopy) after the multiplication, threading the deformation retract data through the tree structure. The number of binary trees with \(n\) leaves is the Catalan number \(C_{n-1}\), so the complexity is \(O(C_{n-1} \cdot \dim(H)^n)\) — exponential in \(n\), but manageable for small \(n\).
References
| Reference Name | Brief Summary | Link to Reference |
|---|---|---|
| Loday & Vallette, Algebraic Operads | The canonical graduate reference; Chapters 6–7 cover the bar/cobar construction and twisting morphisms, Chapters 10–12 cover Koszul duality in full generality. | Springer |
| Ginzburg & Kapranov, “Koszul Duality for Operads” | The foundational paper introducing Koszul duality for operads: quadratic operads, Koszul dual cooperad, the criterion, and the Com–Lie duality. Duke Math. J. 1994. | arXiv:0709.1228 |
| Getzler & Jones, “Operads, Homotopy Algebra and Iterated Integrals” | Extends Koszul duality to the dg setting; introduces the dg bar construction for operads, homotopy algebras over operads, and applications to 2d TFT. | arXiv:hep-th/9403055 |
| Vallette, “Algebra + Homotopy = Operad” | Accessible 43-page survey with figures; §3–4 cover bar/cobar and Koszul duality with many worked examples. Ideal first read before Loday–Vallette. | arXiv:1202.3245 |
| Keller, “Introduction to A∞ Algebras and Modules” | Self-contained lecture notes on \(A_\infty\)-algebras; covers the bar construction, minimal models, and the homotopy transfer theorem in full detail. | arXiv:math/9910179 |
| Markl, Shnider & Stasheff, Operads in Algebra, Topology and Physics | AMS monograph; Chapter 3 covers Koszul duality, and Chapter 4 covers deformation theory of operads and the Kontsevich formality theorem. | AMS |
| nLab: Koszul duality | Encyclopedic treatment with multiple equivalent formulations and connections to rational homotopy theory and derived algebraic geometry. | nLab |
| nLab: bar and cobar construction | Precise definitions of the operadic bar and cobar constructions, with the adjunction and the quasi-isomorphism statement. | nLab |