 By Jean-Louis Loday, Bruno Vallette

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Additional info for Algebraic Operads (version 0.99, draft 2010)

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If C were cocommutative and if A were commutative, then they would be isomorphic. Since the two constructions are symmetric, we will only state the related properties for the right twisted tensor product in the rest of this chapter. This construction is functorial both in C and in A. Let g : A → A be a morphism of dga algebras and f : C → C be a morphism of dga coalgebras. Consider C ⊗α A and C ⊗α A two twisted tensor products. We say that the morphisms f and g are compatible with the twisting morphisms α and α if α ◦ f = g ◦ α.

We denote by Tw(C, A) the set of twisting morphisms from C to A. |b| b a. When 2 is invertible in the ground ring K, we have α α = 21 [α, α], when α has degree −1. Therefore, the ‘associative’ Maurer-Cartan equation, written above, is equivalent to the ‘classical’ Maurer-Cartan equation ∂(α) + 12 [α, α] = 0 in the Lie convolution algebra (Hom(C, A), [−, −]). Until the end of next section, we assume that the characteristic of the ground field is not equal to 2. 4. Twisted structure on the Hom space.

P+q=n 2 It is immediate to verify that d = 0. Let (V, dV ) and (W, dW ) be two differential graded vector spaces. Their tensor product (V ⊗ W )n := p+q=n Vp ⊗ Wq is equipped with the differential dV ⊗W := dV ⊗ IdW + IdV ⊗ dW , that is dV ⊗W (v ⊗ w) := dV (v) ⊗ w + (−1)p v ⊗ dW (w), for v ⊗ w ∈ Vp ⊗ Wq . The suspension sV of the chain complex V is by definition the tensor product of sK with V . Here sK is considered as a chain complex concentrated in degree 1 with 0 differential: (sV )n = Vn−1 and dsV = −dV .