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Algebraic operads by Loday J.-L., Vallette B.

By Loday J.-L., Vallette B.

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For any x ∈ C, let us write ∆(x) = ¯ ⊗ id)∆(x) ¯ (∆ = x(1) ⊗x(2) . We also adopt the notation ¯ ∆(x). ¯ x(1) ⊗ x(2) ⊗ x(3) = (id ⊗ ∆) We have defined d2 (s−1 x) := (−1)|x(1) | s−1 x(1) ⊗ s−1 x(2) ∈ C ⊗2 . Let us prove that d2 ◦ d2 = 0. Let p := |x(1) |, q := |x(2) |, r := |x(3) |. The term ¯ ¯ under d2 ◦d2 comes with the sign s−1 x(1) ⊗s−1 x(2) ⊗s−1 x(3) coming from (∆⊗id) ∆ (−1)p+q (−1)p . Indeed, the first one comes from the application of the first copy of d2 , the second one comes from the application of the second copy of d2 .

Here we have adopted Mac Lane’s notation [a1 | . . | an ] ∈ A¯⊗n . ¯ ⊗n is in degree n. So the Since A¯ is in degree 0, the space sA¯ is in degree 1 and (sA) ⊗n ¯ module of n-chains can be identified with A . Let us identify the boundary map. Since d2 is induced by the product and is a derivation, it has the form indicated in the statement. The signs come from the presence of the shift s. For instance: [a1 | a2 | a3 ] = (sa1 , sa2 , sa3 ) → (d2 (sa1 , sa2 ), sa3 ) − (sa1 , d2 (sa2 , sa3 )) = [µ(a1 , a2 ) | a3 ] − [a1 | µ(a2 , a3 )].

Maurer-Cartan equation, twisting morphism. In the dga algebra Hom(C, A) we consider the Maurer-Cartan equation ∂(α) + α α = 0. By definition a twisting morphism (terminology of John Moore [Moo71], “fonctions tordantes” in H. Cartan [Car58]) is a solution α : C → A of degree −1 of the Maurer-Cartan equation, which is null when composed with the augmentation of A and also when composed with the coaugmentation of C. 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.

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