Download PDF by Jean-Louis Loday, Bruno Vallette: Algebraic Operads (version 0.99, draft 2010)

By Jean-Louis Loday, Bruno Vallette

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Remember young fellow, Ω is left adjoint . . ” Dale Husem¨ oller, MPIM (Bonn), personal communication. In this chapter, we introduce the bar construction and the cobar construction as follows. A twisting morphism is a linear map f : C → A, from a dga coalgebra C to a dga algebra A, which satisfies the Maurer-Cartan equation: ∂(f ) + f f = 0. The set of twisting morphisms Tw(C, A) is shown to be representable both in C and in A. More precisely, the cobar construction is a functor Ω from dga coalgebras to dga algebras and the bar construction is a functor B from dga algebras to dga coalgebras which satisfy the following properties: there are natural isomorphisms Homdga alg (ΩC, A) ∼ = Tw(C, A) ∼ = Homdga coalg (C, BA) .

From convolution algebra to tensor product. 13. Both extended maps are equal to dα := (IdC ⊗ µ) ◦ (IdC ⊗ α ⊗ IdA ) ◦ (∆ ⊗ IdA ) . The following result gives a condition under which dα is a boundary map. 4. For any α, β ∈ Hom(C, A) one has dα β = dα ◦ dβ and du = IdC⊗A . So, d− : (Hom(C, A), ) → (End(C ⊗ A), ◦) is a morphism of associative algebras. If α α = 0, then (dα )2 = 0. Proof. The last assertion follows immediately from the first. 1), the following picture (to be read from top to bottom) is a proof of the first assertion: dα ◦ dβ = t xx tttt xx p  ppp  β pp  pp   α ttt ttxxxx xtt xx ttt x p p  pp  = α β = pp  pp  ttt  ttxxxx xpp xx ppp x xx ppp  xx  α β pp pp xxx  pxp  pp xxx x = dα β .

For any α, β ∈ Hom(C, A) one has dα β = dα ◦ dβ and du = IdC⊗A . So, d− : (Hom(C, A), ) → (End(C ⊗ A), ◦) is a morphism of associative algebras. If α α = 0, then (dα )2 = 0. Proof. The last assertion follows immediately from the first. 1), the following picture (to be read from top to bottom) is a proof of the first assertion: dα ◦ dβ = t xx tttt xx p  ppp  β pp  pp   α ttt ttxxxx xtt xx ttt x p p  pp  = α β = pp  pp  ttt  ttxxxx xpp xx ppp x xx ppp  xx  α β pp pp xxx  pxp  pp xxx x = dα β .

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Algebraic Operads (version 0.99, draft 2010) by Jean-Louis Loday, Bruno Vallette


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