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

By Jean-Louis Loday, Bruno Vallette

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

Example text

10, where fα : C → BA, resp. gα : ΩC → A, is the morphism of wdga coalgebras , resp. algebras, associated to the twisting morphism α and respecting the weight grading. Notice that the universal twisting morphisms π and ι also preserve the weight. (1) ⇔ (3). Consider the tensor map fα ⊗ IdA : C ⊗ A → BA ⊗ A. Since π ◦ fα = α = IdA ◦ α, the map fα ⊗ IdA is a morphism of chain complexes from C ⊗α A to BA ⊗π A. 13 that the augmented bar construction is always acyclic. Therefore, the twisted complex C ⊗α A is acyclic if and only if fα ⊗ IdA is a quasi-isomorphism.

Sak ) = ω(a1 ) + · · · + ω(ak ). 10, where fα : C → BA, resp. gα : ΩC → A, is the morphism of wdga coalgebras , resp. algebras, associated to the twisting morphism α and respecting the weight grading. Notice that the universal twisting morphisms π and ι also preserve the weight. (1) ⇔ (3). Consider the tensor map fα ⊗ IdA : C ⊗ A → BA ⊗ A. Since π ◦ fα = α = IdA ◦ α, the map fα ⊗ IdA is a morphism of chain complexes from C ⊗α A to BA ⊗π A. 13 that the augmented bar construction is always acyclic.

3. Beyond the 2-connected case, the relationship between the cobar construction and quasi-isomorphisms is more subtle. 4. 8. Bar-Cobar adjunction. We show that the bar and cobar constructions form a pair of adjoint functors Ω : {con. dga coalgebras} {aug. dga algebras} : B . More precisely, this adjunction is given by the space of twisting morphisms. When A is augmented and C coaugmented, a twisting morphism between C and A ¯ is supposed to send K to 0 and C to A. 9. For every augmented dga algebra A and every conilpotent dga coalgebra C there exist natural bijections Homdga alg (ΩC, A) ∼ = Tw(C, A) ∼ = Homdga coalg (C, BA) .

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