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

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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) . Proof. Let us make the first bijection explicit. Since ΩC = T (s−1 C) is a free algebra, any morphism of algebras from ΩC to A is characterized by its restriction to C (cf.

As an immediate consequence the functors cobar and bar are adjoint to each other. Then we investigate the twisting morphisms which give rise to quasi-isomorphisms under the aforementioned identifications. We call them Koszul morphisms. The main point is the following characterization of the Koszul morphisms. Any linear map α : C → A gives rise to a map dα : C ⊗A → C ⊗A, which is a differential if and only if α is a twisting morphism. Moreover, α is a Koszul morphism if and only if the chain complex (C ⊗ A, dα ) is acyclic.

1. Graded vector space. A graded vector space V is a family of vector spaces {Vn }n∈Z . The direct sum is denoted by V• := · · · ⊕ V−n ⊕ · · · ⊕ V0 ⊕ V1 ⊕ · · · ⊕ Vn ⊕ · · · , and the product is denoted by V• := Πn∈Z Vn . By abuse of notation we often write V in place of V• (resp. V in place of V• ). The degree of v ∈ Vn is denoted by |v|, so here |v| = n. Most of the time the index will run over N only. A morphism of degree r, say f : V → W , of graded vector spaces is a family of maps fn : Vn → Wn+r for all n.