4 APONTE ROM´ LIVERNET, ROBERTSON, WHITEHOUSE, AND ZIEGENHAGEN 2.2. Twisted chain complexes. The notion of twisted chain complex is im- portant in the theory of derived A∞-algebras. The term multicomplex is also used for a twisted chain complex. Definition 2.2. A twisted chain complex C is a (Z, Z)-bigraded k-module with k-linear maps dC i : C −→ C of bidegree (−i, 1 − i) for i ≥ 0, satisfying ∑ i+p=u (−1)ididp = 0 for u ≥ 0. A map of twisted chain complexes C −→ D is a family of maps fi : C −→ D, for i ≥ 0, of bidegree (−i, −i), satisfying i+p=u (−1)ifidp C = i+p=u di D fp. The composition of maps f : E → F and g : F → G is defined by (gf)u = i+p=u gifp and the resulting category is denoted tChk. A derived A∞-algebra has an underlying twisted chain complex, specified by the maps mi1 for i ≥ 0. 2.3. Vertical bicomplexes and operads in vertical bicomplexes. The underlying category for the operadic view of derived A∞-algebras is the category of vertical bicomplexes. Definition 2.3. An object of the category of vertical bicomplexes BiComplv is a bigraded k-module as above equipped with a vertical differential dA : Aj i −→ Aj+1 i of bidegree (0, 1). The morphisms are those morphisms of bigraded modules com- muting with the vertical differential. We denote by Hom(A, B) the set of morphisms (preserving the bigrading) from A to B. The category BiCompl v is isomorphic to the category of Z-graded chain com- plexes of k-modules. For the suspension s as above, we have dsA(sx) = −s(dAx). The tensor product of two vertical bicomplexes A and B is given by (A ⊗ B)u v = i+p=u, j+q=v Aj i ⊗ Bp,q with dA⊗B = dA ⊗ 1 + 1 ⊗ dB : (A ⊗ B)v u → (A ⊗ B)v+1. u This makes BiCompl v into a symmetric monoidal category. Let A and B be two vertical bicomplexes. We write Homk for morphisms of k-modules. We will denote by Mor(A, B) the vertical bicomplex given by Mor(A, B)v u = α,β Homk(Aα,Bα+u),v+ββ with vertical differential given by ∂Mor(f) = dBf − (−1)jfdA for f of bidegree (l, j). The reason for the change of grading conventions is that, with the convention adopted here, Mor is now an internal Hom on Bicomplv. The following notation will be useful in applying the Koszul sign rule. We denote by |(r, s)||(r , s )| the integer rr + ss .

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