DG (co)algebras, DG Lie algebras and L ∞ algebras Michele Grassi 1 DG Algebras, DG Coalgebras and DG Lie Al- gebras Definition 1.1 A graded complex over the field k is a graded k-vector space C * =  i∈Z C i , together with a differential d C of degree +1 (i.e. d C (C i ) ⊂ C i+1 ). A morphism from the graded complex (C * ,d C ) to the graded complex (D * ,d D ) (over k) is a homogeneous k-linear map φ : C * → D * , such that d D φ = φd C . The category of graded complexes over k is indicated with C (k) . We first recall some operations on graded vector spaces and graded complexes. The base field k is assumed to be fixed unless otherwise stated. Definition 1.2 1) Given two graded complexes of vector spaces V =(V * ,d V ) and W = (W * ,d W ), their tensor product (over the base field) is defined as follows: (V ⊗ W ) r =  p+q = r V p ⊗ W q and the differential is expressed as the sum of its graded components as: For x ∈ V p ,y ∈ W q ,d V ⊗W (x ⊗ y)= d V (x) ⊗ y +(-1) p x ⊗ d W (y) 2) Given a graded complex of vector spaces V =(V * ,d V ), the Twisting map T : V ⊗ V → V ⊗ V is defined as the linear extension of the map defined on homogeneous vectors by T(x ⊗ y)=(-1) deg(x)deg(y) y ⊗ x Remark 1.3 As a rule of thumb to “get the signs right” is formulas like the ones above, which appear frequently when dealing with graded objects, one could use the following: “whenever an object of degree r passes on the other side of an object of degree s, a sign (-1) rs must be inserted”. The proof of the following proposition is elementary, and omitted. 1