1. DERIVED CATEGORY OF DG-MODULES 9
α
HomDG(−,Xα) is fixed. An object Y is called the direct sum of a family of ob-
jects (notation: Y =
α
Xα) if a closed isomorphism of covariant DG-functors
HomDG(Y, −)
α
HomDG(Xα, −) is fixed.
An object Y is called the shift of an object X by an integer i (notation:
Y = X[i]) if a closed isomorphism of contravariant DG-functors HomDG(−,Y )
HomDG(−,X)[i] is fixed, or equivalently, a closed isomorphism of covariant DG-
functors HomDG(Y, −) HomDG(X, −)[−i] is fixed.
An object Z is called the cone of a closed morphism f : X −→ Y (notation:
Z = cone(f)) if a closed isomorphism of contravariant DG-functors HomDG(−,Z)
cone(f∗), where f∗ : HomDG(−,X) −→ HomDG(−,Y ), is fixed, or equivalently, a
closed isomorphism of covariant DG-functors HomDG(Z, −) cone(f
∗)[−1],
where
f

: HomDG(Y, −) −→ HomDG(X, −), is fixed.
Let V be a complex of abelian groups and p: V −→ V be an endomorphism
of degree 1 satisfying the Maurer–Cartan equation d(p) +
p2
= 0. Then one can
define a new differential on V by setting d = d + p; let us denote the complex
so obtained by V (p). Let q HomDG(X,
1
X) be an endomorphism of degree 1
satisfying the equation d(q) +
q2
= 0. An object Y is called the twist of the
object X with respect to q if a closed isomorphism of contravariant DG-functors
HomDG(−,X) HomDG(−,Y )(q∗) is fixed, where q∗(g) = q g for any morphism g
whose target is X, or equivalently, a closed isomorphism of covariant DG-functors
HomDG(Y, −) HomDG(X,
−)(−q∗)
is fixed, where
q∗(g)
=
(−1)|g|g
q for any
morphism g whose source is Y .
As any representing objects of DG-functors, all direct sums, products, shifts,
cones, and twists are defined uniquely up to a unique closed isomorphism. The
direct sum of a finite set of objects is naturally also their product, and vice versa.
Finite direct sums, products, shifts, cones, and twists are preserved by any DG-
functors. One can express the cone of a closed morphism f : X −→ Y as the twist
of the direct sum Y X[1] with respect to the endomorphism q induced by f.
Here is another way to think about cones of closed morphisms in DG-categories.
Let
DG#
denote the category whose objects are the objects of DG and morphisms
are the (not necessarily closed) morphisms in DG of degree 0. Let X −→ X −→ X
be a triple of objects in DG with closed morphisms between them that is split exact
in
DG#.
Then X is the cone of a closed morphism X [−1] −→ X . Conversely, for
any closed morphism X −→ Y in DG with the cone Z there is a natural triple of
objects and closed morphisms Y −→ Z −→ X[1], which is split exact in
DG#.
Let DG be a DG-category with shifts, twists, and infinite direct sums. Let
· · · −→ Xn −→ Xn−1 −→ · · · be a complex of objects of DG with closed differ-
entials ∂n. Then the differentials ∂n induce an endomorphism q of degree 1 on
the direct sum
n
Xn[n] satisfying the equations d(q) = 0 =
q2.
The twist of
this direct sum with respect to this endomorphism is called the total object of the
complex X• formed by taking infinite direct sums and denoted by
Tot⊕(X•).
For
a DG-category DG with shifts, twists, and infinite products, one can consider the
analogous construction with the infinite direct sum replaced by the infinite prod-
uct
n
Xn[n]. Thus one obtains the definition of the total object formed by taking
infinite products Tot (X•).
For a finite complex X •, the two total objects coincide and are denoted simply
by Tot(X•); this total object only requires existence of finite direct sums/products
for its construction. Alternatively, the total objects Tot,
Tot⊕,
and Tot can be
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