92
II Mathematical Interpretation
For graded vector spaces V , W we denote by Lin k (V , W ) the vector space of
degree k morphisms V → W and by Lin(V , W ) the graded vector space
Lin(V , W ) :=
k∈Z
Lin
k (V , W ).
If W is the ground field k, we obtain the graded dual V # := Lin(V , k) of V . 1
Notice that the degree k component of V # equals the standard linear dual (V −k ) # of
the degree −k component of V . A degree k morphism f : V → W defines a map
f # : W # → V # of the same degree by the formula
f
# (ϕ) := (−1)
k|ϕ| ϕ ◦ f.
(2)
A dg-vector space (abbreviating differential graded) is a couple (V , d) of a
graded vector space V with a degree +1 differential d. The graded dual V # of a
dg-vector space is a dg-vector space, too, with the differential d # which is the linear
dual of d.
For dg-vector spaces (V 1 , d 1 ), (V 2 , d 2 ) we define the differential on the tensor
product V 1 ⊗ V 2 by the formula
d(v 1 ⊗ v 2 ) := d 1 (v 1 ) ⊗ v 2 + (−1)
|v 1 | v 1 ⊗ d 2 (v 2 ).
This, together with the flip (commutativity constrain)
τ : V 1 ⊗ V 2 → V 1 ⊗ V 2 , τ(v 1 ⊗ v 2 ) := (−1)
|v 1 ||v 2 | (v 2 ⊗ v 1 ),
(3)
equips the category dgVec of dg-vector spaces and their morphisms of arbitrary
degrees with a structure of symmetric monoidal category enriched over the category
Chain of dg-vector spaces and their morphisms of degree 0 [1]. We will use
a Sweedler-type notation to denote elements of the tensor product V ⊗ V , i.e.
s ∈ V ⊗ V will be written as
s ⊗ s or sometimes even without the summation
symbol as s ⊗ s .
Suppose one has a structure—algebra, module, operad, etc.—with linear operations of the form
α : V
1 ⊗ · · · ⊗ V
s −→ V
1 ⊗ · · · ⊗ V
t ,
where V
1 , . . . , V
s , V
1 , . . . , V
t are graded vector spaces. The dg-version of this
structure—dg-algebra, dg-module, dg-operad, etc.—is the structure with the same
operations, but now we assume that the graded spaces V
1 , . . . , V
s , V
1 , . . . , V
t have
1 We use # instead of the more usual ∗ to avoid confusion with the ∗ indicating a grading.
II Mathematical Interpretation
For graded vector spaces V , W we denote by Lin k (V , W ) the vector space of
degree k morphisms V → W and by Lin(V , W ) the graded vector space
Lin(V , W ) :=
k∈Z
Lin
k (V , W ).
If W is the ground field k, we obtain the graded dual V # := Lin(V , k) of V . 1
Notice that the degree k component of V # equals the standard linear dual (V −k ) # of
the degree −k component of V . A degree k morphism f : V → W defines a map
f # : W # → V # of the same degree by the formula
f
# (ϕ) := (−1)
k|ϕ| ϕ ◦ f.
(2)
A dg-vector space (abbreviating differential graded) is a couple (V , d) of a
graded vector space V with a degree +1 differential d. The graded dual V # of a
dg-vector space is a dg-vector space, too, with the differential d # which is the linear
dual of d.
For dg-vector spaces (V 1 , d 1 ), (V 2 , d 2 ) we define the differential on the tensor
product V 1 ⊗ V 2 by the formula
d(v 1 ⊗ v 2 ) := d 1 (v 1 ) ⊗ v 2 + (−1)
|v 1 | v 1 ⊗ d 2 (v 2 ).
This, together with the flip (commutativity constrain)
τ : V 1 ⊗ V 2 → V 1 ⊗ V 2 , τ(v 1 ⊗ v 2 ) := (−1)
|v 1 ||v 2 | (v 2 ⊗ v 1 ),
(3)
equips the category dgVec of dg-vector spaces and their morphisms of arbitrary
degrees with a structure of symmetric monoidal category enriched over the category
Chain of dg-vector spaces and their morphisms of degree 0 [1]. We will use
a Sweedler-type notation to denote elements of the tensor product V ⊗ V , i.e.
s ∈ V ⊗ V will be written as
s ⊗ s or sometimes even without the summation
symbol as s ⊗ s .
Suppose one has a structure—algebra, module, operad, etc.—with linear operations of the form
α : V
1 ⊗ · · · ⊗ V
s −→ V
1 ⊗ · · · ⊗ V
t ,
where V
1 , . . . , V
s , V
1 , . . . , V
t are graded vector spaces. The dg-version of this
structure—dg-algebra, dg-module, dg-operad, etc.—is the structure with the same
operations, but now we assume that the graded spaces V
1 , . . . , V
s , V
1 , . . . , V
t have
1 We use # instead of the more usual ∗ to avoid confusion with the ∗ indicating a grading.