Symmetric Monoidal Categories for Operads - Index of

1.1 Symmetric Monoidal Categories over a Base 25
we obtain that the category of dg-modules comes equipped with the structure
of a symmetric monoidal category.
The symmetric monoidal category of dg-modules forms obviously a symmetric
monoidal category over k-modules.
The symmetry isomorphism of dg-modules follows the standard sign convention
of differential graded algebra, according to which a permutation of
homogeneous objects of degree p and q produces a sign ± =(−1) pq .Inthe
sequel, we do not make explicit the signs which are determined by this rule.
1.1.5 The Pointwise Representation of dg-Tensors. The pointwise
representation of tensors of §§0.4-0.5 can be applied to the category of dgmodules.
In this representation, the tensor product C ⊗ D is considered as
a dg-module spanned by tensor products x ⊗ y of homogeneous elements
(x, y) ∈ Cp × Dq. To justify this rule, observe that a morphism of dgmodules
φ : C ⊗ D → E is equivalent to a collection of homogeneous maps
φ : Cp × Dq → En, which are multilinear in the usual sense of linear algebra,
together with a commutation relation with respect to differentials. Hence a
morphism φ : C ⊗ D → E is well determined by a mapping x ⊗ y ↦→ φ(x, y)
on the set of generating tensors x ⊗ y, where(x, y) ∈ Cp × Dq.
The definition of the symmetry isomorphism τ(C, D) :X ⊗ Y � −→ Y ⊗ X
gives the commutation rule x ⊗ y ↦→ ±y ⊗ x, with a sign added.
1.1.6 Remark: Symmetric Monoidal Categories of Graded Objects.
Let gr k Mod denote the category of graded k-modules, whose objects consists
simply of k-modules C ∈ k ModequippedwithasplittingC = �
∗∈Z C∗.
Any object in that category can be identified with a dg-module equipped with
a trivial differential. The category gr k Mod inherits a symmetric monoidal
structure from this identification, since a tensor product of dg-modules
equipped with a trivial differential has still a trivial differential. The symmetry
isomorphism of this symmetric monoidal structure includes the sign of
differential graded algebra.
In the context of graded modules, we can drop this sign to obtain another
natural symmetry isomorphism τ(C, D) :C ⊗ D � −→ D ⊗ C. Thisconvention
gives another symmetric monoidal structure on graded modules, which is not
equivalent to the symmetric monoidal structure of differential graded algebra.
These symmetric monoidal categories of graded modules form both symmetric
monoidal categories over k-modules.
1.1.7 Basic Examples: Symmetric Monoidal Structures Based
on the Cartesian Product. The category of sets Set together with the
cartesian product × forms an obvious instance of a symmetric monoidal
category. The cartesian product is also used to define a symmetric monoidal
structure in the usual category of topological spaces Top, and in the usual
category of simplicial sets S. The one-point set, which gives the unit of these
symmetric monoidal categories, is represented by the notation ∗.