Symmetric Monoidal Categories for Operads - Index of

104 5 Definitions and Basic Constructions
where the sum ranges over all partitions I1 ∐···∐Ir = I and is divided out
by a natural action of the symmetric groups Σr. SinceC+(Ik) =k, wealso
have
M ◦ C+(I) =
�
M(r)/ ≡,
I1∐···∐Ir=I
and we can use the natural correspondence between partitions I1∐···∐Ir = I
and functions f : I →{1,...,r}, to obtain an expansion of the form:
M ◦ C+(I) = �
M(J)/ ≡ .
f:I→J
According to this expansion, we obtain readily that a right C+-action ρ :
M ◦ C+ → M determines morphisms f ∗ : M(J) → M(I) for every f ∈ Fin,
so that a right C+-module determines a contravariant functor on the category
Fin. For details, we refer to [33].
On the other hand, one checks that the map n ↦→ A ⊗n ,whereA is any
commutative algebra in a symmetric monoidal category E extends to a covariant
functor A ⊗− : Fin → E. In the point-set context, the morphism
f∗ : A ⊗m → A ⊗n associated to a map f : {1,...,m}→{1,...,n} is defined
by the formula:
m�
f∗( ai) =
i=1
n� � �
j=1
i∈f −1 (j)
The functor SC+ (M) :C+ E→Eassociated to a right C+-module M is also
determined by the coend
ai
�
.
� Fin
SC+ (M,A) = M(I) ⊗ A ⊗I .
This observation follows from a straightforward inspection of definitions.
5.1.7 The Example of Right Modules over Initial Unitary Operads.
Recall that the initial unitary operad ∗C associated to an object C ∈Cis the
operad such that ∗C(0) = C, ∗C(1) = 1, and∗C(n) =0forn>0.
In §5.1.1, we mention that the structure of a right module over an operad
R is fully determined by partial composites
◦i : M(m) ⊗ R(n) → M(m + n − 1), i=1,...,m,
together with natural relations, for which we refer to [46]. In the case of an
initial unitary operad R = ∗C, we obtain that a right ∗C-module structure is
determined by morphisms
∂i : M(m) ⊗ C → M(m − 1), i=1,...,m,
which represent the partial composites ◦i : M(m) ⊗∗C(0) → M(m − 1),
because the term ∗C(n) of the operad is reduced to the unit for n =1and