Symmetric Monoidal Categories for Operads - Index of

4.2 Representations of Algebras over Operads 85
4.2.1 Functors on Pairs. For the moment, we assume A is any object in
a symmetric monoidal category E over the base category C. For any E ∈E,
we set
(A; E) ⊗n n�
= A ⊗···⊗E ⊗···⊗A,
e
e=1
where the sum ranges over tensor products A ⊗···⊗E ⊗···⊗A with n − 1
copies of A ∈E and one copy of E ∈E.
To define the notion of a representation, we use functors S(M,A; −) :E→E,
associated to Σ∗-objects M ∈M, defined by a formula of the form:
S(M,A; E) =
∞� � ⊗n
M(n) ⊗ (A; E) �
n=0
Σn .
For the unit Σ∗-object I, we have an obvious isomorphism
S(I,A; E) =1 ⊗E � E
since I(1) = 1 and I(n) =0forn �= 1.ForΣ∗-objects M,N ∈M,wehave
a natural isomorphism
S(M,S(N,A); S(N,A; E)) � S(M ◦ N,A; E)
whose definition extends the usual composition isomorphisms of functors associated
to Σ∗-objects. These isomorphisms satisfy coherence identities like
the usual composition isomorphisms of functors associated to Σ∗-objects.
4.2.2 Representations of Algebras over Operads. By definition, a representation
of an algebra A over an operad P is an object E ∈Eequipped
with a morphism μ :S(P,A; E) → E so that the following diagrams commute
S(P ◦ P,A; E) �
S(μ,A)
S(P,A; E)
S(I,A; E)
S(η,A;E)
�
S(P,A; E)
E,
S(P, S(P,A); S(P,A; E)) S(P,λ;μ)
S(P,A; E)
where λ :S(P,A) → A refers to the evaluation morphism of the P-algebra A.
The morphism μ :S(P,A; E) → E determines an action of the P-algebra A
on E.
We use the notation RP(A) to refer to the category of representations of a
P-algebra A, where a morphism of representations f : E → F consists obviously
of a morphism in E which commutes with actions on representations.
μ
μ
A,
μ