Symmetric Monoidal Categories for Operads - Index of

68 3 Operads and Algebras in Symmetric Monoidal Categories
3.2.15 Observation. Let P be an operad in C. SetF = F(E, E) and P F =
F(E, PE).
The functor S:M→F restricts to a functor
S:P M→P F
so that for the free P-algebra N = P(M) generated by M ∈Mwe have
S(P(M),X)=P(S(M,X)),
for all X ∈E, where on the right-hand side we consider the free P-algebra
in E generated by the object S(M,X) ∈E associated to X ∈E by the functor
S(M) :E→E.
This construction is functorial in E. Explicitly, for any functor ρ : D→E
of symmetric monoidal categories over C, the diagram
S(N)
D
ρ
E
PD ρ PE
S(N)
commutes up to natural functor isomorphisms, for all N ∈ P M.
3.2.16 Restriction of Functors. Let α : A→Bbe a functor. For any
target category X , we have a functor α ∗ : F(B, X ) →F(A, X ) induced by α,
defined by α ∗ G(A) =G(α(A)), for all G : B→X
In the case X = E, themapG ↦→ α ∗ (G) defines clearly a functor of
symmetric monoidal categories over C
α ∗ :(F(B, E), ⊗, 1) → (F(A, E), ⊗, 1).
By observations of §§3.2.7-3.2.8, the induced functor on P-algebras, obtained
by the construction of observation 3.2.14, is identified with the natural functor
α ∗ :(F(B, PE), ⊗, 1) → (F(A, PE), ⊗, 1),
which is induced by α for the target category X = PE.
3.3 Universal Constructions for Algebras over Operads
The forgetful functor U : PE →E, from a category of algebras over an operad
P to the underlying category E, creates all limits. This assertion is proved
by a straightforward inspection. On the other hand, the example of commutative
algebras shows that the forgetful functor U : PE → E does not