Symmetric Monoidal Categories for Operads - Index of

66 3 Operads and Algebras in Symmetric Monoidal Categories
that satisfy the twisted symmetry relation
μ(a, b) =τ(p, q) · μ(b, a), ∀a ∈ N(p), ∀b ∈ N(q),
and the associativity relation
μ(μ(a, b),c)=μ(a, μ(b, c)), ∀a ∈ N(p), ∀b ∈ N(q), ∀c ∈ N(r).
(b) The structure of an A-algebra in Σ∗-objects consists of a Σ∗-object N
equipped with a collection of Σp × Σq-equivariant morphisms
μ : N(p) ⊗ N(q) → N(p + q), p,q ∈ N,
that satisfy the associativity relation
μ(μ(a, b),c)=μ(a, μ(b, c)), ∀a ∈ N(p), ∀b ∈ N(q), ∀c ∈ N(r).
(c) The structure of an L-algebra in Σ∗-objects consists of a Σ∗-object N
equipped with a collection of Σp × Σq-equivariant morphisms
γ : N(p) ⊗ N(q) → N(p + q), p,q ∈ N,
that satisfy the twisted antisymmetry relation
γ(a, b)+τ(p, q) · γ(b, a) =0, ∀a ∈ N(p), ∀b ∈ N(q),
and the twisted Jacobi relation
γ(γ(a, b),c)+c(p, q, r) · γ(γ(b, c),a)+c 2 (p, q, r) · γ(γ(c, a),b)=0,
where c denotes the 3-cycle c =(123)∈ Σ3.
∀a ∈ N(p), ∀b ∈ N(q), ∀c ∈ N(r),
This proposition illustrates the principle (formalized in §0.5) of generalized
point-tensors: to obtain the definition of a commutative (respectively, associative,
Lie) algebra in Σ∗-objects from the usual pointwise definition in the
context of k-modules, we take simply point-tensors in Σ∗-objects and we add
a permutation, determined by the symmetry isomorphisms of the category
of Σ∗-objects, to every tensor commutation.
3.2.13 Free Objects. Since the category of P-algebras in E is defined by a
monad, we obtain that this category has a free object functor P(−) :E→ PE,
left adjoint to the forgetful functor U : PE →E. Explicitly, the free P-algebra
in E generated by X ∈Eis defined by the object P(X) =S(P,X) ∈Etogether
with the evaluation product λ :S(P, P(X)) → P(X) defined by the morphism
S(P, P(X)) = S(P ◦ P,X) S(μ,X)
−−−−→ S(P,X)
induced by the composition product of the operad.