Symmetric Monoidal Categories for Operads - Index of

3.1 Recollections: Operads 55
In the point-set context, we use the notation p(q1,...,qr) ∈ P(n1+···+nr)
to refer to the composite of the elements p ∈ P(r) andq1 ∈ P(n1),...,qr ∈
P(nr) in the operad. The unit morphism is determined by a unit element
1 ∈ P(1).
3.1.3 Partial Composites. We also use partial composites
defined by composites
P(r) ⊗ P(s) ◦e
−→ P(r + s − 1),
P(r) ⊗ P(s) � P(r) ⊗ (1 ⊗···⊗P(s) ⊗···⊗1)
→ P(r) ⊗ (P(1) ⊗···⊗P(s) ⊗···⊗P(1)) μ −→ P(r + s − 1),
where operad units η : I → P(1) are applied at positions k �= e of the tensor
product. In the point-set context, we have p ◦e q = p(1,...,q,...,1), where
q ∈ P(s) issetattheeth entry of p ∈ P(r).
The unit and associativity relations imply that the composition morphism
of an operad is determined by its partial composition products. The unit
and associativity relations of the composition morphism have an equivalent
formulation in terms of partial composition products (see [46]).
3.1.4 The Intuitive Interpretation of Operads. The elements of an
operad p ∈ P(n) have to be interpreted as operations of n-variables
p = p(x1,...,xn).
The composition morphism of an operad models composites of such operations.
The partial composites represent composites of the form
p ◦e q = p(x1,...,xe−1,q(xe,...,xe+n−1),xi+n,...,xm+n−1),
for p ∈ P(m), q ∈ P(n). The action of permutations w ∈ Σn on P(n) models
permutations of variables:
wp = p(x w(1),...,x w(n)).
The unit element 1 ∈ P(1) has to be interpreted as an identity operation
1(x1) =x1.
3.1.5 Free Operads. The obvious forgetful functor U : O → M from
operads to Σ∗-objects has a left adjoint F : M→Owhich maps any Σ∗object
M to an associated free operad F(M).
In the point-set context, the free operad F(M) onaΣ∗-object M consists
roughly of all formal composites
(···((ξ1 ◦e2 ξ2) ◦e3 ···) ◦er ξr