
212 Algebraic Operads: An Algorithmic Companion
6.4.2 Resolution for monomial relations
Assume that the shuffle operad O = T
X
(X)/(G) is generated by an oper-
ation alphabet X, and that G consists of shuffle tree monomials. We will now
explain how to construct a model of O which is often minimal.
Our first step is to construct a quasi-free shuffle operad A
X
which does
not take into account the relations of O; it is a somewhat universal object for
operads generated by X, various suboperads of A
X
will be used as resolutions
for various choices of G.
Definition 6.4.2.1 (The inclusion–exclusion operad). Let T be a tree mono-
mial, and let Λ(s
1
, . . . , s
q
) be the