related by the Dold-Kan correspondence
for every collection of colours it induces an isomorphism
P(c_1, \cdots, c_n; c) \to Q(f(c_1), \cdots, f(c_n); f(c))
(the operadic analog of being full and faithful).
Call a morphism a fibration if for every isomorphism in and a lift of its source object to there is an isomorphism in covering it under .
Call a morphism a cofibration if it is an injection on objects (on colours)
This is due to (Weiss 07).