Projection onto the Homology of an operad Suppose that $\mathcal{O}$ is a differential graded operad over a field and that $H(\mathcal{O})$ (i.e. taking the arity wise homology) is an operad, too. (If possible, I woul...--prophetes.ai

Projection onto the Homology of an operad Suppose that $\mathcal{O}$ is a differential graded operad over a field and that $H(\mathcal{O})$ (i.e. taking the arity wise homology) is an operad, too. (If possible, I would avoid to restrict to _symmetric_ operads) Is the natural projection $\pi : \mathcal{O} \to H(\mathcal{O})$ necessarily a morpism of operads?

There isn't any such natural projection.