We know that all associative algebra is a pre-Lie algebra, but I could not find an operad morphism between the operad Associative and the operad PreLie. So, this morphism exist? If it exists, can anyone tell me how it looks? If not, why not?
Thanks! :)
This goes the other way around. If you have a morphism of operads $f : \mathtt{P} \to \mathtt{Q}$, then if $A$ is a $\mathtt{Q}$-algebra, it automatically becomes a $\mathtt{P}$-algebra: you have a map $\gamma_A : \mathtt{Q} \circ A \to A$ given by the $\mathtt{Q}$-algebra structure, and you compose it with $f \circ \operatorname{id} : \mathtt{P} \circ A \to \mathtt{Q} \circ A$ to get a map: $$\mathtt{P} \circ A \xrightarrow{f \circ \operatorname{id}} \mathtt{Q} \circ A \xrightarrow{\gamma_A} A$$ to give $A$ a $\mathtt{P}$-algebra structure.
So here what you want is actually a morphism from the preLie operad to the associative operad, not the reverse. This morphism is simply given by sending the generator $\triangleleft \in \mathtt{preLie}(2)$ to the generator $m \in \mathtt{Ass}(2)$. This is a well-defined morphism of operads (not hard to check: $\mathtt{preLie}$ is free on one generator with a single relation, so all you need to check is that $m$ satisfies this relation), hence every associative algebra automatically becomes a pre-Lie algebra.