Associators in mould theory

By developing various techniques of mould theory and establishing a quasi-involutive reformulation of Drinfeld's associator set, we introduce $\mathsf{GARI}(\mathscr{F})_{\mathsf{as}+\mathsf{bal}}$, a mould theoretic formulation of Drinfeld's associator set. We give a mould-theoretical generalization of the result that associator relations imply double shuffle relations, namely, we explain that $\mathsf{GARI}(\mathscr{F})_{\mathsf{as}+\mathsf{bal}}$ is embedded into Ecalle's set $\mathsf{GARI}(\mathscr{F})_{\mathsf{as}\ast\mathsf{is}}$ which is a mould theoretic version of Racinet's double shuffle set.