Homotopy algebras II: The Homotopy Transfer Theorem

Last time we introduced $L_\infty$-algebras and explained how they generalize Lie algebras. Even before their formal definition, $L_\infty$-algebras appeared in disguise as Sullivan's minimal models for rational homotopy types. In this talk, we will show that for every algebra $X$ over $\Omega C$, the cobar construction on some augmented cooperad $C$, we can produce an explicit minimal model, i.e. an equivalent $\Omega C$-algebra structure on its cohomology $H^*X$, from the data of a strong deformation retraction, and use this to show that every quasiisomorphism of $\Omega C$-algebras has a strict quasiinverse. Specializing to $C \in \{\mathrm{Com}^\vee,\mathrm{Ass}^\vee,\mathrm{Lie}^\vee\}$, we obtain the homotopy transfer theorem for $L_\infty$, $A_\infty$, and $C_\infty$-algebras, respectively.