We extend homological perturbation theory to encompass algebraic structures
governed by operads and cooperads. The main difficulty is to find a suitable
notion of algebra homotopy that generalizes to algebras over operads
.
To solve this problem, we introduce thick maps of–algebras
and special thick maps that we call pseudo-derivations that serve as appropriate
generalizations of algebra homotopies for the purposes of homological perturbation
theory.
As an application, we derive explicit formulas for transferring
–algebra structures along
contractions, where
is any connected cooperad in chain complexes. This specializes to transfer formulas for
–algebras for any
Koszul operad , in
particular for –,
–,
– and
–algebras.
A key feature is that our formulas are expressed in terms of the compact description of
–algebras as coderivation
differentials on cofree –coalgebras.
Moreover, we get formulas not only for the transferred structure and a structure on
the inclusion, but also for structures on the projection and the homotopy.