Cohomology in algebra, geometry, physicsand statistics

Properads and Homotopy Algebras Related to Surfaces

Barannikov showed how an algebra over the Cobar construction over a
modular operad can equivalently be described as a solution of a master
 equation for certain generalized BV algebra. We give an analogous
description for properads which were first introduced by B. Vallette
 as connected parts of PROPs.

 In short, we introduce properads along with our main examples, the
closed (commutative) Frobenius properad, and the open (associative)
 Frobenius properad. We consider the construction of the Cobar complex
 of a properad as a free properad over its suspended linear dual
equipped with the differential induced by the duals of the structure
 operations. We describe algebras over the Cobar construction in terms
 of solutions of generalized master equations and recover the
 well-known result that the corresponding algebras for closed Frobenius
properad are $IBL_\infty$-algebras. In addition to this case, we show
 their associative analogues, which we call $IBA$-homotopy algebras.