Workshop "Operads and Homotopy Theory"
"Curved Koszul duality theory"
The classical Koszul duality theory is defined for augmented associative
algebras, operads or properads. To study non-augmented operads or properads, we show
that a curvature appears on the bar-construction and in the Maurer-Cartan equation.
This curvature controls the default of augmentation. We obtain resolutions for the
properad encoding unital and counital Frobenius algebras and for the operad encoding
unital associative algebras. Then we present a definition for homotopy unital
associative algebras, holding good homotopy properties.
by Joan Millès