Ibrahim EL AGAMI  

Homotopy and algebra don’t mix well, which often requires replacing algebraic theories by versions up to homotopy. One such algebraic structure is that of bigebras, a multiplication together with a compatible comultiplication, or a Hopf-algebra without an antipode map. Even though bigebras are simple and have abundant examples, their homotopy theory is still unresolved. A minimal notion of bigebras up to homotopy is known to exist, but an explicit description remains elusive. A more unwieldy, but easier to describe, definition of homotopy bigebras is given by the bar and Cobar constructions coming from operadic theory. Using this definition and recent advancements in operatic theory, a rich homotopy theory can be established with an eye for applications.