CONICET | Buscador de Institutos y Recursos Humanos

We give a definition of weak morphism of T-algebras, for a 2-monad T, with respect to an arbitrary family of 2-cells of the base 2-category. By considering particular choices of this family, we recover the concepts of lax, pseudo and strict morphisms of T-algebras. We give a general notion of weak limit, and define what it means for such a limit to be compatible with another family of 2-cells. These concepts allow us to prove a limit lifting theorem which unifies and generalizes three different previously known results of 2-dimensionalmonad theory. Explicitly, by considering three choices for the family of 2-cells above our theorem has as corollaries the lifting of oplax (resp. sigma, which generalizes lax and pseudo, resp. strict) limits to the 2-categories of lax (resp. pseudo, resp. strict) morphisms of T-algebras.