In Category Theory there is a well developed theory of monads, proved to be very useful for 1-dimensional universal algebra and beyond. The relation between adjunctions and monads was first noticed by Huber (Homotopy Theory in General Categories): every adjunction gives rise to a monad. Then, Eilenberg, Moore and Kleisli realized that every monad comes from an adjunction. In particular, Eilenberg and Moore (Adjoint Functors and Triples) realized that, for every monad T, there is a terminal adjunction (called Eilenberg-Moore adjunction) which gives rise to T. Category Theory can be also developed in a 2-dimensional case, that is, considering not only morphisms between objects but also morphisms (usually called 2-cells) between morphisms themselves. Thereby, one can study lax versions of the theory of monads. In the pseudo version, that is when we replace commutative diagrams by coherent invertible 2-cells, the relation between biadjunctions and pseudomonads has been investigated by F. Lucatelli Nunes in the paper On Biadjoint Triangles as a consequence of the coherent approach to pseudomonads of S. Lack. The next step consists of studying the lax notion of monads, in which the associativity and identity works only up to coherent (not necessarily invertible) 2-cells. In this talk we present a work in progress where we try to generalize to the lax-context the classical result of Eilenberg-Moore. For this purpose, having in mind the notion of lax extension of monads introduced and studied in the context of Monoidal Topology (Metric, topology and multicategory: a common approach - M.M. Clementino and W. Tholen), we use a generalization of Gray’s lax-adjunction (see the monograph Formal Category Theory). Then, we show some steps of the construction leading to the positive answer.

(*) joint work with Fernando Lucatelli Nunes