In mathematical category theory, a pseudomonad is a mathematical generalization of a monad. It is essentially the same notion as a pseudomonoid as introduced in Gray-monoid, and was introduced by for every Gray-category. A pseudomonad on a Gray-category or 2-category (or a more generalized notion weak 2-category) consists of a 2-functor (in particular, if it is a functor on a weak 2-category, it is a pseudo-functor.) equipped with pseudonatural transformations and which satisfy the monad laws up to coherent invertible modifications. In monads, the identity and the associativity of composition hold strictly as equalities, whereas in the axiom of pseudomonads they hold only up to isomorphism, which satisfy the coherent axioms.
The 2-categorical analogue of Beck's monadicity theorem holds for the pseudomonads. Whereas the original theorem gives a necessary and sufficient condition for an adjunction to be monadic, the 2âÂÂcategorical analogue replaces the adjunctions and monads on ordinary categories that are the subject of the original theorem by pseudo-adjunctions and pseudomonads on 2-categories. The formal theory of monads can be developed in arbitrary 2-category, but to develop a formal theory of pseudomonads, move to the Gray-category. The analogue of distributive laws between monads also applies to pseudomonads. This was explicitly introduced by and is called the pseudodistributive law. Initially, it was thought that nine coherence axioms sufficed for the definition of a pseudodistributive law between pseudomonads, but this was later reduced to eight.
Let denotes a Gray-category or 2-category. Then a pseudomonad on consists of:
.
It is possible to define an analogue of adjunctions in Gray categories; these are known as pseudoadjunctions or pseudo-adjunctions. Every pseudoadjunction gives rise to a pseudomonad.
A pseudoadjuction between Gray categories and consists of:
The following pasting diagrams must be equal to the identity: