  Monads are a useful tool for structuring effectful features of computation
such as state, non-determinism, and continuations. In the last decade, several
generalisations of monads have been suggested which provide a more fine-grained
model of effects by replacing the single type constructor of a monad with an
indexed family of constructors. Most notably, graded monads (indexed by a
monoid) model effect systems and parameterised monads (indexed by pairs of pre-
and post-conditions) model program logics. This paper studies the relationship
between these two generalisations of monads via a third generalisation. This
third generalisation, which we call category-graded monads, arises by
generalising a view of monads as a particular special case of lax functors. A
category-graded monad provides a family of functors T f indexed by morphisms f
of some other category. This allows certain compositions of effects to be ruled
out (in the style of a program logic) as well as an abstract description of
effects (in the style of an effect system). Using this as a basis, we show how
graded and parameterised monads can be unified, studying their similarities and
differences along the way.
