A monad on a category 
is a functor 
from the category to itself, together with natural
transformations 
and 
, called the unit and multiplication,
satisfying the associativity and unit identities
 |  |  |
(1)
|
 |  |  |
(2)
|
The associated Kleisli category has morphisms 
represented by morphisms 
in 
.
In older tuple terminology, a monad can instead mean a set containing a single object, or an n-tuple with 
. The yin-yang is also known
as the monad. These meanings are unrelated to the category-theoretic one.
