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.
In nonstandard analysis, an infinitesimal monad is a set of hyperreal numbers. This is a different meaning of the term (Keisler 2007, p. 2).
