The non-descriptive name
"triple"
for this concept has now largely been
superseded by
"monad" ,
although there is an obstinate minority
of category-theorists who continue to use it. A
comonad
(or
cotriple)
on a category
is a monad on
;
in other words, it is a functor
equipped with natural transformations
and
satisfying the duals of the commutative diagrams above. Every adjoint pair of functors
(
)
gives rise to a comonad structure on the composite
,
as well as a monad structure on
.
An important example of a functor which carries a comonad structure is
,
,
or, equivalently, the functor of big Witt vectors, cf.
-ring;
Witt vector.
A special case of the natural transformation
occurs in algebraic number theory as the
Artin–Hasse exponential,
[a5].
Monads in the category of sets can be equivalently described by sets
of
-ary
operations for each cardinal number (or set)
;
gives the projection operations
,
and
gives the rules for composing operations. See
[5]
or
[a1].
This approach extends to monads in arbitrary categories, but it has not
proved useful in general, as it has in or near sets.
Of the two canonical ways of constructing an
adjunction
from a given monad, mentioned in the
main article above, the Eilenberg–Moore construction (or
category of
-algebras)
is by far the more important. Given a monad
on a category
,
a
-algebra
in
is a pair
where
is a morphism such that
commutes. A
homomorphism of
-algebras
is a morphism
in
such that
commutes; thus, one has a category
of
-algebras,
with an evident forgetful functor
.
The functor
has a left adjoint
,
which sends an object
of
to the
-algebra
,
and the monad induced by the adjunction
(
)
is the one originally given.
Now the
Kleisli category
of
is just the full subcategory of
on the objects
:
the
category of free algebras
(cf. also
Category).
For a monad
on
,
in the Kleisli construction the category
has as objects the objects of
,
and as hom-sets the sets
The composition rule for
assigns to
and
the
-composite:
as identity mapping
one uses the
-morphism
.
An adjoint pair
,
is obtained by setting
for
,
for
,
for
,
and
for
.
Then
will serve as unit for the adjunction, while the co-unit
is given by
Co-algebras
are defined in the same manner. In practice, co-algebras
very often occur superposed on algebras; a comonad
will be constructed on a category of algebras of some sort,
,
leading to the category
of
bi-algebras.
An important class of cases involves a monad
and a cotriple
on the same category
.
There is a standard lifting of
to a cotriple
on
.
A
"TG-bi-algebraTG-bi-algebra"
means an object of
;
the reverse order is also possible, but rarely
occurs, and the objects would not be called bi-algebras.
For the role of comonads in (algebraic) cohomology theories see
Cohomology of algebras
and
[a2],
[a3];
particularly
[a2]
for explicit interpretation.
An adjunction is said to be
monadic
(or monadable) if the Eilenberg–Moore construction applied to the
monad it induces yields an adjunction equivalent to the original
one. Many important examples of adjunctions are monadic; for example, for any
variety of universal algebras,
the forgetful functor from the variety to the category of
sets and its left adjoint (the free algebra functor) form a monadic adjunction.
A monad
is said to be
idempotent
if
is an isomorphism. In this case it can be shown that any
-algebra
structure
on an object
is necessarily a two-sided inverse for
,
and hence that
is isomorphic to the full subcategory
consisting of all objects
such that
is an isomorphism.
is a
reflective subcategory
of
,
the left adjoint to the inclusion being given by
itself. Conversely, for any reflective subcategory of
,
the monad on
induced by the inclusion and its left adjoint is idempotent;
thus, the adjunctions corresponding to reflective subcategories are always monadic.