In Category Theory we introduced the language of categories, and in many posts in this blog we have seen how useful it is in describing concepts in modern mathematics, for example in the two most recent posts, The Theory of Motives and Algebraic Spaces and Stacks. In this post, we introduce another important concept in category theory, that of adjoint functors, as well as the closely related notion of monads. Manifestations of these ideas are quite ubiquitous in modern mathematics, and we enumerate a few examples in this post.
An adjunction between two categories and
is a pair of functors,
, and
, such that there exists a bijection
for all objects of
and all objects
of
. We say that
is left-adjoint to
, and that
is right-adjoint to
. We may also write
.
An adjunction determines two natural transformations and
, called the unit and counit, respectively. Conversely, the functors
and
, together with the natural transformations
and
, are enough to determine the adjunction, therefore we can also denote the adjunction by
.
We give an example of an adjunction. Let be a fixed field, and consider the functors
where is the functor which assigns to a set
the vector space
made up of formal linear combinations of elements of
with coefficients in
; in other words, an element of
can be written as
, where
and
, and
is the forgetful functor, which assigns to a vector space
the set
of elements (vectors) of
; in other words it simply “forgets” the vector space structure on
.
For every function in
we have a linear transformation
in
given by
. The correspondence
has an inverse
, given by restricting
to
(so that our only linear transformations are of the form
, and we can obtain set-theoretic functions corresponding to these linear transformations). Hence we have a bijection
.
We therefore see that the two functors and
form an adjunction; the functor
(sometimes called the free functor) is left-adjoint to the forgetful functor
, and
is right-adjoint to
.
As another example, consider now the category of modules over a commutative ring , and the functors
and
(see The Hom and Tensor Functors). For every morphism
we have another morphism
given by
. We actually have a bijection
.
This is called the Tensor-Hom adjunction.
Closely related to the concept of an adjunction is the concept of a monad. A monad is a triple where
is a functor from
to itself,
is a natural transformation from
to
, and
is a natural transformation from
, satisfying the following properties:
Dual to the concept of a monad is the concept of a comonad. A comonad on a category may be thought of as a monad on the opposite category
.
As an example of a monad, we can consider the action of a fixed group on a set (such as the symmetric group permuting the elements of the set, for example). In this case, our category will be
, and
,
, and
are given by
given by
given by
Adjunctions and monads are related in the following way. Let and
be a pair of adjoint functors with unit
and counit
. Then we have a monad on
given by
. We can also obtain a comonad given by
.
Conversely, if we have a monad on the category
, we can obtain a pair of adjoint functors
and
, where
is the Eilenberg-Moore category, whose objects (called
-algebras) are pairs
, where
is an object of
, and
is a morphism
satisfying
,
and whose morphisms are morphisms
in
such that
.
In the example we gave above in the discussion on monads, the -algebras are exactly the sets with the action of the group
. If
is such a set, then the corresponding
-algebra is the pair
, where the function
satisfies
.
For comonads, we have a dual notion of coalgebras. These “dual” ideas are important objects of study in themselves, for example in topos theory. Another reason to consider comonads and coalgebras is that in mathematics there often arises a situation where we have three functors
where is left-adjoint to
, and
is right-adjoint to
(a so-called adjoint triple). As an example, consider the forgetful functor
which assigns to a topological space its underlying set. It has both a left-adjoint
which assigns to a set
the trivial topology (where the only open sets are the empty set and
itself), and a right-adjoint
which assigns to the set
the discrete topology (where every subset of
is an open set). Therefore we have a monad and a comonad on
given by
and
respectively.
Many more examples of adjoint functors and monads can be found in pretty much all areas of mathematics. And according to a principle attributed to the mathematician Saunders Mac Lane (one of the founders of category theory, along with Samuel Eilenberg), such a structure that occurs widely enough in mathematics deserves to be studied for its own sake.
References:
Adjunctions by TheCatsters on YouTube
Monads by TheCatsters on YouTube
Categories for the Working Mathematician by Saunders Mac Lane
Category Theory by Steve Awodey
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