Among the constructions a functor can perform, one pairing recurs so often across the curriculum
that it deserves a name and a theory of its own. Two functors run in opposite directions and
stand in a precise reciprocal relation, each the best possible approximation to an inverse of the
other. Three pairs will recur on this page: the free vector space on a set with the underlying
set of a vector space, the abelianization of a group with the inclusion of abelian groups among
all groups, and the product of sets with the exponential. In each case the two functors are not
inverse, since they change the objects too drastically for that. Yet a single equation binds
them, and that equation turns out to govern an unexpectedly wide range of mathematics. The
relation is called adjunction.
Consider two functors in opposite directions, \(F : \mathscr{A} \to \mathscr{B}\) and
\(G : \mathscr{B} \to \mathscr{A}\). Roughly, \(F\) is adjoint to \(G\) when, for every object
\(A\) of \(\mathscr{A}\) and every object \(B\) of \(\mathscr{B}\), the morphisms
\(F(A) \to B\) in \(\mathscr{B}\) are the same thing as the morphisms \(A \to G(B)\)
in \(\mathscr{A}\). The two sets are not merely equal in number but matched by a
correspondence that respects every morphism in sight. Making "the same thing" precise is the
whole content of the definition.
Definition: Adjunction
Let \(F : \mathscr{A} \to \mathscr{B}\) and \(G : \mathscr{B} \to \mathscr{A}\) be functors.
We say \(F\) is left adjoint to \(G\), and \(G\) is right
adjoint to \(F\), written \(F \dashv G\), if there is a bijection
\[
\mathscr{B}\bigl(F(A), B\bigr) \cong \mathscr{A}\bigl(A, G(B)\bigr)
\]
for each object \(A\) of \(\mathscr{A}\) and each object \(B\) of \(\mathscr{B}\), and this
bijection is natural in \(A\) and \(B\) in the sense made precise below. A
choice of such a natural bijection is an adjunction between \(F\) and \(G\).
The bijection has a standard notation that makes its symmetry visible. Given a morphism
\(g : F(A) \to B\), its image under the bijection is written \(\bar{g} : A \to G(B)\) and called
the transpose of \(g\). Given \(f : A \to G(B)\), its inverse image is
\(\bar{f} : F(A) \to B\). The two operations are mutually inverse, so transposing twice returns
the original morphism: \(\bar{\bar{g}} = g\) and \(\bar{\bar{f}} = f\). One passes freely between
a morphism out of \(F(A)\) and a morphism into \(G(B)\), and the bar is the device for doing so.
Naturality is the requirement that this passage be compatible with composition on both
sides. Stated in full, it has two parts: for all morphisms \(g : F(A) \to B\) and
\(q : B \to B'\) in \(\mathscr{B}\),
\[
\overline{q \circ g} = G(q) \circ \bar{g},
\]
and for all morphisms \(p : A' \to A\) in \(\mathscr{A}\) and \(f : A \to G(B)\),
\[
\overline{f \circ p} = \bar{f} \circ F(p).
\]
The first says that postcomposing in \(\mathscr{B}\) and then transposing agrees with transposing
and then applying \(G\). The second says the analogous thing for precomposition in
\(\mathscr{A}\) through \(F\). Together they pin down the correspondence so tightly that no
arbitrary choices remain, and this is exactly what one means, informally, by calling the matching
between the two sets of morphisms natural.
A single consequence shows the naturality axiom at work. Suppose we are given a chain of
morphisms in \(\mathscr{A}\) ending at some object, a single morphism out of its \(F\)-image, and
a chain in \(\mathscr{B}\) leading away:
\[
A_0 \to \cdots \to A_n, \quad F(A_n) \to B_0, \quad B_0 \to \cdots \to B_m.
\]
From this data the adjunction produces exactly one morphism \(A_0 \to G(B_m)\). To obtain it, we
compose the \(\mathscr{B}\)-chain onto the middle morphism, transpose the result across the
bijection, and precompose with the \(\mathscr{A}\)-chain. The two naturality equations are
precisely the guarantee that the order of these operations does not matter. Transposing first and
then composing sends the \(\mathscr{B}\)-chain through \(G\), since a transpose lands in
\(G(B_0)\) and only the \(G\)-image of the \(\mathscr{B}\)-chain can be composed onto it.
Composing first and then transposing sends the \(\mathscr{A}\)-chain through \(F\). Both orders
yield the same arrow \(A_0 \to G(B_m)\). Naturality is what makes "the morphism obtained from
this data" a well-defined phrase rather than an ambiguous recipe.
Naturality as a coming attraction
The naturality requirement is the categorical meaning of "natural" we met for transformations
between functors: a construction defined without arbitrary choices. This is no analogy. The
full force of that earlier notion surfaces again once we recognize the adjunction bijection
itself as a
natural isomorphism
between two functors built from the hom-sets. At that point the two naturality equations
above become a single statement, namely that one natural transformation is invertible.
Free Constructions as Left Adjoints
Once one knows to look for them, adjoint functors turn up everywhere. The reliable signal is a
pair of functors running in opposite directions between two categories, and the phenomenon is
common enough to serve as a working heuristic. Told that some construction turns every object
of one kind into an object of another, and that a construction in the reverse direction also
exists, one should suspect at once that the two are adjoint. The suspicion is usually correct.
A mathematician describing, say, a way to turn any Lie algebra into an associative algebra and
a way back again is, whether or not the word is used, describing an adjunction. Recognizing the
adjunction spares one the labor of learning either construction in detail, since each
determines the other.
The most transparent family pairs a forgetful functor with a free
construction. A forgetful functor discards structure. A group, a vector space, or a
topological space goes to its underlying set, and a structure-preserving map to the underlying
function. The left adjoint of a forgetful functor, when it exists, builds the most economical
structured object on a given set: the one that imposes no relations beyond those forced by the
axioms. The adjunction equation expresses precisely the sense in which the free object is
universal among all maps into structured objects.
The Free Vector Space
Let \(k\) be a field. The forgetful functor \(U : \mathbf{Vect}_k \to \mathbf{Set}\) sends a
vector space
to its underlying set of vectors and a
linear map
to its underlying function. Its left adjoint \(F : \mathbf{Set} \to \mathbf{Vect}_k\) sends a set
\(S\) to the vector space \(F(S)\) with basis \(S\). Its elements are the formal finite linear
combinations \(\sum_{s \in S} \lambda_s\, s\) with coefficients in \(k\). We claim
\(F \dashv U\), and unlike the heuristic above, this claim can be checked by hand.
Proposition: The Free-Forgetful Adjunction on Vector Spaces
Let \(k\) be a field, let \(U : \mathbf{Vect}_k \to \mathbf{Set}\) be the forgetful functor,
and let \(F : \mathbf{Set} \to \mathbf{Vect}_k\) send a set to the vector space with that set
as basis. For each set \(S\) and vector space \(V\) there is a bijection
\[
\mathbf{Vect}_k\bigl(F(S), V\bigr) \cong \mathbf{Set}\bigl(S, U(V)\bigr),
\]
natural in \(S\) and \(V\). That is, \(F \dashv U\).
Proof:
We exhibit the two transpose operations and verify they are mutually inverse. Given a linear
map \(g : F(S) \to V\), define a function \(\bar{g} : S \to U(V)\) by restricting \(g\) to
the basis: \(\bar{g}(s) = g(s)\) for each \(s \in S\). This is a well-defined map of sets.
Conversely, given a function \(f : S \to U(V)\), define a linear map
\(\bar{f} : F(S) \to V\) by extending \(f\) linearly off the basis:
\[
\bar{f}\Bigl(\textstyle\sum_{s \in S} \lambda_s\, s\Bigr) = \sum_{s \in S} \lambda_s\, f(s).
\]
Because \(S\) is a basis of \(F(S)\), every element of \(F(S)\) is a unique finite linear
combination of basis vectors, so \(\bar{f}\) is well defined and linear, and it is the unique
linear map agreeing with \(f\) on \(S\).
These two operations are mutually inverse. Starting from a linear map \(g\), the map
\(\bar{\bar{g}}\) extends \(\bar{g} = g|_S\) linearly. Since \(g\) is itself linear and
agrees with \(\bar{g}\) on the basis, uniqueness of linear extension gives
\(\bar{\bar{g}} = g\). Starting from a function \(f : S \to U(V)\), the function
\(\bar{\bar{f}}\) restricts \(\bar{f}\) to the basis, and \(\bar{f}(s) = f(s)\) there, so
\(\bar{\bar{f}} = f\). Hence the two operations are inverse bijections between
\(\mathbf{Vect}_k(F(S), V)\) and \(\mathbf{Set}(S, U(V))\) for each \(S\) and \(V\).
Naturality is the statement that these bijections commute with precomposition by functions
\(S' \to S\) and postcomposition by linear maps \(V \to V'\). For postcomposition by a linear
map \(q : V \to V'\): transposing \(q \circ g\) restricts it to the basis and gives
\(s \mapsto q(g(s)) = q(\bar{g}(s))\), which is \(U(q) \circ \bar{g}\) since \(U(q)\) is the
underlying function of \(q\). The precomposition identity is verified the same way, using
that \(F\) sends a function \(p : S' \to S\) to the linear map extending it on bases. Both
\(\overline{f \circ p}\) and \(\bar{f} \circ F(p)\) are linear maps \(F(S') \to V\) sending
each basis vector \(s'\) to \(f(p(s'))\), so they coincide. Both naturality equations
therefore hold, completing the proof.
The proof is worth dwelling on because it exposes the general mechanism. The bijection rests on a
single fact about \(F(S)\): a linear map out of it is determined by, and may be freely prescribed
by, its values on \(S\). This is the universal property of the free object, and it is
what every free construction has in common. We isolate that property in the next section, where
it acquires its categorical name.
Free Groups and the Pattern in General
The same adjunction holds with
groups
in place of vector spaces. The forgetful functor \(\mathbf{Grp} \to \mathbf{Set}\) has a left
adjoint sending a set to the free group on it, the group of reduced words in the elements of the
set and their formal inverses. The free group is harder to construct explicitly than the free
vector space, since there is no basis to extend along so transparently. All the same, the
adjunction characterizes it completely. A homomorphism out of the free group on \(S\) is the same
thing as a function from \(S\) into the underlying set of the target. Forgetful functors between
categories of algebraic structures almost always admit such left adjoints, a fact that can be
established once and for all rather than rediscovered case by case.
A left adjoint, when it exists, is unique up to natural isomorphism. We indicate the reason in
the next section and prove it in full once the Yoneda lemma is available. The phrase "the free
group on \(S\)" is therefore unambiguous even before any construction is given. Knowing that a
functor is left adjoint to the forgetful functor pins it down, which is often all one needs.
Monoids, and a Forgetful Functor with Two Adjoints
Groups are not the only structure built from an associative binary operation. Dropping the
requirement of inverses, while keeping associativity and an identity element, gives the notion of
a monoid. We will need monoids repeatedly, since a monoid is exactly a category with a single
object, its elements being the morphisms and its operation the composition.
Definition: Monoid
A monoid is a set \(M\) equipped with an associative binary operation
\(M \times M \to M\), written multiplicatively, and an identity element
\(e \in M\) satisfying \(e\,m = m = m\,e\) for all \(m \in M\). A monoid
homomorphism is a function preserving the operation and the identity. Monoids and
their homomorphisms form a category \(\mathbf{Mon}\).
Every group is a monoid in which each element happens to have an inverse, so there is an
inclusion functor \(U : \mathbf{Grp} \to \mathbf{Mon}\) viewing a group as a monoid and a
homomorphism as itself. Forgetting that inverses exist loses no data. The functor simply declines
to record a property. The morphisms are genuinely unchanged, not merely relabeled. A monoid
homomorphism \(\phi\) between two groups preserves the product and the identity, and it
automatically preserves inverses as well. Indeed,
\(\phi(g)\,\phi(g^{-1}) = \phi(g\,g^{-1}) = \phi(e) = e\) forces \(\phi(g^{-1}) = \phi(g)^{-1}\),
so \(\phi\) is already a group homomorphism.
We therefore have
\[
\mathbf{Grp}(G, H) = \mathbf{Mon}\bigl(U(G), U(H)\bigr),
\]
which exhibits \(\mathbf{Grp}\) as a
full subcategory
of \(\mathbf{Mon}\). This inclusion has adjoints on both sides:
\[
F \dashv U \dashv R,
\]
a configuration we have not seen before.
The left adjoint \(F : \mathbf{Mon} \to \mathbf{Grp}\) freely adjoins inverses. It sends a
monoid \(M\) to the group obtained by formally throwing in an inverse for every element, subject
only to the relations already holding in \(M\). When \(M\) is the additive monoid of natural
numbers, for instance, \(F(M)\) is the group of integers, the natural numbers with subtraction
made possible. This is the group completion of \(M\), and adjointness \(F \dashv U\) says
exactly that a monoid homomorphism \(M \to U(H)\) into (the monoid underlying) a group \(H\) is
the same thing as a group homomorphism \(F(M) \to H\). Maps out of the completion thus
correspond to maps out of the original monoid.
The right adjoint \(R : \mathbf{Mon} \to \mathbf{Grp}\) goes the other way, extracting rather
than adjoining. It sends a monoid \(M\) to its group of units. The units are the
elements that do possess an inverse, and they form a submonoid that is a group. Adjointness
\(U \dashv R\) says that a group homomorphism \(H \to R(M)\) is the same thing as a monoid
homomorphism \(U(H) \to M\). Since every element of a group is invertible, any monoid
homomorphism out of a group must land among the invertible elements of \(M\), which is to say in
\(R(M)\). The two adjoints capture the two canonical ways to mediate between monoids and groups:
complete a monoid into a group, or carve out the largest submonoid that is a group.
A two-sided rarity
Forgetful functors routinely have left adjoints, and the free constructions of this section
are exactly those. Having a right adjoint as well is rare. The inclusion
\(\mathbf{Grp} \hookrightarrow \mathbf{Mon}\) is the unusual case where both exist, which in
the standard terminology makes \(\mathbf{Grp}\) simultaneously a reflective
and a coreflective subcategory of \(\mathbf{Mon}\). A subcategory is
reflective when its inclusion admits a left adjoint and coreflective when its inclusion
admits a right adjoint, and here both hold at once. We will meet this two-sided pattern
\(F \dashv U \dashv R\) again for topological spaces.
Universal Properties and Initial Objects
The free constructions above all rested on a property of the form: a map out of the constructed
object is freely determined by less data. Such characterizations single out "the unique object
through which every map of a certain kind factors." They pervade mathematics under the name
universal property, and we have met many already, each proved in its own setting
by its own hands. The categorical viewpoint reveals them as instances of one structure, definable
purely in terms of arrows.
Abelianization as a Left Adjoint
A free construction need not start from a bare set. The inclusion functor
\(U : \mathbf{Ab} \to \mathbf{Grp}\), viewing an abelian group as a group, has a left adjoint
\(F : \mathbf{Grp} \to \mathbf{Ab}\) that forces a group to become abelian in the most economical
way. It sends a group \(G\) to its abelianization \(G^{\mathrm{ab}}\), the
quotient of \(G\) by the smallest
normal subgroup
containing all commutators
\(xyx^{-1}y^{-1}\). Together with the quotient homomorphism \(\eta : G \to G^{\mathrm{ab}}\), this
construction has a universal property that is the very shape of the adjunction.
The universal property of abelianization. Any homomorphism \(\varphi\) from \(G\) to an
abelian group \(A\) (purple) factors as \(\varphi = \bar{\varphi} \circ \eta\) through the
quotient \(\eta : G \to G^{\mathrm{ab}}\) (blue), and the factoring map \(\bar{\varphi}\)
(dashed) is unique.
Every homomorphism
\(\varphi : G \to A\) into an abelian group \(A\) factors uniquely as
\(\varphi = \bar{\varphi} \circ \eta\) through a homomorphism
\(\bar{\varphi} : G^{\mathrm{ab}} \to A\). The reason is that \(\varphi\), having an
abelian target, sends every commutator to the identity, so the
kernel
of \(\varphi\) contains the commutator subgroup. The homomorphism \(\varphi\) therefore
descends to the quotient, and the descended map is forced. This factoring is precisely
the adjunction bijection
\[
\mathbf{Ab}\bigl(G^{\mathrm{ab}}, A\bigr) \cong \mathbf{Grp}\bigl(G, U(A)\bigr),
\]
sending \(\bar{\varphi}\) on the left to \(U(\bar{\varphi}) \circ \eta\) on the right.
The functor \(U\) appears because \(\bar{\varphi}\) is a morphism of \(\mathbf{Ab}\) while
\(\eta\) is a morphism of \(\mathbf{Grp}\). To compose them, one first views \(\bar{\varphi}\) as
a group homomorphism, which is what \(U\) does. Since \(U\) is the inclusion and changes nothing,
it is routinely suppressed, and one writes simply \(\bar{\varphi} \circ \eta\). That
\(F = (-)^{\mathrm{ab}}\) is left adjoint to \(U\) is the statement that this factoring exists,
is unique, and is natural in \(G\) and \(A\). The quotient map \(\eta\), the universal
homomorphism out of \(G\), is the structural heart of the construction. It is an instance of a
phenomenon we now name.
Definition: Initial and Terminal Object
Let \(\mathscr{C}\) be a category. An object \(I\) of \(\mathscr{C}\) is
initial if for every object \(A\) there is exactly one morphism
\(I \to A\). An object \(T\) is terminal if for every object \(A\) there is
exactly one morphism \(A \to T\).
The empty set is initial in \(\mathbf{Set}\) and a one-element set is terminal. The trivial
group is both initial and terminal in \(\mathbf{Grp}\). The defining requirement of exactly
one morphism is a universal property in its barest form, and it is rigid enough to determine
the object completely.
Lemma: Uniqueness of Initial Objects
Let \(I\) and \(I'\) be initial objects of a category. Then there is a unique isomorphism
\(I \to I'\). In particular, \(I \cong I'\). The same holds for terminal objects.
Proof:
Since \(I\) is initial there is a unique morphism \(f : I \to I'\), and since \(I'\) is
initial there is a unique morphism \(f' : I' \to I\). The composite \(f' \circ f\) is a
morphism \(I \to I\). Since \(I\) is initial, there is exactly one such morphism, and the
identity \(1_I\) is one. Hence \(f' \circ f = 1_I\). Symmetrically
\(f \circ f' = 1_{I'}\), so \(f\) is an
isomorphism.
Its uniqueness as a morphism \(I \to I'\) is built in, since initiality of \(I\) allows
only one morphism to \(I'\) at all. The statement for terminal objects follows by
reversing every arrow.
This is the categorical source of a phrase used freely throughout the curriculum: "the" object
with a given universal property. Whenever an object is pinned down by a universal property, that
property exhibits it as initial or terminal in a suitable category of "objects equipped with the
relevant data," and the lemma then guarantees uniqueness up to a single canonical isomorphism.
The same argument settles the uniqueness of left adjoints promised earlier. A left adjoint to
\(G\) assigns to each object \(A\) an initial object in an associated category, whose objects
are the morphisms \(A \to G(B)\) out of \(A\) into a \(G\)-image. Any two left adjoints
therefore agree at each object up to a canonical isomorphism, and the full proof, given once
the Yoneda lemma is available, checks that these isomorphisms are natural. (The initial
morphism has the form \(A \to G(F(A))\), and its initiality is precisely the universal property
defining the free object.)
The universal properties already proved across the curriculum
The constructions whose universal properties were established earlier are exactly initial or
terminal objects in disguise. The
universal property of the tensor product
states that bilinear maps out of \(V \times W\) correspond to linear maps out of a single
object \(V \otimes W\). That object is initial among "vector spaces receiving a bilinear map
from \(V \times W\)." The
universal property of the quotient topology
and the universal property of the product topology
have the identical shape, one initial and one terminal. The abelianization just discussed is
the same phenomenon once more, since \(G^{\mathrm{ab}}\) is initial among abelian groups
receiving a homomorphism from \(G\). Each was proved by an argument of its own, and the
categorical definition names the one pattern they all instantiate.
Initial and terminal objects are themselves adjoints, which closes the circle. There is exactly
one functor from any category \(\mathscr{C}\) to the terminal category \(\mathbf{1}\). This
category has one object and only its identity morphism. A functor from \(\mathbf{1}\) to
\(\mathscr{C}\) is the same thing as a choice of object of \(\mathscr{C}\). Viewing objects of
\(\mathscr{C}\) this way, a left adjoint to the unique functor \(\mathscr{C} \to \mathbf{1}\) is
precisely an initial object of \(\mathscr{C}\), and a right adjoint is precisely a terminal
object. The two most basic universal properties are thus the two simplest adjunctions.
Adjunctions Beyond Algebra
The free-forgetful pattern might suggest that adjunctions are a feature of algebra in particular.
They are not. The same relation organizes topology and the elementary set theory of functions,
and two further examples show its range while introducing structures we will need later.
Topology: A Forgetful Functor Between Two Adjoints
The forgetful functor \(U : \mathbf{Top} \to \mathbf{Set}\), sending a
topological space
to its underlying set of points, sits between adjoints on both sides:
\[
D \dashv U \dashv I.
\]
The left adjoint \(D : \mathbf{Set} \to \mathbf{Top}\) equips a set with the discrete
topology, in which every subset is open. The right adjoint
\(I : \mathbf{Set} \to \mathbf{Top}\) equips it with the indiscrete topology, in
which only the empty set and the whole space are open.
The adjunctions express two familiar facts about continuity. Adjointness \(D \dashv U\) says that
a continuous map out of a discrete space \(D(S)\) is the same thing as an arbitrary function out
of \(S\). From a discrete space every function is continuous, so prescribing a continuous map
\(D(S) \to X\) is prescribing a bare function \(S \to U(X)\). Dually, \(U \dashv I\) says that a
continuous map into an indiscrete space \(I(S)\) is the same thing as an arbitrary function into
\(S\). Into an indiscrete space every function is continuous. The discrete and indiscrete
topologies are the finest and coarsest topologies on a set, and adjointness is the precise
expression of their extremal character.
The configuration \(D \dashv U \dashv I\) is the same shape \(F \dashv U \dashv R\) met for
groups inside monoids: a single structure-forgetting functor with a free construction on its left
and a cofree one on its right. The pattern will appear once more, decisively, when categories of
functors take the place of categories of spaces.
Sets: The Adjunction Behind Currying
Fix a set \(B\). Forming the product with \(B\) is a functor
\(- \times B : \mathbf{Set} \to \mathbf{Set}\), sending \(A\) to \(A \times B\). Forming the set
of functions out of \(B\) is another. Writing \(C^B\) for the set of functions \(B \to C\), the
assignment \(C \mapsto C^B\) is a functor \((-)^B : \mathbf{Set} \to \mathbf{Set}\). These two
are adjoint, \(- \times B \dashv (-)^B\), and the bijection is one used constantly, usually
without naming it.
Proposition: The Currying Adjunction
Let \(B\) be a set. For all sets \(A\) and \(C\) there is a bijection
\[
\mathbf{Set}\bigl(A \times B,\, C\bigr) \cong \mathbf{Set}\bigl(A,\, C^B\bigr),
\]
natural in \(A\) and \(C\). That is, \(- \times B \dashv (-)^B\).
Proof:
Given a function \(g : A \times B \to C\) of two arguments, define \(\bar{g} : A \to C^B\) by
holding the first argument fixed, so that \(\bar{g}(a)\) is the function
\(b \mapsto g(a, b)\). Conversely, given \(f : A \to C^B\), define
\(\bar{f} : A \times B \to C\) by \(\bar{f}(a, b) = \bigl(f(a)\bigr)(b)\). The two operations
are mutually inverse. Applied in succession, they return
\(\bar{\bar{g}}(a, b) = \bigl(\bar{g}(a)\bigr)(b) = g(a, b)\) and
\(\bigl(\bar{\bar{f}}(a)\bigr)(b) = \bar{f}(a, b) = \bigl(f(a)\bigr)(b)\), recovering \(g\)
and \(f\) respectively. Naturality in \(A\) and \(C\) follows by tracking the two
constructions through precomposition by a function \(A' \to A\) and postcomposition by a
function \(C \to C'\), each of which commutes with fixing and releasing the argument \(b\).
The bijection is the formal content of currying: a function of two arguments is
the same thing as a function of the first argument whose values are functions of the second.
Reading a map \(A \times B \to C\) as assigning to each point of \(A\) a map \(B \to C\) is the
set-theoretic shadow of a structure that a category may or may not possess, the
exponential object \(C^B\). This object internalizes the morphisms \(B \to C\).
A category with finite products in which every such exponential exists, with the currying
adjunction holding throughout, is called
cartesian closed.
We do not develop the notion here, but it is the categorical home of function types and of the
composition of processes, and the reference point against which the categorical accounts of
learning systems are later positioned.
When a Left Adjoint Fails to Exist
Adjunction is not automatic, and the exceptions are as informative as the rules. Let
\(\mathbf{Field}\) be the category of
fields
with ring homomorphisms
that send \(1\) to \(1\) as morphisms. The forgetful functor \(\mathbf{Field} \to \mathbf{Set}\)
has no left adjoint. There is no "free field" on a set.
Why fields admit no free construction
The failure can be seen directly. If a left adjoint \(F\) existed, then \(F(\varnothing)\)
would be an initial object of \(\mathbf{Field}\), because there is exactly one function from
the empty set into any set and hence exactly one homomorphism from \(F(\varnothing)\) into
any field. But no field is initial. A homomorphism between fields is injective, since its
kernel is an ideal of a field not containing \(1\) and is therefore zero, so it preserves the
characteristic. Hence a homomorphism between fields exists only when the two fields have the
same characteristic, and since fields of different characteristics exist, no field admits a
homomorphism into every other field.
The conceptual reason is the feature distinguishing a field from a group or a vector space.
For groups, vector spaces, and monoids, the defining data are operations that are
everywhere defined on the underlying set, subject to equations that hold
everywhere. For instance, every element of a group has an inverse, and
\(x \cdot x^{-1} = e\) holds for all \(x\). Call a structure of this kind a model of an
algebraic theory. Such a structure always admits a free construction,
because one can generate freely and then impose only the universal equations.
A field violates the condition. Its inverse operation \(x \mapsto x^{-1}\) is defined only
for \(x \neq 0\), a partial operation rather than a total one. The theory of fields is
therefore not algebraic in this sense, and the guarantee of a left adjoint that algebraic
theories enjoy is lost. The possibility of failure is what makes adjunction meaningful. When
two functors are adjoint, the relation is genuine information about how two kinds of
structure interlock, not a formality available for the asking.
Duality and Composition
Two structural features of adjunctions deserve recording before we put the relation to work, both
of which amplify its reach. The first is that the entire theory comes in mirror-image pairs. The
second is that adjunctions chain together.
Duality
The notions of this page are organized by the
opposite category
into dual pairs. Terminal is dual to initial. An object is terminal in \(\mathscr{C}\) exactly
when it is initial in \(\mathscr{C}^{\mathrm{op}}\). Right adjoint is dual to left adjoint in the
same way, since reversing all arrows in the defining bijection
\(\mathscr{B}(F(A), B) \cong \mathscr{A}(A, G(B))\) interchanges the roles of \(F\) and \(G\).
Consequently every theorem about initial objects or left adjoints yields, for free, a theorem
about terminal objects or right adjoints. The uniqueness lemma above was stated once and applied
to both cases by exactly this principle. Duality halves the labor of the subject.
Composition of Adjunctions
Adjunctions compose along a chain of categories. Suppose \(F \dashv G\) between \(\mathscr{A}\)
and \(\mathscr{B}\), and \(F' \dashv G'\) between \(\mathscr{B}\) and \(\mathscr{C}\). Then the
composite functors satisfy \(F' \circ F \dashv G \circ G'\), because for objects \(A\) of
\(\mathscr{A}\) and \(C\) of \(\mathscr{C}\) the two adjunction bijections compose:
\[
\begin{align*}
\mathscr{C}\bigl(F'(F(A)), C\bigr)
&\cong \mathscr{B}\bigl(F(A), G'(C)\bigr) \\\\
&\cong \mathscr{A}\bigl(A, G(G'(C))\bigr),
\end{align*}
\]
naturally in \(A\) and \(C\). The free-forgetful adjunctions stack in exactly this way. Composing
the free group functor \(\mathbf{Set} \to \mathbf{Grp}\) with abelianization
\(\mathbf{Grp} \to \mathbf{Ab}\) gives a left adjoint to the composite forgetful functor
\(\mathbf{Ab} \to \mathbf{Grp} \to \mathbf{Set}\), so the abelianization of the free group on
\(S\) has the universal property of a free abelian group on \(S\).