The Preservation Theorem
Two threads of the preceding pages are about to meet. One thread developed
adjunctions,
pairs of functors in opposite directions related by a natural bijection of
hom-sets. The other developed limits. Along the way the two threads produced facts
that already point at each other. A
set-valued functor with a left adjoint is representable,
and representables preserve limits.
Set side by side, the two results strongly suggest a third, that a set-valued
functor with a left adjoint should itself preserve limits. The suggestion is
correct, and the restriction to set-valued functors is unnecessary. Possessing a
left adjoint forces a functor to preserve limits, whatever category it lands in,
and possessing a right adjoint forces it to preserve colimits. That is the theorem
of this page, proved here for locally small categories.
The theorem repays a promissory note issued when the
limit functor
was exhibited as a right adjoint, and it opens a question of its own. If every
right adjoint preserves limits, when does the converse hold, so that preserving
limits guarantees the existence of a left adjoint? The closing section takes up
that question, proves the answer in full for ordered sets, and describes the shape
of the answer, the adjoint functor theorems, for categories at large.
Convention. Unless a statement introduces its own functors,
\(F \dashv G\) is an adjunction between functors
\(F : \mathscr{A} \to \mathscr{B}\) and \(G : \mathscr{B} \to \mathscr{A}\).
The bar \(\bar{(\,\cdot\,)}\) denotes the transpose of a morphism across the
adjunction, in either direction, so that \(\bar{g} : A \to G(B)\) for
\(g : F(A) \to B\) and conversely. Both categories are assumed locally small,
so that hom-collections are sets and the machinery of set-valued functors
applies to them.
One preliminary comes first. It is stated on its own because the proof below turns
on it, and because it is the kind of step that is easier to check once than to
repeat. Limit cones can be carried across an isomorphism of diagrams.
Lemma: Transport of a Limit Cone along an Isomorphism of Diagrams
Let \(\mathbf{I}\) be a small category, let
\(E, E' : \mathbf{I} \to \mathscr{C}\) be diagrams in a category
\(\mathscr{C}\), and let \(\kappa : E \to E'\) be a natural
transformation all of whose components \(\kappa_I\) are isomorphisms. If
\(\big(L \xrightarrow{q_I} E(I)\big)_{I \in \mathbf{I}}\) is a limit cone
on \(E\), then
\(\big(L \xrightarrow{\kappa_I \circ q_I} E'(I)\big)_{I \in \mathbf{I}}\)
is a limit cone on \(E'\).
Proof
Fix an object \(S\) of \(\mathscr{C}\). Sending a family
\((f_I : S \to E(I))_{I \in \mathbf{I}}\) to
\((\kappa_I \circ f_I)_{I \in \mathbf{I}}\) takes cones on \(E\) with vertex
\(S\) to cones on \(E'\) with vertex \(S\). Indeed, naturality of \(\kappa\)
turns the cone equation \(E(u) \circ f_I = f_J\) into
\(E'(u) \circ (\kappa_I \circ f_I) = \kappa_J \circ f_J\) for every
\(u : I \to J\) of \(\mathbf{I}\). Composing with the inverses
\(\kappa_I^{-1}\) is an assignment in the other direction, which also respects
the cone equations, since naturality of \(\kappa\) rearranges to
\(E(u) \circ \kappa_I^{-1} = \kappa_J^{-1} \circ E'(u)\). The two assignments
are mutually inverse, so this is a bijection.
Let \((g_I : S \to E'(I))_{I \in \mathbf{I}}\) be a cone on \(E'\) and
let \(h : S \to L\). Then \(\kappa_I \circ q_I \circ h = g_I\) holds for
every \(I\) exactly when \(q_I \circ h = \kappa_I^{-1} \circ g_I\) holds
for every \(I\), and the family on the right is the cone on \(E\) matched
with \((g_I)\) by the bijection. The universal property of \((q_I)\)
supplies one such \(h\) and no other, which is the universal property
required of \((\kappa_I \circ q_I)\).
Theorem: Adjoints and the Preservation of (Co)limits
Let \(\mathscr{A}\) and \(\mathscr{B}\) be locally small categories and let
\(F \dashv G\) be an adjunction between functors
\(F : \mathscr{A} \to \mathscr{B}\) and \(G : \mathscr{B} \to \mathscr{A}\).
Then \(G\)
preserves limits
and \(F\) preserves colimits.
Proof
Right adjoints preserve limits. Let \(\mathbf{I}\) be a small
category, \(D : \mathbf{I} \to \mathscr{B}\) a diagram, and
\(\big(L \xrightarrow{p_I} D(I)\big)_{I \in \mathbf{I}}\) a
limit cone
on \(D\). We must show that
\(\big(G(L) \xrightarrow{G(p_I)} G(D(I))\big)_{I \in \mathbf{I}}\) is a limit
cone on \(G \circ D\). It is a cone, since applying the functor \(G\) to the
cone equations \(Du \circ p_I = p_J\), one for each map \(u : I \to J\) of
\(\mathbf{I}\), yields \(G(Du) \circ G(p_I) = G(p_J)\). The work is in the
universal property.
The plan is to assemble, for each object \(A\) of \(\mathscr{A}\), an
isomorphism between the set \(\mathscr{A}(A, G(L))\) and the set of cones on
\(G \circ D\) with vertex \(A\), naturally in \(A\), and then to recognize the
result as a representation.
Two limit cones in \(\mathbf{Set}\). Fix \(A \in \mathscr{A}\). Since
representables preserve limits,
applying the hom-functor \(\mathscr{B}(F(A), -)\) to the limit cone
\((p_I)_{I \in \mathbf{I}}\) shows that the family
\[
\Big(\mathscr{B}(F(A), L)
\xrightarrow{\mathscr{B}(F(A), p_I)}
\mathscr{B}(F(A), D(I))\Big)_{I \in \mathbf{I}}
\]
is a limit cone in \(\mathbf{Set}\) on the diagram \(\mathscr{B}(F(A), D)\).
Independently, by the lemma exhibiting
the set of cones as a limit,
applied to the diagram \(G \circ D : \mathbf{I} \to \mathscr{A}\),
the evaluation maps
\[
\Big(\operatorname{Cone}(A, G \circ D)
\xrightarrow{e_I}
\mathscr{A}(A, G(D(I)))\Big)_{I \in \mathbf{I}},
\quad e_I\big((x_J)_{J \in \mathbf{I}}\big) = x_I,
\]
form a limit cone on the diagram \(\mathscr{A}(A, G \circ D)\).
Transposition matches the two diagrams. The two limit cones just
constructed sit over different diagrams, and the adjunction is exactly the
bridge between them. For each \(I\), transposition is a bijection
\(\mathscr{B}(F(A), D(I)) \to \mathscr{A}(A, G(D(I)))\),
\(g \mapsto \bar{g}\), and the first of the two
naturality equations of the adjunction
reads, for \(u : I \to J\) in \(\mathbf{I}\) and \(g : F(A) \to D(I)\),
\[
\overline{Du \circ g} = G(Du) \circ \bar{g} .
\]
The transposition bijections therefore constitute an isomorphism between the
diagrams \(\mathscr{B}(F(A), D)\) and \(\mathscr{A}(A, G \circ D)\), commuting
with every map of either diagram.
Transport along an isomorphism of diagrams
now applies, the transposition bijections being the components of the
isomorphism. Composing them with the limit cone on \(\mathscr{B}(F(A), D)\),
we obtain the family
\[
\Big(\mathscr{B}(F(A), L)
\xrightarrow{\gamma_I}
\mathscr{A}(A, G(D(I)))\Big)_{I \in \mathbf{I}},
\quad \gamma_I(g) = \overline{p_I \circ g},
\]
a limit cone on \(\mathscr{A}(A, G \circ D)\).
Comparing the two limit cones. The cones \((\gamma_I)\) and \((e_I)\)
are limit cones on the same diagram, so by the
uniqueness of limits
there is a unique isomorphism
\(\theta_A : \mathscr{B}(F(A), L) \to \operatorname{Cone}(A, G \circ D)\)
satisfying \(e_I \circ \theta_A = \gamma_I\) for every \(I\). The evaluations
\(e_I\) are jointly injective, a cone being by definition the family of its
components, so this characterization determines \(\theta_A\) pointwise. The
isomorphism \(\theta_A\) sends \(g : F(A) \to L\) to the cone
\(\big(\overline{p_I \circ g}\big)_{I \in \mathbf{I}}\). Composing with
transposition of the adjunction itself, in the direction
\(\mathscr{A}(A, G(L)) \to \mathscr{B}(F(A), L)\), gives a bijection
\(\alpha_A : \mathscr{A}(A, G(L)) \to \operatorname{Cone}(A, G \circ D)\) with
\[
\begin{align*}
\alpha_A(f)
&= \theta_A(\bar{f}) \\\\
&= \big(\overline{p_I \circ \bar{f}}\big)_{I \in \mathbf{I}} .
\end{align*}
\]
Naturality in the vertex. Both the source and the target of
\(\alpha\) are contravariant set-valued functors of \(A\). For the target,
precomposition with a map acts on a cone legwise, as recorded when
cones were identified with natural transformations,
and each evaluation \(e_I\) is
natural in the vertex.
To verify that the maps \(\alpha_A\) form a natural transformation, let
\(s : A' \to A\) be a morphism of \(\mathscr{A}\) and \(f : A \to G(L)\) an
element. By the joint injectivity of the evaluations noted above, it suffices
to check the naturality square after composing with each \(e_I\). The second
naturality equation of the adjunction
states that \(\overline{f \circ s} = \bar{f} \circ F(s)\). Since transposition
is a bijection, the same equation read in the inverse direction states that
\(\overline{g \circ F(s)} = \bar{g} \circ s\) for \(g : F(A) \to B\). Using
the first form and then the second,
\[
\begin{align*}
e_I\big(\alpha_{A'}(f \circ s)\big)
&= \overline{p_I \circ \overline{f \circ s}} \\\\
&= \overline{p_I \circ \bar{f} \circ F(s)} \\\\
&= \overline{p_I \circ \bar{f}} \circ s \\\\
&= e_I\big(\alpha_A(f)\big) \circ s \\\\
&= e_I\big(\alpha_A(f) \circ s\big),
\end{align*}
\]
the final equality because precomposition of cones acts legwise.
Hence \(\alpha\) is a natural isomorphism
\[
\alpha : \mathscr{A}(-, G(L)) \cong \operatorname{Cone}(-, G \circ D) ,
\]
a representation of the cone functor of the diagram \(G \circ D\).
From representation to limit cone. The passage from a representation
of the cone functor back to a limit cone is the correspondence of the
proposition that
limits are representations.
We carry the passage out explicitly, in order to identify the resulting cone.
Set \(w = \alpha_{G(L)}(1_{G(L)}) \in \operatorname{Cone}(G(L), G \circ D)\).
For any \(A\) and any \(f : A \to G(L)\), the naturality square of \(\alpha\)
at \(f\), evaluated at the identity, gives
\[
\operatorname{Cone}(f, G \circ D)(w) = \alpha_A(f),
\]
where the left-hand side composes every leg of \(w\) with \(f\). Since
\(\alpha_A\) is a bijection, every cone on \(G \circ D\) with vertex \(A\) is
obtained from \(w\) by legwise composition with a unique map
\(f : A \to G(L)\). This is verbatim the universal property of a
limit
cone, so \(w\) is a limit cone on \(G \circ D\).
It remains to compute \(w\) and to recognize it as
\(\big(G(p_I)\big)_{I \in \mathbf{I}}\). By the definition of the
counit,
the transpose \(\overline{1_{G(L)}}\) is the component
\(\varepsilon_L : FG(L) \to L\). The leg of the cone \(w\) at \(I\) is
therefore \(\overline{p_I \circ \varepsilon_L}\), and by the
transpose formulae
followed by the
triangle identities,
\[
\begin{align*}
\overline{p_I \circ \varepsilon_L}
&= G(p_I \circ \varepsilon_L) \circ \eta_{G(L)} \\\\
&= G(p_I) \circ G(\varepsilon_L) \circ \eta_{G(L)} \\\\
&= G(p_I) .
\end{align*}
\]
Hence \(w = \big(G(L) \xrightarrow{G(p_I)} G(D(I))\big)_{I \in \mathbf{I}}\),
and this cone is a limit cone on \(G \circ D\). Since \(\mathbf{I}\), \(D\),
and the limit cone \((p_I)\) were arbitrary, \(G\) preserves limits.
Left adjoints preserve colimits. This half is the mirror image of the
first, and the dualization is already installed in the definitions. Reversing
all arrows in the defining bijection
\(\mathscr{B}(F(A), B) \cong \mathscr{A}(A, G(B))\) reads it as a bijection
\(\mathscr{A}^{\mathrm{op}}(G(B), A) \cong \mathscr{B}^{\mathrm{op}}(B, F(A))\),
natural in both variables. The opposite functors therefore form an adjunction
\(G^{\mathrm{op}} \dashv F^{\mathrm{op}}\) in which
\(F^{\mathrm{op}} : \mathscr{A}^{\mathrm{op}} \to \mathscr{B}^{\mathrm{op}}\)
is the right adjoint. This exchange of roles is recorded in
the duality discussion of the adjunction page.
The opposite categories are locally small along with the originals, so the
half already proved applies to \(F^{\mathrm{op}}\).
Now let \(D : \mathbf{I} \to \mathscr{A}\) be a diagram and
\(\big(D(I) \xrightarrow{q_I} C\big)_{I \in \mathbf{I}}\) a
colimit cocone
on \(D\). By definition this is a limit cone on the diagram
\(D^{\mathrm{op}} : \mathbf{I}^{\mathrm{op}} \to \mathscr{A}^{\mathrm{op}}\),
whose shape \(\mathbf{I}^{\mathrm{op}}\) is small along with \(\mathbf{I}\).
The right adjoint \(F^{\mathrm{op}}\) carries it to a limit cone on
\(F^{\mathrm{op}} \circ D^{\mathrm{op}} = (F \circ D)^{\mathrm{op}}\), and a
limit cone on \((F \circ D)^{\mathrm{op}}\), read back in \(\mathscr{B}\), is
a colimit cocone on \(F \circ D\) with legs \(F(q_I)\) and vertex \(F(C)\).
Hence \(F\) preserves colimits.
We pause for a remark on the hypotheses. Local smallness enters only because the
proof routes the universal property through hom-sets and set-valued functors, the
same route that carried the representability theory of the preceding pages. Every
category appearing in the examples of this page, and every category of algebraic
or topological structures we have met, is locally small, so the hypothesis costs
nothing in practice.
This proof is worth a second look, because it is a reunion. The adjunction
supplied the transposition bijections and, at the very last step, the counit and a
triangle identity. The representability theory supplied the passage from a natural
isomorphism to a limit cone. The limit theory of the presheaf pages supplied both
limit cones in \(\mathbf{Set}\) that the transposition matched. Each of the three
formalisms for universal properties contributed the step the others could not, and
the theorem is their joint product.
What Preservation Buys
The preservation theorem
is a machine for converting adjunctions into limit computations. Every adjunction
catalogued on the earlier pages now emits a preservation statement for free, and
the contrapositive runs the machine backwards. A functor that fails to preserve
some colimit cannot have a right adjoint, and a functor that fails to preserve
some limit cannot have a left one, so failures of preservation become proofs of
nonexistence. This section works through both directions.
Forgetful functors
The free-forgetful adjunction on vector spaces
exhibits the forgetful functor \(U : \mathbf{Vect}_k \to \mathbf{Set}\) as a
right adjoint. By the preservation theorem, \(U\) preserves limits. Products of
vector spaces, equalizers of linear maps, and limits generally are therefore
computed on underlying sets.
The study of forgetful functors
obtained this by lifting limits from sets, writing the argument out for products
of groups and asserting it for the other algebraic categories. Here the
adjunction delivers it in one line. The same conclusion applies to groups,
rings, and modules the moment a free construction is exhibited as a left adjoint
to the corresponding forgetful functor. The question of when such left adjoints
exist, answered affirmatively for all these categories, is precisely the
business of the closing section.
On the colimit side the machine explains the asymmetry noted in that study as
well. Forgetful functors out of algebraic categories hardly ever possess right
adjoints, and they preserve all limits but rarely colimits. The
underlying set of a direct sum of vector spaces is not the disjoint union of the
underlying sets, and now the failure has a structural reading, the absence of a
right adjoint.
Products and exponentials
The
currying adjunction
\(- \times B \dashv (-)^B\) on \(\mathbf{Set}\), with the
exponential
\(C^B\) the set of functions \(B \to C\), feeds the machine in both directions at
once. The functor \(- \times B\) is a left adjoint, so it preserves colimits, and
in particular
sums,
finite and infinite alike. For the empty sum and a binary sum this reads
\[
0 \times B \cong 0,
\quad
(A_1 + A_2) \times B \cong (A_1 \times B) + (A_2 \times B) .
\]
The functor \((-)^B\) is a right adjoint, so it preserves limits, and in
particular
products.
For the empty product and a binary product this reads
\[
1^B \cong 1,
\quad
(A_1 \times A_2)^B \cong A_1^B \times A_2^B .
\]
Restricted to finite sets and read through cardinality, these four isomorphisms
are the annihilation of zero, the distributive law, and two rules of exponents of
ordinary arithmetic, for the natural numbers are exactly the isomorphism classes
of finite sets. The arithmetic of the natural numbers is, from this vantage point,
the shadow of an adjunction. The same adjunction pattern, a right adjoint
\((-)^B\) to the product \(- \times B\) demanded in an arbitrary category with
finite products rather than found in \(\mathbf{Set}\), defines the
cartesian closed categories.
The limit functor itself
When a locally small category \(\mathscr{A}\) has all limits of a small shape
\(\mathbf{I}\), the
limit functor is right adjoint to the diagonal,
\(\Delta \dashv \lim\). The functor category \([\mathbf{I}, \mathscr{A}]\) is
locally small along with \(\mathscr{A}\), the shape being small. Feeding this
adjunction to the preservation theorem therefore shows that
\(\lim : [\mathbf{I}, \mathscr{A}] \to \mathscr{A}\) preserves limits. Since
limits in the functor category are computed pointwise,
unwinding the statement retraces the isomorphism between the two iterated limits
in the theorem that
limits commute with limits,
this time as a direct consequence of adjointness.
The adjunction itself needs only the limits of shape \(\mathbf{I}\). With them
alone, \(\lim\) preserves every limit that exists in
\([\mathbf{I}, \mathscr{A}]\), whatever its shape. Recovering the interchange
isomorphism in the form of the theorem uses the completeness hypotheses of that
theorem, all limits of both shapes. The double limit over the product shape, which
the interchange theorem places between the iterated ones, still needs the original
argument, so the adjunction gives a second road to the outer isomorphism rather
than a replacement for the theorem.
With that road in hand, a promise is kept. The page that first established the
diagonal adjunction recorded that the preservation theorem, applied to \(\lim\)
itself, would one day explain in one stroke why a limit of one shape can be
exchanged with a limit of another. This is that stroke.
A nonexistence proof
The contrapositive direction is best seen in an example already met when
adjunctions were introduced, now read through colimits. Write
\(\mathbf{Field}\) for the category whose objects are
fields,
required here to satisfy \(1 \neq 0\), and whose maps are
ring homomorphisms
sending \(1\) to \(1\). The forgetful functor
\(U : \mathbf{Field} \to \mathbf{Set}\) has no left adjoint, and the proof
runs through colimits.
Suppose some \(F : \mathbf{Set} \to \mathbf{Field}\) satisfied \(F \dashv U\). As
a left adjoint, \(F\) would preserve colimits. The empty set is an
initial object
of \(\mathbf{Set}\), the empty function into each set being the unique map out of
it, and an initial object is the colimit of the empty diagram, as
the tour of basic colimits
recorded. Preservation of colimits would therefore force \(F(\emptyset)\) to be an
initial object of \(\mathbf{Field}\).
No such object exists. An initial field \(K\) would admit a homomorphism into
every field, in particular into \(\mathbb{Z}_2\) and \(\mathbb{Z}_3\), the
fields of integers modulo a prime.
Consider \(\varphi : K \to \mathbb{Z}_2\) and the element
\(2 \cdot 1_K = 1_K + 1_K\). If it were nonzero it would be a unit of the field
\(K\), and a homomorphism preserving \(1\) carries units to units, since
\(\varphi(x)\varphi(x^{-1}) = \varphi(1) = 1\). But
\(\varphi(2 \cdot 1_K) = 2 \cdot 1_{\mathbb{Z}_2} = 0\), and \(0\) is a unit of
no field, since \(0 \cdot x = 0 \neq 1\) for every \(x\). Hence
\(2 \cdot 1_K = 0\). The same argument with \(\psi : K \to \mathbb{Z}_3\) gives
\(3 \cdot 1_K = 0\), and subtracting,
\[
1_K = 3 \cdot 1_K - 2 \cdot 1_K = 0 ,
\]
contradicting \(1 \neq 0\).
The obstruction is the familiar incompatibility of
characteristics,
which partitions the fields into islands no initial object could bridge. So
\(\mathbf{Field}\) has no initial object, no left adjoint to \(U\) can exist, and
a purely structural question, the existence of an adjoint, is settled by
elementary arithmetic in \(\mathbb{Z}_2\) and \(\mathbb{Z}_3\).
From Preservation to Existence
Every functor with a left adjoint preserves limits. The converse fails, and it
fails already in the smallest possible example. For any category \(\mathscr{B}\),
the unique functor \(\mathscr{B} \to \mathbf{1}\) into the one-object category
preserves limits, every cone in \(\mathbf{1}\) being a limit cone for want of
alternatives. A left adjoint \(F : \mathbf{1} \to \mathscr{B}\), on the other
hand, amounts to a choice of object \(B_0 = F(*)\) with
\(\mathscr{B}(B_0, B) \cong \mathbf{1}(*, *)\) a one-element set for every \(B\),
which says exactly that \(B_0\) is an
initial object.
A category without an initial object therefore supplies a limit-preserving functor
with no left adjoint.
Preservation is necessary but not sufficient, and the question becomes what must
be added to make it sufficient. The answers are the adjoint functor theorems, and
they all live over the same base, the
complete categories,
those with all limits.
The adjoint functor theorems share a single template. Let \(\mathscr{A}\) be a
category, \(\mathscr{B}\) a complete category, and
\(G : \mathscr{B} \to \mathscr{A}\) a functor satisfying certain further
conditions. Then \(G\) has a left adjoint if and only if \(G\) preserves limits.
The forwards implication costs nothing, being the
preservation theorem
of this page. All of the content sits in the backwards implication, and all of
the variety among the theorems sits in the further conditions, which invariably
police the boundary between small and large. Before facing that boundary, we
prove the theorem in a setting where it vanishes.
The theorem for ordered sets
An ordered set, that is, a poset, regarded as a
thin category with a
unique morphism \(B \to B'\) exactly when \(B \leq B'\), is a small category, and
we write it in bold accordingly. Limits in an ordered set are
meets.
In a thin category the commuting conditions on a cone are automatic, so a diagram
matters only through the family of objects in its image.
For a family \((B_i)_{i \in I}\) of elements, a cone with vertex \(B\) is nothing
but a lower bound \(B \leq B_i\) for all \(i\), and a limit cone is a greatest
lower bound, the meet \(\bigwedge_{i \in I} B_i\) extending the binary one. The
meet is unique when it exists, since two meets of the same family are lower bounds
of each other and \(\leq\) is antisymmetric.
An ordered set is therefore complete exactly when every family of its elements has
a meet, and a map \(G : \mathbf{B} \to \mathbf{A}\) between ordered sets is a
functor exactly when it is order-preserving, functoriality on the unique morphisms
being monotonicity.
Proposition: Adjoint Functor Theorem for Ordered Sets
Let \(\mathbf{A}\) be an ordered set, \(\mathbf{B}\) a complete ordered set,
and \(G : \mathbf{B} \to \mathbf{A}\) an order-preserving map. Then \(G\) has
a left adjoint if and only if \(G\) preserves meets, that is, if and only if
\[
G\Big(\bigwedge_{i \in I} B_i\Big) = \bigwedge_{i \in I} G(B_i)
\]
for every family \((B_i)_{i \in I}\) of elements of \(\mathbf{B}\), the
equation asserting in particular that the meet on the right exists.
Proof
If \(G\) has a left adjoint, then \(G\) preserves limits by the
preservation theorem.
A meet is the limit of the family regarded as a diagram over a discrete
shape, so \(G\) carries the limit cone with vertex
\(\bigwedge_i B_i\) to a limit cone with vertex
\(G\big(\bigwedge_i B_i\big)\) over the family \((G(B_i))_i\). The vertex of
a limit cone over a family in an ordered set is a meet of that family, and
meets are unique, so the displayed equation holds.
Now suppose \(G\) preserves meets. Since a
left adjoint exists precisely when every comma category has an initial object,
it suffices to fix \(A \in \mathbf{A}\) and produce an initial object of the
comma category
\((A \Rightarrow G)\). In the ordered setting the comma category simplifies
completely. An object of \((A \Rightarrow G)\) is an element
\(B \in \mathbf{B}\) together with a morphism \(A \to G(B)\), and a morphism
of \(\mathbf{A}\) carries no data beyond the relation it asserts, so
\((A \Rightarrow G)\) is the subset
\[
S = \{\, B \in \mathbf{B} \mid A \leq G(B) \,\}
\]
with the order inherited from \(\mathbf{B}\), the commuting condition on
morphisms of the comma category holding automatically in a thin category.
An initial object of an ordered set is a least element. Define
\[
B_0 = \bigwedge_{B \in S} B ,
\]
which exists because \(\mathbf{B}\) is complete. Every element of \(S\)
satisfies \(A \leq G(B)\), so \(A\) is a lower bound of the family
\((G(B))_{B \in S}\), vacuously so when \(S\) is empty. Since \(G\) preserves
meets, \(G(B_0)\) is the greatest lower bound of that family, and therefore
\(A \leq G(B_0)\). Hence \(B_0\) belongs to \(S\), and as the meet of \(S\) it
lies below every element of \(S\), so it is the least element of \(S\) and the
required initial object of \((A \Rightarrow G)\).
The proof does more than settle existence. It writes the left adjoint down. The
value \(F(A)\) is the initial object just constructed,
\[
F(A) = \bigwedge \{\, B \in \mathbf{B} \mid A \leq G(B) \,\} ,
\]
the least element of \(\mathbf{B}\) whose image dominates \(A\). Best
approximation from above, computed as a meet, is the whole content of
adjointness in the ordered world.
Two special cases repay attention. Taking \(\mathbf{A} = \mathbf{1}\), the unique
map \(\mathbf{B} \to \mathbf{1}\) preserves meets, so a complete ordered set has a
least element, namely \(\bigwedge_{B \in \mathbf{B}} B\). This is not quite a
triviality. Completeness asks for all meets, while a least element is an empty
join, and the proposition converts the one currency into the other.
The exchange is in fact total. For any subset \(T\) of a complete ordered set,
the meet of the set of upper bounds of \(T\) is itself an upper bound of \(T\).
Each element of \(T\) is a lower bound of the upper bounds, and the meet is
their greatest lower bound. The same meet lies below every upper bound, so it is
a least upper bound. A join is quite literally the meet of the upper bounds, and
an ordered set with all meets has all joins. Complete ordered sets are
cocomplete for free.
The general theorems
The proof for ordered sets suggests a general recipe. Given a complete category
\(\mathscr{B}\) and a limit-preserving \(G : \mathscr{B} \to \mathscr{A}\), one
would like to define
\[
F(A) = \lim_{(A \Rightarrow G)} P_A ,
\]
the limit of the functor \(P_A : (A \Rightarrow G) \to \mathscr{B}\) projecting
each object \((B, f)\) of the comma category to \(B\), the categorical analogue of
the meet over \(S\).
The obstruction is size. Completeness supplies limits over small categories
only, and the comma category \((A \Rightarrow G)\) is in general large, so the
limit above need neither exist nor, if it exists, be preserved by \(G\). Nor can
the problem be dodged by working with small categories throughout, since a
classical size argument, not reproduced here, forces a small category with all
small limits to be, up to equivalence, a complete ordered set, and the ordered
case is already settled. The further conditions of the adjoint functor theorems
exist to tame exactly this largeness, by arranging for the large limit to be
replaceable by a small one.
The general adjoint functor theorem does so with a smallness condition on the
comma categories. Call a set \(\mathcal{S}\) of objects of a category
weakly initial if every object of the category admits at least
one morphism from some member of \(\mathcal{S}\), a set-sized supply of starting
points with existence demanded and uniqueness dropped.
The theorem then states that if \(\mathscr{B}\) is complete and locally small and
each comma category \((A \Rightarrow G)\) has a weakly initial set, then \(G\) has
a left adjoint if and only if \(G\) preserves limits. We do not prove it here, and
on this page it plays the role of a landmark rather than a tool.
A typical application is worth tracing once. The category of groups is locally
small. It is also complete, and its forgetful functor to \(\mathbf{Set}\)
preserves limits, both facts from
the study of forgetful functors.
The weakly initial sets are supplied by a routine cardinality bound on how
large a group generated by a given set can be. The theorem produces a left
adjoint, the free group functor, and with it every free group, without a single
word or relation being written down. An identical scheme covers rings, modules,
and the other algebraic categories. The theorem thereby answers the question
left open earlier on this page, and it is the once-and-for-all argument
anticipated when the free constructions
opened the study of adjunctions.
The price is stated in the theorem's own terms. Adjointness describes the maps out
of \(F(A)\), while an element of \(F(A)\) is a map into it, so the theorem grants
existence while withholding any description of elements. Explicit presentations
must still be built by hand when they are wanted.
The special adjoint functor theorem trades breadth for cleanliness. Under stronger
structural hypotheses on the categories involved, it removes the condition on
weakly initial sets altogether, and its classical application belongs to topology.
Let \(\mathbf{CptHff}\) be the category of compact Hausdorff
topological spaces
and continuous maps, and let \(U : \mathbf{CptHff} \to \mathbf{Top}\) be the
inclusion of this full subcategory into all topological spaces. Together,
compactness
and the Hausdorff separation property carve out the best-behaved spaces of
classical analysis. The special theorem applies and yields a left adjoint \(F\),
which turns an arbitrary space canonically into a compact Hausdorff one. The value
\(F(X)\) is the Stone-Čech compactification of \(X\), an object whose
existence is far from obvious by bare hands, and the proof of the special theorem,
not given here, even pays out an explicit formula.
The space \(F(X)\) is the closure of the image of the canonical evaluation map
\[
X \longrightarrow [0,1]^{\mathbf{Top}(X,\,[0,1])} ,
\]
sending a point to its values under all continuous maps \(X \to [0,1]\), the
codomain being a power of the unit interval. When \(X\) satisfies mild separation
hypotheses, the unit of the adjunction embeds \(X\) into \(F(X)\) as a subspace,
so every sufficiently separated space sits densely inside a canonical compact
Hausdorff envelope.
The page closes where it began, on the two-way traffic between adjunctions and
limits. Possession of an adjoint forces preservation, always and everywhere.
Preservation forces possession only across the small-large boundary, at the price
of the size conditions that the adjoint functor theorems administer, and in
exchange it delivers free groups and compactifications without building either by
hand. Construction proves what it can build. Adjointness proves what must exist.