From Products and Equalizers to a Conjecture
The previous page made the limit operation into a functor and exhibited that functor as a
right adjoint. This page reverses the direction of the questions. Instead of asking what
structure the operation \(\lim\) itself carries, we ask how limits interact with the other
principal objects of this series: with the
hom-functors
\(\mathscr{A}(A, -)\), and with the
functor categories
in which diagrams live. Two theorems answer these questions. First, every hom-functor
preserves limits. Second, limits in a functor category exist whenever the target category
has them, and are computed pointwise. Each proof is a direct payoff of one of the two
central results of the previous page, and together the theorems prepare the study of
limits and colimits in
presheaf
categories that occupies the remainder of this stretch of the series.
Throughout, \(\mathbf{I}\) denotes a small category, \(\mathscr{A}\) a category, and
\(D : \mathbf{I} \to \mathscr{A}\) a diagram of shape \(\mathbf{I}\). When a limit of
\(D\) exists we write it \(\lim D\), with limit cone
\(\big(\lim D \xrightarrow{p_I} D(I)\big)_{I \in \mathbf{I}}\). Hypotheses of smallness
and
local smallness
are stated explicitly in each formal statement where they are needed.
Two computations with hom-sets
For the two computations of this subsection, let \(\mathscr{A}\) be a locally small
category with the relevant products or equalizers, so that each \(\mathscr{A}(A, X)\) is
a set and \(\mathscr{A}(A, -)\) is a genuine \(\mathbf{Set}\)-valued functor.
Recall what the definition of
product
says about maps into a product: composition with the two projections
\(p_1 : X \times Y \to X\) and \(p_2 : X \times Y \to Y\) puts maps \(A \to X \times Y\)
in bijection with pairs consisting of a map \(A \to X\) and a map \(A \to Y\). Collected
into a single statement about hom-sets, the bijection reads
\[
\mathscr{A}(A, X \times Y) \cong \mathscr{A}(A, X) \times \mathscr{A}(A, Y),
\]
the map from left to right being \(f \mapsto (p_1 \circ f, p_2 \circ f)\). The
right-hand side is itself a product, formed in \(\mathbf{Set}\). Applying the hom-functor
\(\mathscr{A}(A, -)\) to a product in \(\mathscr{A}\) has produced a product of hom-sets,
and the bijection between the two sides is composition with the images
\(\mathscr{A}(A, p_1)\) and \(\mathscr{A}(A, p_2)\) of the projections.
Is this a special feature of products? Let us try the same computation with an
equalizer.
Suppose \(\mathscr{A}\) has equalizers, and for a parallel pair \(s, t : X \to Y\) write
\(\operatorname{Eq}(s, t)\) for a chosen equalizer, with structure map
\(i : \operatorname{Eq}(s, t) \to X\). The universal property states that composition
with \(i\) puts maps \(A \to \operatorname{Eq}(s, t)\) in bijection with the maps
\(f : A \to X\) satisfying \(s \circ f = t \circ f\):
\[
\mathscr{A}\big(A, \operatorname{Eq}(s, t)\big)
\cong
\{ f \in \mathscr{A}(A, X) \mid s \circ f = t \circ f \}.
\]
Now rewrite the condition on the right in terms of hom-sets alone. The maps \(s\) and
\(t\) induce functions \(s_* = \mathscr{A}(A, s)\) and \(t_* = \mathscr{A}(A, t)\)
from \(\mathscr{A}(A, X)\) to \(\mathscr{A}(A, Y)\), the actions of the hom-functor
\(\mathscr{A}(A, -)\) on \(s\) and \(t\), and the condition
\(s \circ f = t \circ f\) reads \(s_*(f) = t_*(f)\). The set on the right of
the bijection is therefore the subset of \(\mathscr{A}(A, X)\) on which \(s_*\) and
\(t_*\) agree. By the explicit description of
equalizers in \(\mathbf{Set}\),
that subset, with its inclusion, is exactly an equalizer of \(s_*\) and \(t_*\). The
computation ends in the same shape as before:
\[
\mathscr{A}\big(A, \operatorname{Eq}(s, t)\big) \cong \operatorname{Eq}(s_*, t_*).
\]
Once again the hom-functor has converted a universal construction in \(\mathscr{A}\)
into the same universal construction in \(\mathbf{Set}\).
The conjecture
Products and equalizers are the two building blocks from which
all limits are assembled,
so the two bijections invite a single common generalization. Stating it
requires one piece of notation. Let \(\mathscr{A}\) be locally small, so that each
\(\mathscr{A}(A, X)\) is a set, and let \(D : \mathbf{I} \to \mathscr{A}\) be a diagram.
Write \(\mathscr{A}(A, D)\) for the composite functor
\[
\mathbf{I} \xrightarrow{D} \mathscr{A} \xrightarrow{\mathscr{A}(A, -)} \mathbf{Set},
\]
so that \(\mathscr{A}(A, D)\) sends an object \(I\) to the hom-set
\(\mathscr{A}(A, D(I))\) and a map \(u : I \to J\) to the function
\(\mathscr{A}(A, Du) = (Du) \circ -\). This is a diagram of shape \(\mathbf{I}\) in
\(\mathbf{Set}\), and since \(\mathbf{I}\) is small it has a limit, by the
explicit construction of limits in \(\mathbf{Set}\).
The conjecture suggested by the two computations is that whenever \(\lim D\) exists in
\(\mathscr{A}\),
\[
\mathscr{A}\big(A, \lim D\big) \cong \lim_{\mathbf{I}} \mathscr{A}(A, D).
\]
For \(\mathbf{I}\) the discrete category with two objects this is the product bijection,
and for \(\mathbf{I}\) the shape of a parallel pair it is the equalizer bijection.
The slogan form of the conjecture is that representables preserve limits, and the next
section proves it. The first step computes the right-hand side by itself, before any
limit of \(D\) enters the picture.
Representables Preserve Limits
The conjectured isomorphism has two halves, and the previous page already supplied one of
them. Choosing a limit cone identifies maps \(A \to \lim D\) with
cones on \(D\) with vertex \(A\).
The lemma below supplies the other half: cones with vertex \(A\) are, in turn, exactly the
elements of the limit of the hom-sets. The lemma needs no hypothesis on \(D\) whatsoever.
Whether or not \(D\) has a limit in \(\mathscr{A}\), the diagram \(\mathscr{A}(A, D)\) of
hom-sets always has a limit in \(\mathbf{Set}\), and that limit is the set of cones.
Lemma: The Set of Cones as a Limit
Let \(\mathbf{I}\) be a small category, \(\mathscr{A}\) a locally small category,
\(D : \mathbf{I} \to \mathscr{A}\) a diagram, and \(A \in \mathscr{A}\). Write
\(\operatorname{Cone}(A, D)\) for the set of
cones
on \(D\) with vertex \(A\), and for each \(I \in \mathbf{I}\) let
\[
e_I : \operatorname{Cone}(A, D) \to \mathscr{A}(A, D(I)),
\quad
(x_J)_{J \in \mathbf{I}} \mapsto x_I ,
\]
be evaluation of a cone at \(I\). Then:
(a) The family \((e_I)_{I \in \mathbf{I}}\) is a limit cone on the diagram
\(\mathscr{A}(A, D)\). In particular
\[
\operatorname{Cone}(A, D) \cong \lim_{\mathbf{I}} \mathscr{A}(A, D).
\]
(b) Each evaluation is natural in the vertex: for every map \(s : A' \to A\) in
\(\mathscr{A}\) and every \(I \in \mathbf{I}\), the square
\[
\begin{array}{ccc}
\operatorname{Cone}(A, D) & \xrightarrow{e_I} & \mathscr{A}(A, D(I)) \\\\
{\scriptstyle \operatorname{Cone}(s, D)}\big\downarrow & & \big\downarrow{\scriptstyle \mathscr{A}(s, D(I))} \\\\
\operatorname{Cone}(A', D) & \xrightarrow[e_I]{} & \mathscr{A}(A', D(I))
\end{array}
\]
commutes, where \(\operatorname{Cone}(s, D)\) composes every leg of a cone with \(s\)
and \(\mathscr{A}(s, D(I)) = - \circ s\) is the action of the
contravariant hom-functor
\(\mathscr{A}(-, D(I))\).
Proof
Since \(\mathbf{I}\) is small and \(\mathscr{A}\) is locally small, a cone with vertex
\(A\) is an element of the product of the sets \(\mathscr{A}(A, D(I))\) over the
objects of \(\mathbf{I}\), so \(\operatorname{Cone}(A, D)\) is a genuine set and the
maps \(e_I\) are well defined.
(a) First, \((e_I)_{I \in \mathbf{I}}\) is a cone on \(\mathscr{A}(A, D)\). For a map
\(u : I \to J\) in \(\mathbf{I}\) and a cone \(x = (x_K)_{K \in \mathbf{I}}\) with
vertex \(A\),
\[
\big(\mathscr{A}(A, Du)\big)\big(e_I(x)\big) = Du \circ x_I = x_J = e_J(x),
\]
the middle equality being the cone condition satisfied by \(x\). So
\(\mathscr{A}(A, Du) \circ e_I = e_J\), which is the cone condition for the family
\((e_I)\).
We now verify the universal property. Let
\(\big(S \xrightarrow{g_I} \mathscr{A}(A, D(I))\big)_{I \in \mathbf{I}}\) be any cone
on \(\mathscr{A}(A, D)\) in \(\mathbf{Set}\), so that
\(\mathscr{A}(A, Du) \circ g_I = g_J\) for every \(u : I \to J\). Evaluating this
equality of functions at an element \(w \in S\) gives
\[
Du \circ g_I(w) = g_J(w)
\quad \text{for every } u : I \to J \text{ in } \mathbf{I}.
\]
For fixed \(w\), the family \(\big(g_I(w)\big)_{I \in \mathbf{I}}\) therefore consists
of maps \(A \to D(I)\) satisfying exactly the cone condition, and so is an element of
\(\operatorname{Cone}(A, D)\). Define
\[
\bar{g} : S \to \operatorname{Cone}(A, D),
\quad
\bar{g}(w) = \big(g_I(w)\big)_{I \in \mathbf{I}} .
\]
By construction \(e_I \circ \bar{g} = g_I\) for every \(I\). Conversely, suppose
\(h : S \to \operatorname{Cone}(A, D)\) also satisfies \(e_I \circ h = g_I\) for every
\(I\). Then for each \(w \in S\) the \(I\)-th leg of the cone \(h(w)\) is \(g_I(w)\),
so \(h(w) = \bar{g}(w)\). Hence \(\bar{g}\) is the unique factorization of the cone
\((g_I)\) through \((e_I)\), and \(\big(\operatorname{Cone}(A, D), (e_I)\big)\) is a
limit cone. Any other limit of \(\mathscr{A}(A, D)\) is isomorphic to it, by the
uniqueness of limits.
The verification just given is the
explicit construction of limits in \(\mathbf{Set}\),
with the coordinates of a compatible family relabelled as the legs of a cone.
(b) Note first that \(\operatorname{Cone}(s, D)\) does carry cones to cones: if
\((x_J)\) satisfies the cone condition, then
\(Du \circ (x_I \circ s) = (Du \circ x_I) \circ s = x_J \circ s\), so
\((x_J \circ s)_{J \in \mathbf{I}}\) is a cone with vertex \(A'\). Both routes around
the square send a cone \(x = (x_J)_{J \in \mathbf{I}}\) to the same map. The route
along the top and down sends \(x \mapsto x_I \mapsto x_I \circ s\), and the route
down and along the bottom sends
\(x \mapsto (x_J \circ s)_{J \in \mathbf{I}} \mapsto x_I \circ s\). The square
commutes.
Part (b) says that the identification of part (a) respects change of vertex. The set of
cones, as the vertex varies, is the cone functor
\(\operatorname{Cone}(-, D) : \mathscr{A}^{\mathrm{op}} \to \mathbf{Set}\) of the previous
page, and the evaluations are natural transformations from it to the presheaves
\(\mathscr{A}(-, D(I))\). The lemma computes the limit of a diagram of hom-sets once for
every vertex simultaneously, in a way compatible with all restriction maps.
Assembling the two halves now proves the conjecture, in the exact cone-level sense demanded
by the definition of
preservation of limits.
One general observation makes the assembly frictionless. Let
\(\big(L \xrightarrow{q_I} E(I)\big)_{I \in \mathbf{I}}\) be a limit cone on a diagram
\(E\) in any category, and let \(\varphi : S \to L\) be an isomorphism. Then
\(\big(S \xrightarrow{q_I \circ \varphi} E(I)\big)_{I \in \mathbf{I}}\) is again a limit
cone. It is a cone, since the equations \(Eu \circ q_I = q_J\) persist after composing
both sides with \(\varphi\) on the right. And for any cone \((g_I)\) with vertex \(S'\), a
map \(k : S' \to L\) satisfies \(q_I \circ k = g_I\) for all \(I\) exactly when
\(m = \varphi^{-1} \circ k\) satisfies \((q_I \circ \varphi) \circ m = g_I\) for all
\(I\). Since \(k \mapsto \varphi^{-1} \circ k\) is a bijection between maps \(S' \to L\)
and maps \(S' \to S\), the unique factorization through \((q_I)\) supplies a unique
factorization through \((q_I \circ \varphi)\).
Proposition: Representables Preserve Limits
Let \(\mathscr{A}\) be a locally small category and \(A \in \mathscr{A}\). Then the
hom-functor
\(\mathscr{A}(A, -) : \mathscr{A} \to \mathbf{Set}\) preserves limits. Explicitly, for
every small category \(\mathbf{I}\), every diagram \(D : \mathbf{I} \to \mathscr{A}\),
and every limit cone \(\big(X \xrightarrow{p_I} D(I)\big)_{I \in \mathbf{I}}\) on
\(D\), the family
\[
\Big(\mathscr{A}(A, X) \xrightarrow{\mathscr{A}(A, p_I)} \mathscr{A}(A, D(I))\Big)_{I \in \mathbf{I}}
\]
is a limit cone on the diagram \(\mathscr{A}(A, D)\).
Proof
By the
representation of limits
and the isomorphism recorded with it, composition with the limit cone is a bijection
\[
\varphi : \mathscr{A}(A, X) \to \operatorname{Cone}(A, D),
\quad
\varphi(f) = (p_I \circ f)_{I \in \mathbf{I}},
\]
whose inverse sends a cone to its unique factorization through
\(\big(X \xrightarrow{p_I} D(I)\big)\). By the
lemma, the pair
\(\big(\operatorname{Cone}(A, D), (e_I)\big)\) is a limit cone on
\(\mathscr{A}(A, D)\). Now trace the two through one another. For
\(f \in \mathscr{A}(A, X)\),
\[
e_I\big(\varphi(f)\big) = p_I \circ f = \big(\mathscr{A}(A, p_I)\big)(f),
\]
so \(\mathscr{A}(A, p_I) = e_I \circ \varphi\) for every \(I \in \mathbf{I}\). The
displayed family is therefore obtained from the limit cone \((e_I)\) by composing with
the isomorphism \(\varphi\), and by the observation preceding the proposition it is a
limit cone.
In isomorphism form the proposition states that whenever \(\lim D\) exists,
\[
\mathscr{A}\big(A, \lim D\big) \cong \lim_{\mathbf{I}} \mathscr{A}(A, D),
\]
which is the conjecture of the opening section. The isomorphism is natural in \(A\).
Computing the right-hand side by the lemma identifies it, vertex by vertex and compatibly
with change of vertex by part (b), with the cone functor value
\(\operatorname{Cone}(A, D)\), and the representation isomorphism
\(\mathscr{A}(A, \lim D) \cong \operatorname{Cone}(A, D)\) is natural in \(A\), as recorded with
the representation of limits.
Applying the proposition in \(\mathscr{A}^{\mathrm{op}}\), which is locally small exactly
when \(\mathscr{A}\) is, yields the dual statement. The
covariant hom-functor of \(\mathscr{A}^{\mathrm{op}}\) based at \(A\) is the contravariant
hom-functor \(\mathscr{A}(-, A) : \mathscr{A}^{\mathrm{op}} \to \mathbf{Set}\) of
\(\mathscr{A}\), and a
colimit
of \(D\) is by definition a limit of \(D^{\mathrm{op}}\). The dual therefore reads:
\(\mathscr{A}(-, A)\) carries colimits in \(\mathscr{A}\) to limits in \(\mathbf{Set}\).
Whenever \(\operatorname{colim} D\) exists,
\[
\mathscr{A}\big(\operatorname{colim} D, A\big) \cong \lim_{\mathbf{I}^{\mathrm{op}}} \mathscr{A}(D, A),
\]
where \(\mathscr{A}(D, A)\) denotes the composite of \(D^{\mathrm{op}}\) with
\(\mathscr{A}(-, A)\). Look closely at the right-hand side: it is a limit, not a colimit.
Hom-functors of either variance produce limits in \(\mathbf{Set}\), and in this precise
sense limits hold the upper hand over colimits.
The smallest instance of the dual is already familiar. For a
sum
\(X + Y\), a map \(X + Y \to A\) amounts, by the defining universal property, to a pair
consisting of a map \(X \to A\) and a map \(Y \to A\):
\[
\mathscr{A}(X + Y, A) \cong \mathscr{A}(X, A) \times \mathscr{A}(Y, A).
\]
A colimit went in on the left, and a product came out on the right.
Limits in Functor Categories Are Computed Pointwise
The second theorem of the page moves from hom-sets to
functor categories,
and the cast changes with it. From now on \(\mathbf{A}\) denotes a small category,
\(\mathscr{S}\) a locally small category, and the objects of \(\mathbf{A}\) are written
\(A\). We study limits in the functor category \([\mathbf{A}, \mathscr{S}]\), whose
objects are functors \(X, Y : \mathbf{A} \to \mathscr{S}\) regarded as things in their own
right rather than as maps between categories. A diagram of shape \(\mathbf{I}\) here is a
functor \(D : \mathbf{I} \to [\mathbf{A}, \mathscr{S}]\), so each \(D(I)\) is itself a
functor \(\mathbf{A} \to \mathscr{S}\) and each \(Du\) is a natural transformation.
Presheaf categories, the destination of the pages just ahead, are the special case
\(\mathscr{S} = \mathbf{Set}\) with \(\mathbf{A}\) an opposite category.
Under the standing hypotheses \([\mathbf{A}, \mathscr{S}]\) is itself locally small. A
natural transformation
\(X \to Y\) assigns to each object \(A\) a component in the set
\(\mathscr{S}(X(A), Y(A))\), and since \(\mathbf{A}\) is small its objects form a set. The
transformations \(X \to Y\) therefore form a subclass of the product of the sets
\(\mathscr{S}(X(A), Y(A))\) over all \(A\), hence a set.
How should a limit of such a diagram look? A guess is suggested by the simplest case.
If \(\mathscr{S}\) has binary products, one expects the product of two functors
\(X, Y : \mathbf{A} \to \mathscr{S}\) to be the functor \(X \times Y\) with
\[
(X \times Y)(A) = X(A) \times Y(A),
\]
the product formed in \(\mathscr{S}\) separately at each object. The theorem below shows
that this guess is correct, that it is forced, and that nothing about it is special to
products. The statement needs one small piece of language for passing from the functor
category down to \(\mathscr{S}\).
Definition: Evaluation Functor
Let \(\mathbf{A}\) and \(\mathscr{S}\) be categories and \(A\) an object of
\(\mathbf{A}\). The evaluation functor at \(A\),
\[
\operatorname{ev}_A : [\mathbf{A}, \mathscr{S}] \to \mathscr{S},
\]
sends a functor \(X\) to its value \(X(A)\) and a natural transformation
\(\alpha : X \to Y\) to its component \(\alpha_A : X(A) \to Y(A)\). Functoriality is
immediate. Identities and composites in \([\mathbf{A}, \mathscr{S}]\) are computed
componentwise, so \(\operatorname{ev}_A\) sends \(1_X\) to \(1_{X(A)}\) and
\(\beta \circ \alpha\) to \(\beta_A \circ \alpha_A\).
For a diagram \(D : \mathbf{I} \to [\mathbf{A}, \mathscr{S}]\) and an object
\(A \in \mathbf{A}\), write
\[
D(-)(A) = \operatorname{ev}_A \circ D : \mathbf{I} \to \mathscr{S},
\]
the diagram in \(\mathscr{S}\) obtained by evaluating everything at \(A\). It sends \(I\)
to \(D(I)(A)\) and \(u : I \to J\) to the component \((Du)_A\). Likewise, the image under
\(\operatorname{ev}_A\) of a cone \(\big(X \xrightarrow{q_I} D(I)\big)_{I \in \mathbf{I}}\)
on \(D\) is the family \(\big(X(A) \xrightarrow{q_{I,A}} D(I)(A)\big)_{I \in \mathbf{I}}\),
a cone on \(D(-)(A)\): applying \(\operatorname{ev}_A\) to the equations
\(Du \circ q_I = q_J\) gives \((Du)_A \circ q_{I,A} = q_{J,A}\).
Theorem: Limits in Functor Categories Are Pointwise
Let \(\mathbf{A}\) and \(\mathbf{I}\) be small categories, \(\mathscr{S}\) a locally
small category, and \(D : \mathbf{I} \to [\mathbf{A}, \mathscr{S}]\) a diagram.
Suppose that for every \(A \in \mathbf{A}\) the diagram
\(D(-)(A) : \mathbf{I} \to \mathscr{S}\) has a limit in \(\mathscr{S}\). Then:
(a) There is a cone on \(D\) whose image under
\(\operatorname{ev}_A\)
is a
limit cone
on \(D(-)(A)\) for every \(A \in \mathbf{A}\).
(b) Every cone on \(D\) whose image under \(\operatorname{ev}_A\) is a limit cone on
\(D(-)(A)\) for every \(A \in \mathbf{A}\) is itself a limit cone on \(D\).
In particular, \(D\) has a limit in \([\mathbf{A}, \mathscr{S}]\).
Proof
(a) For each \(A \in \mathbf{A}\) choose a limit cone
\(\big(L(A) \xrightarrow{p_{I,A}} D(I)(A)\big)_{I \in \mathbf{I}}\) on \(D(-)(A)\).
We first make the assignment \(A \mapsto L(A)\) into a functor. Let \(f : A \to A'\)
be a map in \(\mathbf{A}\), and consider the family
\[
D(-)(f) = \big(D(I)(f)\big)_{I \in \mathbf{I}},
\quad
D(I)(f) : D(I)(A) \to D(I)(A').
\]
This family is a natural transformation \(D(-)(A) \to D(-)(A')\) between diagrams of
shape \(\mathbf{I}\). Indeed, for \(u : I \to J\) in \(\mathbf{I}\), the naturality
square of the transformation \(Du : D(I) \to D(J)\) at the map \(f\) reads
\[
(Du)_{A'} \circ D(I)(f) = D(J)(f) \circ (Du)_A ,
\]
which is exactly the required commuting square. By part (a) of the
induced map of limits,
there is therefore a unique map \(L(f) : L(A) \to L(A')\) satisfying the equations
\[
p_{I,A'} \circ L(f) = D(I)(f) \circ p_{I,A}
\quad \text{for every } I \in \mathbf{I}.
\]
Uniqueness forces functoriality. Since \(D(I)(1_A) = 1_{D(I)(A)}\), the identity
\(1_{L(A)}\) satisfies the defining equations of \(L(1_A)\), so
\(L(1_A) = 1_{L(A)}\). For composable maps \(f : A \to A'\) and \(g : A' \to A''\),
\[
\begin{align*}
p_{I,A''} \circ L(g) \circ L(f)
&= D(I)(g) \circ p_{I,A'} \circ L(f) \\\\
&= D(I)(g) \circ D(I)(f) \circ p_{I,A} \\\\
&= D(I)(g \circ f) \circ p_{I,A},
\end{align*}
\]
so \(L(g) \circ L(f)\) satisfies the defining equations of \(L(g \circ f)\), and
uniqueness gives \(L(g \circ f) = L(g) \circ L(f)\). Hence
\(L : \mathbf{A} \to \mathscr{S}\) is a functor.
Now read the defining equations of \(L(f)\) a second time. They say precisely that for
each \(I\) the family \(p_I = (p_{I,A})_{A \in \mathbf{A}}\) satisfies the naturality
square at every map \(f\), so \(p_I : L \to D(I)\) is a natural transformation, a map
in \([\mathbf{A}, \mathscr{S}]\). The family
\(\big(L \xrightarrow{p_I} D(I)\big)_{I \in \mathbf{I}}\) is moreover a cone on \(D\).
Equality of natural transformations is componentwise, and for every \(u : I \to J\)
and every \(A\),
\[
(Du)_A \circ p_{I,A} = p_{J,A}
\]
holds because \((p_{I,A})_{I \in \mathbf{I}}\) is a cone on \(D(-)(A)\). Hence
\(Du \circ p_I = p_J\) in \([\mathbf{A}, \mathscr{S}]\). The image of this cone under
\(\operatorname{ev}_A\) is the chosen limit cone at \(A\), which proves (a). The
functor structure was forced as well: any functor structure on \(A \mapsto L(A)\)
making every \(p_I\) natural satisfies the defining equations, hence coincides with
\(L\).
(b) Let \(\big(L \xrightarrow{p_I} D(I)\big)_{I \in \mathbf{I}}\) now be any cone on
\(D\) whose image under every \(\operatorname{ev}_A\) is a limit cone, and let
\(\big(X \xrightarrow{q_I} D(I)\big)_{I \in \mathbf{I}}\) be an arbitrary cone on
\(D\). Fix \(A \in \mathbf{A}\). The image of the second cone under
\(\operatorname{ev}_A\) is a cone on \(D(-)(A)\), so the limit cone
\(\big(L(A) \xrightarrow{p_{I,A}} D(I)(A)\big)_{I \in \mathbf{I}}\) supplies a unique
map \(\bar{q}_A : X(A) \to L(A)\) with
\[
p_{I,A} \circ \bar{q}_A = q_{I,A}
\quad \text{for every } I \in \mathbf{I}.
\]
It remains to show that the family \((\bar{q}_A)_{A \in \mathbf{A}}\) is natural, for then
\(\bar{q} : X \to L\) is a map in \([\mathbf{A}, \mathscr{S}]\).
Fix \(f : A \to A'\) in \(\mathbf{A}\). Naturality of each \(p_I\) at \(f\) states
that \(p_{I,A'} \circ L(f) = D(I)(f) \circ p_{I,A}\) for every \(I\), which is
exactly the characterization of the induced map of limits along the transformation
\(D(-)(f)\) between the limit cones at \(A\) and at \(A'\). Naturality of each
\(q_I\) at \(f\) states that
\[
D(I)(f) \circ q_{I,A} = q_{I,A'} \circ X(f)
\quad \text{for every } I \in \mathbf{I}.
\]
These are exactly the hypotheses of part (b) of the
induced map of limits,
applied to the transformation \(D(-)(f)\) (natural, as verified at the start of the
proof), the evaluated cones with vertices \(X(A)\) and \(X(A')\), and the
connecting map \(X(f)\). Its conclusion is
\[
L(f) \circ \bar{q}_A = \bar{q}_{A'} \circ X(f),
\]
the naturality square of \(\bar{q}\) at \(f\).
So \(\bar{q} : X \to L\) is a map in \([\mathbf{A}, \mathscr{S}]\), and the
componentwise identities \(p_{I,A} \circ \bar{q}_A = q_{I,A}\) assemble into
\(p_I \circ \bar{q} = q_I\) for every \(I \in \mathbf{I}\). Finally, if
\(r : X \to L\) also satisfies \(p_I \circ r = q_I\) for every \(I\), then at each
object \(A\) the component \(r_A\) satisfies \(p_{I,A} \circ r_A = q_{I,A}\) for
every \(I\), so \(r_A = \bar{q}_A\) by the uniqueness of the factorization at \(A\),
and \(r = \bar{q}\). Every cone on \(D\) therefore factors uniquely through
\(\big(L \xrightarrow{p_I} D(I)\big)_{I \in \mathbf{I}}\), which is a limit cone on
\(D\). Combining (a) and (b), \(D\) has a limit.
The theorem is summarized in a slogan: limits in a functor category are computed
pointwise. The points are the objects of \(\mathbf{A}\). To form a limit of a
diagram of functors, form the limit at each point separately, and the induced maps of
limits knit the pointwise vertices into a single functor. The guess about binary products
is redeemed: when \(\mathscr{S}\) has binary products, so does
\([\mathbf{A}, \mathscr{S}]\), and \((X \times Y)(A) = X(A) \times Y(A)\). The
theorem also dualizes. Colimits in a functor category are computed pointwise as well,
with cocones in place of cones and colimit cocones chosen at each object, and the dual
statement will be put to work when presheaf categories are treated in earnest.
Everything above required each pointwise diagram \(D(-)(A)\) to have a limit in
\(\mathscr{S}\). What the theorem yields when \(\mathscr{S}\) has all limits of the
relevant shape, and what can go wrong without that hypothesis, are the subject of the
closing section.
Evaluation Functors and a Warning
The pointwise theorem is sharpest as stated, cone by cone. Its most common use, however,
is a cruder consequence. When \(\mathscr{S}\) has all limits of a given shape, the
pointwise hypothesis holds automatically for every diagram of that shape, and evaluation
then interacts with limits as well as one could hope.
Corollary: Evaluation Functors Preserve Limits
Let \(\mathbf{A}\) and \(\mathbf{I}\) be small categories, and let \(\mathscr{S}\) be
a locally small category with all limits of shape \(\mathbf{I}\). Then the functor
category \([\mathbf{A}, \mathscr{S}]\) has all limits of shape \(\mathbf{I}\), and for
every \(A \in \mathbf{A}\) the
evaluation functor
\(\operatorname{ev}_A : [\mathbf{A}, \mathscr{S}] \to \mathscr{S}\)
preserves limits
of shape \(\mathbf{I}\).
Proof
Existence is immediate. For any diagram
\(D : \mathbf{I} \to [\mathbf{A}, \mathscr{S}]\), each pointwise diagram
\(D(-)(A)\) has a limit in \(\mathscr{S}\) by hypothesis, so \(D\) has a limit by the
pointwise theorem.
For preservation, let \(\big(L \xrightarrow{p_I} D(I)\big)_{I \in \mathbf{I}}\) be any
limit cone on such a diagram \(D\), and fix \(A \in \mathbf{A}\). By parts (a) and (b)
of the pointwise theorem, \(D\) also carries a limit cone
\(\big(L' \xrightarrow{p'_I} D(I)\big)_{I \in \mathbf{I}}\) whose image under every
evaluation functor is a limit cone. The
uniqueness of limits
supplies a unique isomorphism \(\theta : L \to L'\) with \(p'_I \circ \theta = p_I\)
for all \(I \in \mathbf{I}\). Applying \(\operatorname{ev}_A\) to these equations
gives \(p'_{I,A} \circ \theta_A = p_{I,A}\) for all \(I\), and \(\theta_A\) is an
isomorphism in \(\mathscr{S}\), with inverse the component of \(\theta^{-1}\) at
\(A\), since \(\operatorname{ev}_A\) is a functor. The image of the primed cone under
\(\operatorname{ev}_A\) is a limit cone on \(D(-)(A)\), and the image of the given
cone is obtained from it by composing with the isomorphism \(\theta_A\). By the
observation on composing limit cones with isomorphisms, recorded before the
preservation proposition, that image is a
limit cone on \(D(-)(A)\).
The proof used the completeness of \(\mathscr{S}\) only to guarantee, for the one diagram
at hand, that its pointwise diagrams have limits. So the argument establishes a
diagram-by-diagram refinement: whenever every pointwise diagram \(D(-)(A)\) of a single
diagram \(D\) has a limit in \(\mathscr{S}\), every limit cone on \(D\) evaluates to a
limit cone at every object. Under the pointwise hypothesis there are no exotic limits in
the functor category. The colimit counterparts of the corollary and of this refinement hold by duality.
The completeness hypothesis is not decorative. If \(\mathscr{S}\) lacks limits of shape
\(\mathbf{I}\), the functor category \([\mathbf{A}, \mathscr{S}]\) can still contain
diagrams of shape \(\mathbf{I}\) that happen to have limits. Such accidental limits
need not be computed pointwise: some evaluation functor can fail to send the limit
cone to a limit cone. The refinement above constrains the failure sharply. It requires a
diagram \(D\) whose limit exists in \([\mathbf{A}, \mathscr{S}]\) while some pointwise
diagram \(D(-)(A)\) has no limit in \(\mathscr{S}\) at all. At such an object \(A\) the
evaluated cone lies on a diagram with no limit whatsoever, so it cannot be a limit cone.
Explicit examples of this phenomenon exist, but their construction takes machinery beyond
the scope of this series, and we record the warning without one.
Both theorems of this page feed forward. The preservation of limits by hom-functors is
the germ of a far stronger statement about adjoint functors. The pointwise computation
of limits is the engine behind the completeness of presheaf categories, the
setting in which the Yoneda machinery of the earlier pages does its deepest work.
Pointwise Constructions and Structured Feature Spaces
The previous page closed by noting that the categorical deep learning programme
proposes to specify architectures by functors, natural transformations, and universal
properties. It promised that preservation theorems would turn such specifications
into statements about what a constrained model can and cannot represent. The first
two preservation theorems are now on the table. In this reading, a functor
\(X : \mathbf{A} \to \mathscr{S}\) is a structured family of feature spaces, one for
each object of \(\mathbf{A}\), and a natural transformation is a map of families that
respects the structure by construction. The pointwise theorem then says that
universal ways of combining such families, products, equalizers, and general limits,
can be carried out one object at a time. The induced maps then assemble the local
constructions into a coherent global whole, with the coherence supplied by uniqueness
rather than by design.
The preservation of limits by representables complements this from the side of
observation. Probing a family by maps out of a fixed object converts universal
constructions in the model category into universal constructions on plain sets, so
that what holds of the probes holds of the structure. The warning tempers both
readings. The guarantees carry hypotheses, and when the ambient category lacks the
needed limits, pointwise reasoning about structured families can silently break. A
specification language built on universal properties inherits both the theorems and
their fine print.