The Diagonal Functor and Cones as Natural Transformations
Across this series we have described universal properties in three different languages. An
adjunction
expresses a universal property as an isomorphism of hom-sets between two functors. A
representable functor
expresses it as an isomorphism between an abstract functor and a hom-functor. A
limit
expresses it as a universal cone. Each language can in principle say what the others say, but
each says certain things more gracefully.
This page carries out the translation for limits. We rephrase the definition of
limit first in terms of representability, then in terms of adjointness. Both rephrasings will
pay off immediately in the pages that follow, where they drive the main theorems about how
limits interact with functors and with adjunctions.
The key to both rephrasings is the observation that a cone is itself a natural transformation
of a special kind. Making this precise requires one new functor. Throughout, \(\mathbf{I}\) is
a small category and \(\mathscr{A}\) a category, and we write
\([\mathbf{I}, \mathscr{A}]\)
for the category of functors \(\mathbf{I} \to \mathscr{A}\) and natural transformations
between them, so that a diagram \(D : \mathbf{I} \to \mathscr{A}\) of shape \(\mathbf{I}\) is
exactly an object of \([\mathbf{I}, \mathscr{A}]\).
Definition: Diagonal Functor
Let \(\mathbf{I}\) be a small category and \(\mathscr{A}\) a category. The
diagonal functor
\[
\Delta : \mathscr{A} \to [\mathbf{I}, \mathscr{A}]
\]
sends an object \(A \in \mathscr{A}\) to the constant functor \(\Delta A\), which takes
the value \(A\) on every object of \(\mathbf{I}\) and the value \(1_A\) on every map of
\(\mathbf{I}\). It sends a map \(f : A \to A'\) in \(\mathscr{A}\) to the natural
transformation \(\Delta f : \Delta A \to \Delta A'\) whose every component is \(f\).
Two small verifications keep the definition honest. First, \(\Delta A\) is a functor. It
preserves identities by construction, and for composable maps \(u, v\) in \(\mathbf{I}\) it
sends \(v \circ u\) to \(1_A = 1_A \circ 1_A\). Second, \(\Delta f\) is natural. For any map
\(u : I \to J\) in \(\mathbf{I}\), the
naturality square
demands \(f \circ (\Delta A)(u) = (\Delta A')(u) \circ f\), which reads
\(f \circ 1_A = 1_{A'} \circ f\) and holds trivially. Functoriality of \(\Delta\) itself is
equally direct, since components compose pointwise.
The name is explained by the smallest interesting case. Let \(\mathbf{I}\) be the discrete
category with two objects and only identity maps. A functor \(\mathbf{I} \to \mathscr{A}\) is
then just a pair of objects, a natural transformation is just a pair of maps, and so
\([\mathbf{I}, \mathscr{A}]\) is the
product category
\(\mathscr{A} \times \mathscr{A}\). Under this identification,
\[
\Delta A = (A, A),
\]
the diagonal embedding of \(\mathscr{A}\) into \(\mathscr{A} \times \mathscr{A}\).
Cones as Natural Transformations
Now let \(D : \mathbf{I} \to \mathscr{A}\) be a diagram and \(A \in \mathscr{A}\) an object.
A natural transformation \(\alpha : \Delta A \to D\) assigns to each object
\(I \in \mathbf{I}\) a component \(\alpha_I : A \to D(I)\), one map out of \(A\) for each
vertex of the diagram. Its naturality square at a map \(u : I \to J\) of \(\mathbf{I}\)
demands
\[
Du \circ \alpha_I = \alpha_J \circ (\Delta A)(u) = \alpha_J \circ 1_A = \alpha_J .
\]
The condition \(Du \circ \alpha_I = \alpha_J\) is word for word the compatibility condition
defining a
cone
on \(D\) with vertex \(A\). A cone, then, is not merely analogous to a natural
transformation. It is one, with the constant functor as its domain. Writing
\(\operatorname{Cone}(A, D)\) for the set of cones on \(D\) with vertex \(A\), we have
established the identification
\[
\operatorname{Cone}(A, D) = [\mathbf{I}, \mathscr{A}](\Delta A, D),
\]
an equality of sets. The cones on \(D\) with vertex \(A\) are exactly the maps
\(\Delta A \to D\) in the functor category.
The right-hand side is a hom-set, and hom-sets are functorial in each variable. The set
\(\operatorname{Cone}(A, D)\) therefore varies contravariantly in \(A\). A map
\(s : A' \to A\) in \(\mathscr{A}\) induces \(\Delta s : \Delta A' \to \Delta A\), and
precomposition with \(\Delta s\) carries a cone with vertex \(A\) to a cone with vertex
\(A'\). Concretely, it composes every leg of the cone with \(s\). The set likewise varies
covariantly in \(D\), by postcomposition. The contravariant dependence on the vertex is the
one we exploit next. Fixing \(D\) and letting the vertex vary produces a functor
\[
\operatorname{Cone}(-, D) : \mathscr{A}^{\mathrm{op}} \to \mathbf{Set},
\]
a presheaf on \(\mathscr{A}\), and asking whether this presheaf is representable will turn
out to be asking whether \(D\) has a limit.
The two-object discrete case again makes everything concrete. There a diagram \(D\) is a pair
of objects \((X, Y)\), and a natural transformation \(\Delta A \to D\) is a pair of maps
\((A \to X, A \to Y)\): precisely the data of a cone on the pair, whose universal instance
is the
product
\(X \times Y\) with its two projections. Every statement of this page can profitably be read
twice, first in general and then specialized to this case. Here the limit is a binary
product and \(\operatorname{Cone}(A, D) = \mathscr{A}(A, X) \times \mathscr{A}(A, Y)\).
Limits as Representations
The functor \(\operatorname{Cone}(-, D)\) is a presheaf, and the fundamental question one asks
of a presheaf is whether it is
representable.
The first rephrasing of the definition of limit is that this question has the answer we would
hope for. Representing \(\operatorname{Cone}(-, D)\) is the same thing as finding a limit of
\(D\). Here the work invested in the Yoneda lemma and its consequences is repaid in full. The
proof below is a single application of the correspondence between representations and
universal elements, with no new computation at all.
Proposition: Limits are Representations
Let \(\mathbf{I}\) be a small category, \(\mathscr{A}\) a
locally small
category, and \(D : \mathbf{I} \to \mathscr{A}\) a diagram. Then there is a one-to-one
correspondence between limit cones on \(D\) and representations of the functor
\[
\operatorname{Cone}(-, D) : \mathscr{A}^{\mathrm{op}} \to \mathbf{Set},
\]
under which the representing objects are exactly the limit objects of \(D\), the vertices
of the limit cones.
Proof
Since \(\mathbf{I}\) is small and \(\mathscr{A}\) is locally small, each
\(\operatorname{Cone}(A, D)\) is a subset of the product of the hom-sets
\(\mathscr{A}(A, D(I))\) over the objects of \(\mathbf{I}\), hence a genuine set, and the
displayed presheaf is well defined. By the correspondence between
representations and universal elements,
to give a representation of \(\operatorname{Cone}(-, D)\) is to give an object
\(L \in \mathscr{A}\) together with a universal element
\(u \in \operatorname{Cone}(L, D)\). Such an element is a cone
\(\big(L \xrightarrow{u_I} D(I)\big)_{I \in \mathbf{I}}\) on \(D\) with the following
property: for every \(A \in \mathscr{A}\) and every \(x \in \operatorname{Cone}(A, D)\)
there is a unique map \(\bar{x} : A \to L\) with
\[
\big(\operatorname{Cone}(\bar{x}, D)\big)(u) = x .
\]
By the description of functoriality in the previous section,
\(\operatorname{Cone}(\bar{x}, D)\) composes every leg of a cone with \(\bar{x}\), so this
condition says that for every cone \(\big(A \xrightarrow{x_I} D(I)\big)_{I \in \mathbf{I}}\)
on \(D\) there is a unique map \(\bar{x} : A \to L\) with \(u_I \circ \bar{x} = x_I\) for
all \(I \in \mathbf{I}\). This is precisely the universal property defining a
limit
cone. Universal elements of \(\operatorname{Cone}(-, D)\) are therefore exactly limit
cones on \(D\), and the correspondence matches representations with limit cones, carrying
the representing object to the vertex.
Briefly put, a limit of \(D\) is a representation of
\([\mathbf{I}, \mathscr{A}](\Delta -, D)\). In particular, when \(D\) has a limit, choosing a
limit cone produces an isomorphism
\[
\operatorname{Cone}(A, D) \cong \mathscr{A}(A, \lim D),
\]
natural in \(A \in \mathscr{A}\). Maps into the limit are the same thing as cones. From right
to left the correspondence sends \(g : A \to \lim D\) to the cone \((p_I \circ g)_{I \in \mathbf{I}}\)
obtained by composing with the projections, and from left to right it sends a cone
\((x_I)_{I \in \mathbf{I}}\) to the unique factorization \(\bar{x}\) supplied by the universal
property. We will upgrade this isomorphism to one natural in \(D\) as well, once we have made
\(\lim D\) functorial in \(D\), the task of the next two sections.
The rephrasing immediately yields a second route to the
uniqueness of limits,
proved earlier by a direct cone-chasing argument. Any two limit objects of \(D\)
represent the same presheaf \(\operatorname{Cone}(-, D)\), and any two representing objects of
a single presheaf are isomorphic by the
isomorphism of representables.
The earlier proposition is sharper, since it also makes the isomorphism unique and compatible
with the projections. On the shared core, that limit objects are isomorphic, the elementary
route and the Yoneda route agree, and this agreement is a first sign that the translation
between the formalisms is faithful.
Maps between Diagrams
The identification \(\operatorname{Cone}(A, D) = [\mathbf{I}, \mathscr{A}](\Delta A, D)\)
invites a change of perspective. Since diagrams are objects of a category, we may vary the
diagram as well as the vertex. Given a map of diagrams, that is, a natural transformation
\(\alpha : D \to D'\), it is natural to ask whether it induces a map
\(\lim D \to \lim D'\) between their limits, and whether the induced map interacts correctly
with the factorizations supplied by the universal property. The following lemma answers both
questions, and it is the engine behind everything in the next section. Part (a) will make
\(\lim\) into a functor, and part (b) is exactly the naturality computation that the adjunction
will require.
Lemma: Induced Map of Limits
Let \(\mathbf{I}\) be a small category, \(\mathscr{A}\) a category, and
\(\alpha : D \to D'\) a natural transformation between diagrams
\(D, D' : \mathbf{I} \to \mathscr{A}\). Suppose that
\(\big(\lim D \xrightarrow{p_I} D(I)\big)_{I \in \mathbf{I}}\) and
\(\big(\lim D' \xrightarrow{p'_I} D'(I)\big)_{I \in \mathbf{I}}\) are limit cones.
Then:
(a) There is a unique map \(\lim \alpha : \lim D \to \lim D'\) such that the square
\[
\begin{array}{ccc}
\lim D & \xrightarrow{p_I} & D(I) \\\\
{\scriptstyle \lim \alpha}\big\downarrow & & \big\downarrow{\scriptstyle \alpha_I} \\\\
\lim D' & \xrightarrow[p'_I]{} & D'(I)
\end{array}
\]
commutes for every \(I \in \mathbf{I}\).
(b) Given cones \(\big(A \xrightarrow{f_I} D(I)\big)_{I \in \mathbf{I}}\) on \(D\) and
\(\big(A' \xrightarrow{f'_I} D'(I)\big)_{I \in \mathbf{I}}\) on \(D'\), and a map
\(s : A \to A'\) such that the square
\[
\begin{array}{ccc}
A & \xrightarrow{f_I} & D(I) \\\\
{\scriptstyle s}\big\downarrow & & \big\downarrow{\scriptstyle \alpha_I} \\\\
A' & \xrightarrow[f'_I]{} & D'(I)
\end{array}
\]
commutes for every \(I \in \mathbf{I}\), the square
\[
\begin{array}{ccc}
A & \xrightarrow{\bar{f}} & \lim D \\\\
{\scriptstyle s}\big\downarrow & & \big\downarrow{\scriptstyle \lim \alpha} \\\\
A' & \xrightarrow[\bar{f}']{} & \lim D'
\end{array}
\]
also commutes, where \(\bar{f}\) and \(\bar{f}'\) are the unique factorizations of the two
cones through the respective limit cones.
Proof
(a) We first check that the family
\(\big(\lim D \xrightarrow{\alpha_I \circ p_I} D'(I)\big)_{I \in \mathbf{I}}\) is a
cone on \(D'\). For any map \(u : I \to J\) in \(\mathbf{I}\),
\[
\begin{align*}
D'u \circ \alpha_I \circ p_I
&= \alpha_J \circ Du \circ p_I \\\\
&= \alpha_J \circ p_J ,
\end{align*}
\]
the first equality by naturality of \(\alpha\) at \(u\) and the second by the cone
condition for the limit cone on \(D\). Since
\(\big(\lim D' \xrightarrow{p'_I} D'(I)\big)\) is a limit cone, there is a unique map
\(\lim \alpha : \lim D \to \lim D'\) with
\(p'_I \circ \lim \alpha = \alpha_I \circ p_I\) for all \(I \in \mathbf{I}\), which is
exactly the asserted commuting square.
(b) We claim first that maps into a limit are determined by their composites with the
projections. If \(g, h : B \to \lim D'\) satisfy \(p'_I \circ g = p'_I \circ h\) for all
\(I \in \mathbf{I}\), then \(g = h\). Indeed, the family \((p'_I \circ g)_{I \in \mathbf{I}}\)
is a cone on \(D'\), since for \(u : I \to J\) the cone condition for \((p'_I)\) gives
\(D'u \circ p'_I \circ g = p'_J \circ g\). The
uniqueness clause in the definition of
limit
then allows only one map \(B \to \lim D'\) whose composites with the projections realize
this cone. Both \(g\) and \(h\) do, so \(g = h\).
Now compute, for each \(I \in \mathbf{I}\),
\[
\begin{align*}
p'_I \circ (\lim \alpha) \circ \bar{f}
&= \alpha_I \circ p_I \circ \bar{f} \\\\
&= \alpha_I \circ f_I \\\\
&= f'_I \circ s \\\\
&= p'_I \circ \bar{f}' \circ s ,
\end{align*}
\]
the four equalities holding, in order, by the defining square of \(\lim \alpha\) from
part (a), the factorization property of \(\bar{f}\), the hypothesis square, and the
factorization property of \(\bar{f}'\). The maps \((\lim \alpha) \circ \bar{f}\) and
\(\bar{f}' \circ s\) from \(A\) to \(\lim D'\) thus have equal composites with every
projection, so by the previous paragraph they are equal.
Part (a) should be read as the statement that limits are functorial in the diagram.
Part (b) says that the factorization \(\bar{f}\), the very bijection underlying
\(\operatorname{Cone}(A, D) \cong \mathscr{A}(A, \lim D)\), is compatible with maps of
diagrams on one side and maps of vertices on the other. In the two-object discrete case the
lemma is a familiar fact about
products.
A pair of maps \(X \to X'\), \(Y \to Y'\) induces the map
\(X \times Y \to X' \times Y'\) acting coordinatewise, and it commutes with pairing. The next
section assembles these observations into the second rephrasing of the definition of limit.
The Limit Functor as a Right Adjoint
Everything is now in place for the second rephrasing. Suppose \(\mathscr{A}\) has
all limits of shape \(\mathbf{I}\), so that every diagram \(D \in [\mathbf{I}, \mathscr{A}]\)
admits a limit cone. The previous section produced, from any map of diagrams, a map between
chosen limits. The section before that produced, for each fixed diagram, an isomorphism
between cones and maps into the limit. Assembled, these two statements say that
\(\lim\) is a functor and that it participates in an
adjunction
with the diagonal functor, the climax of the page.
Proposition: The Limit Functor is Right Adjoint to the Diagonal
Let \(\mathbf{I}\) be a small category and \(\mathscr{A}\) a locally small category with
all limits of shape \(\mathbf{I}\). Then a choice of limit cone for each diagram makes the
assignment \(D \mapsto \lim D\) into a functor
\[
\lim : [\mathbf{I}, \mathscr{A}] \to \mathscr{A},
\]
and this functor is right adjoint to the diagonal functor:
\(\Delta \dashv \lim\), with adjunction bijection
\[
[\mathbf{I}, \mathscr{A}](\Delta A, D) \cong \mathscr{A}(A, \lim D),
\]
natural in \(A \in \mathscr{A}\) and \(D \in [\mathbf{I}, \mathscr{A}]\).
Proof
Choose for each diagram \(D\) a limit cone
\(\big(\lim D \xrightarrow{p^D_I} D(I)\big)_{I \in \mathbf{I}}\), and for a map
\(\alpha : D \to D'\) of diagrams let
\(\lim \alpha : \lim D \to \lim D'\) be the induced map of the
previous lemma,
the unique map satisfying \(p^{D'}_I \circ \lim \alpha = \alpha_I \circ p^D_I\) for all
\(I\).
Functoriality. Both checks are applications of that uniqueness. For the identity
\(1_D\), the map \(1_{\lim D}\) satisfies
\(p^D_I \circ 1_{\lim D} = p^D_I = (1_D)_I \circ p^D_I\), so
\(\lim 1_D = 1_{\lim D}\). For composable maps
\(\alpha : D \to D'\) and \(\beta : D' \to D''\), we compute
\[
\begin{align*}
p^{D''}_I \circ (\lim \beta) \circ (\lim \alpha)
&= \beta_I \circ p^{D'}_I \circ (\lim \alpha) \\\\
&= \beta_I \circ \alpha_I \circ p^D_I \\\\
&= (\beta \circ \alpha)_I \circ p^D_I ,
\end{align*}
\]
so \((\lim \beta) \circ (\lim \alpha)\) satisfies the property that characterizes
\(\lim(\beta \circ \alpha)\) uniquely, and the two are equal.
The bijection, natural in \(A\). Both sides of the displayed bijection are sets:
the right side because \(\mathscr{A}\) is locally small, the left side because it equals
\(\operatorname{Cone}(A, D)\), shown to be a set in the proof that
limits are representations.
That proposition, applied to the chosen limit cone on \(D\), gives the bijection
\(\Phi_{A,D} : [\mathbf{I}, \mathscr{A}](\Delta A, D) \to \mathscr{A}(A, \lim D)\) sending
a cone \(f = (f_I)_{I \in \mathbf{I}}\) to its unique factorization \(\bar{f}\) through the
limit cone, and this bijection is natural in \(A\).
Naturality in \(D\). Let \(\alpha : D \to D'\) be a map of diagrams. We must show
that \(\Phi_{A,D'}(\alpha \circ f) = (\lim \alpha) \circ \Phi_{A,D}(f)\) for every cone
\(f : \Delta A \to D\), that is,
\[
\overline{\alpha \circ f} = (\lim \alpha) \circ \bar{f} .
\]
The composite \(\alpha \circ f : \Delta A \to D'\) is a cone on \(D'\) with vertex \(A\)
and legs \(\alpha_I \circ f_I\). Apply part (b) of the previous lemma with
\(A' = A\), \(s = 1_A\), \(f'_I = \alpha_I \circ f_I\). The hypothesis square
\(\alpha_I \circ f_I = f'_I \circ 1_A\) holds by construction, so the conclusion gives
\((\lim \alpha) \circ \bar{f} = \overline{\alpha \circ f} \circ 1_A = \overline{\alpha \circ f}\),
as required. The bijection is therefore natural in both variables, and by the definition
of adjunction, \(\Delta \dashv \lim\).
A limit, in summary, is three things at once: a universal cone, a representation of
\(\operatorname{Cone}(-, D)\), and the value at \(D\) of a right adjoint of the diagonal
functor. Dually, the same translations exhibit the
colimit
of a diagram
as a representation of the covariant cocone functor and \(\operatorname{colim}\) as a
left adjoint of the diagonal, \(\operatorname{colim} \dashv \Delta\), by running
every argument of this page in \(\mathscr{A}^{\mathrm{op}}\).
A Remark on Choice
To define the functor \(\lim\) we had to choose, for every diagram at once, one
limit cone among all of them. The step is genuinely non-canonical, since nothing in the
universal property singles out a preferred cone. The damage this choice can do is, however,
strictly bounded. At the level of a single diagram, any two choices of vertex are both representing
objects of the presheaf \(\operatorname{Cone}(-, D)\), hence isomorphic by the
isomorphism of representables.
At the level of functors, any two versions of \(\lim\) built from different systems of choices
are both right adjoint to the same functor \(\Delta\). Adjoints on a given side are
unique up to natural isomorphism. This is the uniqueness established through the Yoneda
machinery in
the consequences of the Yoneda lemma. The
functor \(\lim\) is thus canonical exactly to the degree that category theory measures
canonicity: not as a single object, but as an object determined up to coherent isomorphism.
Why the Adjunction Formulation Earns Its Keep
The rephrasings of this page are investments. Once \(\lim\) is known to be a right
adjoint, general facts about adjoints apply to limits wholesale. In the pages ahead we
will prove that every functor with a left adjoint preserves limits, and applying that
theorem to \(\lim\) itself will explain, in one stroke, why limits commute with limits.
A limit of products, for instance, is a product of limits. The representable
formulation will work just as hard, driving the result that hom-functors and, more
broadly, presheaf categories treat limits as transparently as sets do.
The same shapes recur in the categorical view of machine learning. Naturality
squares of the kind manipulated throughout this page are the formal template for
equivariance constraints on network layers, and the categorical deep learning programme
proposes to specify architectures by exactly such structure. A specification by
functors, natural transformations, and universal properties turns statements like the
preservation theorems ahead into statements about what a constrained network can and
cannot represent. The diagonal-limit adjunction is a first, fully worked instance of the
pattern that programme builds on.