Limits as Adjoints and as Representables

Three formalisms express the same universal property: adjunctions, representable functors, and limits. Anything sayable in one is sayable in the others. Rewriting the definition of a limit as a representation, and again as an adjoint, recovers uniqueness, functoriality, and the dual statement for colimits from general facts about representations and adjoints.¹

A cone is a natural transformation whose source is constant.

The diagonal functor

Fix a small category $I$ (the shape) and a category $A$ . For each object $A \in A$ there is a functor $Δ A : I \to A$ that is constant: it sends every object of $I$ to $A$ and every map to $1_{A}$ . Sending $A$ to this constant diagram, and a map $g : A \to A^{'}$ to the natural transformation with every component equal to $g$ , defines a functor into the functor category.

The name comes from the simplest shape. When $I$ is the discrete category on two objects, $[I, A] = A \times A$ and $Δ (A) = (A, A)$ , the diagonal of the product. For a general shape, $Δ A$ is the constant diagram at $A$ .

The diagonal functor

Δ

carries an object

A

to the diagram that is constant at

A

, with every internal map the identity.

Cones as natural transformations

A cone on a diagram $D : I \to A$ with vertex $A$ is a family of maps $f_{I} : A \to D (I)$ , one for each object $I \in I$ , compatible with the maps of the diagram: $D (u) \circ f_{I} = f_{J}$ for every $u : I \to J$ in $I$ . Compare this to the naturality condition for a transformation $Δ A \Rightarrow D$ . A component at $I$ is a map $(Δ A) (I) = A \to D (I)$ , and naturality at $u : I \to J$ is exactly $D (u) \circ f_{I} = f_{J} \circ 1_{A} = f_{J}$ . The two conditions coincide.

A cone with vertex

A

(left) is precisely a natural transformation from the constant diagram

Δ A

D

(right); the leg conditions are the naturality squares.

Because $Cone (A, D) = [I, A] (Δ A, D)$ , the set of cones is functorial in both arguments: contravariantly in the vertex $A$ (precompose a cone with a map $A^{'} \to A$ ) and covariantly in the diagram $D$ (postcompose along a map $D \to D^{'}$ ). Fixing $D$ and letting the vertex vary gives the functor of interest,

Cone (-, D) : A^{op} ⟶ Set .

Limits as representations

A limit cone on $D$ is a cone through which every other cone factors uniquely. That universal property is just the condition for a representation of the cone functor: a universal element of $Cone (-, D)$ .²

Unwinding the representation gives a natural isomorphism. If $D$ has a limit, then

Cone (A, D) ≅ A (A, lim D) naturally in A .

Left to right, a cone $(f_{I})_{I}$ maps to its unique mediating map $m : A \to lim D$ through the limit cone; right to left, a map $g : A \to lim D$ maps to the cone $(p_{I} \circ g)_{I}$ , where $p_{I} : lim D \to D (I)$ are the limit projections. The two assignments are mutually inverse by the universal property.

The representation of the cone functor: cones with vertex

A

correspond to maps

A \to lim D

, recovered by composing with the projections.

Two corollaries drop out. First, a representing object is unique up to isomorphism, so limits are too.

Second, a terminal object is the limit of the empty diagram and a product is the limit over a discrete shape, so both inherit the representation description without separate argument.

Functoriality of the limit

The representation isomorphism suggests varying $D$ as well as $A$ . A map of diagrams $α : D \to D^{'}$ (a natural transformation, with components $α_{I} : D (I) \to D^{'} (I)$ ) should induce a map on limits, and it does.³

The induced map on limits is the unique arrow making every projection square commute; here the map of diagrams

α : D \Rightarrow D^{'}

is written

t

, the second diagram

D^{'}

is written

E

, and its projections are

q_{I}

The construction respects identities and composition, so $lim$ is a functor.

Limits as a right adjoint

Suppose $A$ has all limits of shape $I$ . Choose a limit cone for each diagram $D$ and write its vertex $lim D$ . By the lemma, each map $α : D \to D^{'}$ induces $lim α$ , and these choices assemble into a functor

lim : [I, A] ⟶ A .

The representation isomorphism now reads as an adjunction. On the one hand, $[I, A] (Δ A, D) = Cone (A, D)$ ; on the other, $Cone (A, D) ≅ A (A, lim D)$ naturally in $A$ . Taking the diagram argument into account (via the second part of the lemma) makes the isomorphism natural in $D$ as well.⁴

The choice of limit cones is non-canonical, but by uniqueness of adjoints a different choice changes $lim$ only up to natural isomorphism.

Dualizing reverses every arrow. A cocone on $D$ is a natural transformation $D \Rightarrow Δ A$ , a colimit is a representation of the cocone functor, and when all colimits of shape $I$ exist the assignment $colim : [I, A] \to A$ is left adjoint to the diagonal.

The diagonal has $colim$ as a left adjoint and $lim$ as a right adjoint: $colim ⊣ Δ ⊣ lim$ .

When a category has all limits and colimits of shape

I

, the diagonal functor

Δ

sits between two adjoints:

colim ⊣ Δ ⊣ lim

Completeness in adjoint form

A category is complete when it has all small limits. The adjoint rephrasing turns completeness into a statement about the diagonal functor.

Formalism	A limit of $D$ is	Existence of all limits of shape $I$ means
Explicit	a universal cone on $D$	every diagram has a universal cone
Representable	a representation of $Cone (-, D)$	$Cone (-, D)$ is representable for every $D$
Adjoint	the value $lim D$ of the right adjoint to $Δ$	$Δ : A \to [I, A]$ has a right adjoint

The third row reduces has all limits of shape $I$ to the diagonal functor has a right adjoint, which lets general theorems about adjoints act on limits. Applying the fact that right adjoints preserve limits to $lim$ itself shows that limits commute with limits, and the question of when a limit-preserving functor is a right adjoint is answered by the adjoint functor theorem.

Leinster, Basic Category Theory, Ch. 6 opening — the three formalisms (adjointness, representability, limits) as three coordinate systems for universal properties, with the chapter devoted to translating between them. ↩
Leinster, Basic Category Theory, §6.1, Proposition 6.1.1 — a limit of $D$ is a representation of $Cone (-, D)$ , proved by identifying a representation with a universal cone via the corollary on representations as universal elements. ↩ ↩²
Leinster, Basic Category Theory, §6.1, Lemma 6.1.3 — a map of diagrams induces a unique map of limits commuting with the projections, and the induced square on factorizations commutes. ↩
Leinster, Basic Category Theory, §6.1, Proposition 6.1.4 — when all limits of shape $I$ exist, $lim$ is a functor and is right adjoint to $Δ$ ; the dual gives $colim ⊣ Δ$ . ↩

The diagonal functor

Cones as natural transformations

Limits as representations

Functoriality of the limit

Limits as a right adjoint

Completeness in adjoint form

Footnotes