Cones and Limits

Products, terminal objects, and pullbacks each attach a new object to a configuration of old ones, and the universal property has the same shape each time: a best object equipped with maps to the given data. The limit is the single construction that contains all of them. Wherever a method takes some objects and arrows and builds a new object with maps out to the originals, that method is almost always a limit; the dual method, building an object with maps in, is a colimit.¹ Kernels in group theory, intersections of subsets, inverse limits of chains, and greatest common divisors in the divisibility order are all limits of one shape or another.

Diagrams and shape categories

A commutative square, a parallel pair of arrows, a descending chain of objects — each of these configurations inside $A$ is a functor from a small category that fixes its shape.

A functor sends each node $I$ to an object $D (I)$ and each edge $u : I \to J$ to an arrow $D (u) : D (I) \to D (J)$ , respecting composition. So a diagram is exactly a labelling of the shape by objects and arrows of $A$ that commutes wherever the shape does. The three shapes that generate the classical constructions are tiny.²

$T$ — two objects, no non-identity arrows (the discrete category on two points). A diagram is a bare pair $(X, Y)$ ; its limit is a product.
$E$ — two objects with two parallel arrows $∙ ⇉ ∙$ . A diagram is a parallel pair $X ⇉ Y$ ; its limit is an equalizer.
$P$ — three objects in a corner, $∙ \to ∙ \leftarrow ∙$ . A diagram is a cospan $X \to Z \leftarrow Y$ ; its limit is a pullback.

A shape category (left) instantiated as a diagram (right): each node becomes an object, each edge an arrow, and the corner shape P produces a cospan.

Simmons calls the shape a template and the functor its instantiation; the two descriptions agree once any commuting conditions on the template are read as functoriality of $D$ .³ When the shape has no non-identity arrows the distinction between a directed graph and a category disappears, which is why products need only an index set, not a full category.

Cones

The name cone is geometric — a vertex sits above the plane of the diagram and its legs slope down to every node, as in the figure below.

For the product shape $T$ there are no edges and no condition, so a cone is any pair of arrows $A \to X$ , $A \to Y$ ; for the equalizer shape a cone is a fork; for the pullback shape a cone is a commutative square. Each edge $u : I \to J$ costs one equation and determines the leg $f_{J}$ from the leg $f_{I}$ , which is why forks and squares carry fewer free legs than their shapes have nodes.

A cone with vertex A over a three-node diagram. Every leg agrees with the diagram's arrows, so the two slanted triangles commute.

Cones on a fixed $D$ form a category. An arrow from a cone $(A, (f_{I}))$ to a cone $(A^{'}, (f_{I}^{'}))$ is a map $g : A \to A^{'}$ commuting with all the legs, $f_{I}^{'} \circ g = f_{I}$ . Identities and composites are inherited from $A$ , so this is the category of cones on $D$ , written $Cone (D)$ . The limit is a distinguished object of it.

The universal cone

Equivalently, a limit of $D$ is a terminal object of $Cone (D)$ .⁴ The word limit here is the order-theoretic one, an extremal cone, not the limiting process of analysis; the two senses coincide only for chains indexed by the natural numbers.

The limit cone (thick) is universal: any cone with vertex A factors through it by a unique mediating arrow, drawn dashed.

Any arrow $g : A \to L$ composes with the projections to give a cone $(p_{I} \circ g)_{I}$ with vertex $A$ , and the limit condition says that this assignment is a bijection:

A (A, lim D) ≅ {cones on D with vertex A}, natural in A .

A map into $lim D$ is the same thing as a cone on $D$ . The adjoint and representable descriptions of limits build on this bijection, and it makes the uniqueness of limits automatic.

Products and terminal objects as limits

On the two smallest shapes the definition reproduces products and terminal objects.

More generally, let $I$ be any set and $I$ the discrete category on it. A diagram $I \to A$ is an $I$ -indexed family $(X_{i})_{i \in I}$ , and its limit is the product $\prod_{i \in I} X_{i}$ . Shrinking the index set to the extreme gives the terminal object.

The product

P = X \times Y

is the limit over the discrete two-object shape: with no edges to satisfy, a cone is just a pair of legs.

The terminal object is the product of no objects, which explains the shared notation $1$ : it is the empty product just as the number $1$ is the empty product of integers. The small shapes and their limits, in one table:

Shape $I$	Diagram	Cone	Limit
$\emptyset$ (empty)	nothing	an object	terminal object $1$
$T$ (two points)	pair $(X, Y)$	two legs	product $X \times Y$
discrete on $I$	family $(X_{i})$	family of legs	product $\prod_{i} X_{i}$
$E$ ( $∙ ⇉ ∙$ )	parallel pair	fork	equalizer
$P$ ( $∙ \to ∙ \leftarrow ∙$ )	cospan	commutative square	pullback
$(N, \leq)^{op}$	descending chain	compatible family	inverse limit

Equalizers and pullbacks occupy the two middle rows. The last row, an inverse limit, is the case that makes contact with the analytic meaning of limit: the limit of a decreasing chain of subsets is their intersection.

Uniqueness up to isomorphism

A limit is defined by a universal property, so like every universal construction it is determined up to a unique compatible isomorphism.

The projections need not be monic individually, but they are jointly monic: no two maps into $L$ can agree on all projections without being equal.

Two limit cones L and M each mediate the other; the round trips are identities by collective monicity, so the mediators are inverse isomorphisms.

Because the isomorphism is unique and compatible with the projections, it is harmless to speak of the limit and to name it $lim D$ . The same argument, run in the opposite category, gives uniqueness of colimits.

Existence

Limits do not always exist. In the discrete two-object category $T$ itself, viewed as its own ambient category, the two objects have no cone over them at all, so no product.⁷ Existence is a property of the ambient category and its shapes.

The categories $Set$ , $Grp$ , $Ring$ , $Vect_{k}$ , and $Top$ are all complete, and their limits can be written down explicitly; the construction appears in computing limits in concrete categories. A single structural fact reduces every limit to two basic kinds: a category with all products and all equalizers has all limits, and a category with binary products, a terminal object, and equalizers has all finite limits. The reason is already visible in $Set$ , where any limit is carved out of a product by a system of equations, and equations are equalizers.

Leinster, Basic Category Theory, Ch. 5 opening — limits and colimits as the third route to universal properties, with the kernel and lowest-common-multiple examples framing the chapter. ↩
Leinster, Basic Category Theory, §5.1 — the categories $T$ , $E$ , $P$ of Diagram (5.14) and Definition 5.1.18 of a diagram of shape $I$ . ↩
Simmons, An Introduction to Category Theory, §4.1 — templates and their instantiation as diagrams; §4.2 recasts a template as an indexing category and a diagram as a functor. ↩
Barr & Wells, Category Theory for Computing Science, §9.2 — the general concept of limit as a terminal object in the category of commutative cones over a diagram. ↩
Simmons, An Introduction to Category Theory, §4.4, Lemma 4.4.2 — a limit cone is collection-wise monic, and its Corollary 4.4.3. ↩
Simmons, An Introduction to Category Theory, §4.4, Theorem 4.4.4 — essential uniqueness of limits via the mediator argument; Leinster, §5.1, Remark 5.1.20 and Lemma 2.1.8. ↩
Leinster, Basic Category Theory, §5.1, Remarks 5.1.2 and Definition 5.1.25 — non-existence of some products, and the definitions of having limits of a shape, completeness, and finite completeness; Proposition 5.1.26 reduces all limits to products and equalizers. ↩