Colimits: Coproducts, Coequalizers, Pushouts

Every concept of the theory of limits has a mirror image obtained by reversing all arrows, and the duality principle says the mirrored theory comes for free. A limit gathers maps into a universal object; a colimit gathers maps out of one. The constructions dual to products, equalizers, and pullbacks are coproducts, coequalizers, and pushouts, and where limits build objects as subobjects of products (solution sets, fibered products), colimits build them as quotients of sums (disjoint unions, gluings, identifications). Whenever a construction takes given objects and produces a new object receiving maps from them, a colimit is the likely formalism; the lowest common multiple in the divisibility order and the gluing of coordinate patches into a manifold are both examples.¹

Cocones and the initial cocone

Unwound into $A$ itself, a cocone is an object $A$ (the vertex) with arrows $f_{I} : D (I) \to A$ such that $f_{J} \circ D (u) = f_{I}$ for every edge $u : I \to J$ , and a colimit is a cocone $(D (I) p_{I} C)_{I}$ through which every cocone factors by a unique mediating arrow $\overset{ˉ}{f} : C \to A$ with $\overset{ˉ}{f} \circ p_{I} = f_{I}$ .² Where the limit was the terminal cone, the colimit is the initial cocone. Everything proved for limits dualizes on the spot: colimits are unique up to a unique compatible isomorphism, the coprojections are collectively epic, and a map out of $colim D$ is the same thing as a cocone on $D$ :

A (colim D, A) ≅ {cocones on D with vertex A}, natural in A .

Maps out of a colimit are easy to describe; that asymmetry is why colimit-style presentations dominate in geometry, where one mostly needs to construct maps off a space.

A cocone under a diagram: the legs point from the nodes up to the vertex, and the colimit C is the cocone every other cocone factors through.

Naming conventions for duals are irregular: sometimes a co- is added or removed (limit/colimit, product/coproduct), sometimes the pair is initial/terminal or pullback/pushout. The table keeps the pairs straight.

Limit-side	Shape	Colimit-side
terminal object $1$	$\emptyset$	initial object $0$
product $\prod_{i} X_{i}$	discrete	coproduct (sum) $∐_{i} X_{i}$
equalizer	$∙ ⇉ ∙$	coequalizer
pullback $X \times_{Z} Y$	corner	pushout $X +_{Z} Y$
inverse limit of a chain	$(N, \leq)^{op}$	direct limit of a chain

Coproducts

In $Set$ the coproduct is the disjoint union: the injections have disjoint images covering $X_{1} + X_{2}$ , so any element lies in exactly one image and $\overset{ˉ}{f}$ is defined case by case, $\overset{ˉ}{f} (p_{1} (x_{1})) = f_{1} (x_{1})$ and $\overset{ˉ}{f} (p_{2} (x_{2})) = f_{2} (x_{2})$ .³ Elsewhere the coproduct can look quite different from a disjoint union:

$Vect_{k}$ . The direct sum $X_{1} \oplus X_{2}$ with $i_{1} (x) = (x, 0)$ , $i_{2} (y) = (0, y)$ is the coproduct, the same object that serves as the product. Finite products and coproducts of vector spaces coincide, a degeneracy special to additive settings and false in $Set$ .⁴
Ordered sets. In a poset viewed as a category, the coproduct of a family is its least upper bound (join) $⋁_{i} x_{i}$ ; the initial object is a least element. In $(P (S), \subseteq)$ the join is union; in $(N, ∣)$ it is the lowest common multiple, and the least element is $1$ .
$Grp$ . The coproduct is the free product $G * H$ , generated by words alternating between the two groups; nothing like the underlying disjoint union.

The underlying set of $G * H$ , or of $X_{1} \oplus X_{2}$ , is not the disjoint union of the underlying sets; forgetful functors to $Set$ tend to preserve limits but not colimits.

Coequalizers

Every coequalizer is epic, dually to every equalizer is monic. The coequalizer is the categorical quotient, and in $Set$ it is a literal one. Given $s, t : X \to Y$ , let $\sim$ be the equivalence relation on $Y$ generated by $s (x) \sim t (x)$ for all $x \in X$ — the smallest equivalence relation containing the relation $R = {(s (x), t (x)) ∣ x \in X}$ , obtained by symmetrizing $R$ and chaining zigzags. The quotient map

p : Y ⟶ Y / \sim

is the coequalizer: maps out of $Y / \sim$ correspond exactly to maps $h$ out of $Y$ with $h (s (x)) = h (t (x))$ for all $x$ , which is the universal property.⁵ The pair $(s, t)$ lists the identifications to impose, and the coequalizer imposes them and nothing more.

A quotient as a coequalizer in Set: the pair s, t lists which elements of Y to identify, and C is Y with exactly those identifications.

In algebraic categories the same idea runs through the relevant quotient construction. For $s, t : A \to B$ in $Ab$ , the difference $t - s$ is a homomorphism, and the coequalizer is the canonical quotient

B ⟶ B / im (t - s),

mirroring the equalizer $ker (t - s)$ on the limit side.⁶ In $Grp$ one quotients by the normal subgroup generated by the elements $s (a) t (a)^{- 1}$ ; in $Top$ one takes the $Set$ coequalizer with the quotient topology. Coequalizers also expose the asymmetry of epic: in $Ring$ the inclusion $Z ↪ Q$ is epic without being surjective, so epics, unlike coequalizers of concrete quotients, need not be onto.⁷

Pushouts

The pushout square: the span through X is glued into P, and any other commutative square on the span factors through P uniquely.

The pushout glues $Y$ and $Z$ along the common part $X$ . The $Set$ formula makes the gluing literal: $Y +_{X} Z = (Y + Z) / \sim$ where $\sim$ is generated by $s (x) \sim t (x)$ , i.e. take the disjoint union, then identify the two images of each point of $X$ .⁸ Two special cases organize the picture.

Union along an intersection. For subsets $Y, Z \subseteq A$ , the square of inclusions with $Y \cap Z$ in the top corner and $Y \cup Z$ in the bottom is a pushout in $Set$ ; the formula places $Y$ and $Z$ side by side and glues the copy of $Y \cap Z$ in each. (The same square is also a pullback, a coincidence special to sets.)
Coproducts as pushouts. If the category has an initial object $0$ , the pushout of the unique span $Y \leftarrow 0 \to Z$ recovers the coproduct $Y + Z$ : gluing along nothing is placing side by side.

Attaching a cell to a space, gluing two spaces along a common subspace, and assembling surfaces from patches are all pushouts in $Top$ . The van Kampen theorem has exactly this shape: under suitable hypotheses, the fundamental-group functor sends a pushout square of spaces to a pushout square of groups.⁹

Colimits in Set and the two-sided picture

The general colimit formula in $Set$ dualizes the limit-as-subset-of-a-product formula. For any diagram $D : I \to Set$ ,

colim D = (I \in I ∐ D (I)) / \sim,

where $\sim$ is the equivalence relation generated by $x \sim (D u) (x)$ for every edge $u : I \to J$ and every $x \in D (I)$ . A limit in $Set$ is a subset of a product; a colimit is a quotient of a sum.¹⁰

A sphere admits both presentations (Leinster's example). As a limit-style object, the sphere is an equalizer: $S^{2} ↪ R^{3} ⇉ R$ , where the parallel maps are $(x, y, z) \mapsto x^{2} + y^{2} + z^{2}$ and the constant $1$ — an equation captured by an equalizer, at the cost of a choice of coordinates. As a colimit-style object, the sphere is a coequalizer: two open disks glued along a cylindrical belt,

S^{1} \times (0, 1) ⇉ D + D ⟶ S^{2},

the two parallel maps being the inclusions of the belt into each disk — a gluing captured by a coequalizer, at the cost of a choice of decomposition. An atlas presents a manifold as a colimit of Euclidean balls, and the colimit view dominates modern geometry because maps out of a gluing amount to compatible families of maps on the patches.¹¹

One sphere, two presentations: cut out of a product by an equation (limit style), or glued from two disks along a belt (colimit style).

Directed and confluent colimits

Colimits of chains, and more generally of directed diagrams, were studied under the name direct limits before category theory supplied their general definition.

For a diagram $A$ over a confluent poset in $Set$ — write $x ∣_{j} \in A (j)$ for the image of $x \in A (i)$ under the connecting map $A (i) \to A (j)$ , $i \leq j$ — the generated equivalence relation collapses to a single condition:

(x, i) \sim (y, j) ⟺ \exists k \geq i, j : x ∣_{k} = y ∣_{k} .

Two tagged elements are identified precisely when they eventually agree. Confluence is what makes this relation transitive without any zigzag closure: given agreement at $l$ and at $m$ , the nodes $l, m$ share the lower bound $j$ , so confluence supplies $n \geq l, m$ where both agreements combine. The colimit is the set of eventual-agreement classes, with coprojections $x \mapsto [x, i]$ .¹³ For an increasing chain of sets $X_{0} \subseteq X_{1} \subseteq \dots$ with inclusion maps, the direct limit is just the union $⋃_{n} X_{n}$ : every element is eventually present, and eventual agreement is equality. Dually, the inverse limit of a decreasing chain is the intersection.

The same eventual-agreement construction computes directed colimits in $Grp$ , $Ring$ , and the other algebraic categories, because the operations of any finite set of classes can be evaluated at a common upper-bound stage. This finite-data property is the seed of the theory of filtered colimits and finitely presentable objects; here it is enough to know that directed colimits in $Set$ exist and have elements that are elements at some stage, up to eventual agreement.

Which categories have all colimits? $Set$ , $Grp$ , $Vect_{k}$ , $Top$ all do (cocomplete is the term dual to complete), and the generation theorem dualizes: coproducts plus coequalizers give all colimits. But computing a colimit in an algebraic category is generally harder than computing a limit, precisely because the forgetful functor no longer does the work. Both computations, worked out underneath a forgetful functor, appear in computing limits in concrete categories.

Leinster, Basic Category Theory, Ch. 5 opening and §5.2 — colimits as the ubiquitous dual, with lcm-in-divisibility as a first example. ↩
Leinster, Basic Category Theory, §5.2, Definition 5.2.1 — cocone and colimit by dualization, with the explicit unwound form and coprojections. ↩
Leinster, Basic Category Theory, §5.2, Definition 5.2.2 and Example 5.2.4 — sums as colimits over discrete shapes and the disjoint-union verification in $Set$ . ↩
Leinster, Basic Category Theory, §5.2, Example 5.2.5 — the direct sum of vector spaces is both product and coproduct. ↩
Leinster, Basic Category Theory, §5.2, Remarks 5.2.8 and Example 5.2.9 — the equivalence relation generated by a relation, and the quotient map as the coequalizer in $Set$ ; Simmons §2.6, Examples 2.6.6–2.6.7. ↩
Leinster, Basic Category Theory, §5.2, Example 5.2.10 — coequalizers in $Ab$ via $B / im (t - s)$ ; the $Grp$ case is Simmons, Exercise 2.6.3(b). ↩
Leinster, Basic Category Theory, §5.2, Definition 5.2.17 and Example 5.2.19 — epics as dual monics; $Z ↪ Q$ epic in $Ring$ . ↩
Leinster, Basic Category Theory, §5.2, Definition 5.2.11 and Example 5.2.12 — pushouts, the $(Y + Z) / \sim$ formula, the union-along-intersection square, and coproducts as pushouts from an initial object (Example 5.2.13). ↩
Leinster, Basic Category Theory, §5.2, Example 5.2.14 — the van Kampen theorem as a pushout-preservation statement. ↩
Leinster, Basic Category Theory, §5.2, Example 5.2.16 — the colimit of any small diagram in $Set$ as a quotient of the sum, verified through the cocone correspondence. ↩
Leinster, Basic Category Theory, §5.2, Figure 5.2 and surrounding discussion — the sphere as an equalizer in coordinates and as a coequalizer of glued disks, and why maps out of colimits favour the gluing view. ↩
Simmons, An Introduction to Category Theory, §4.7, Definition 4.7.1 — directed and confluent posets. ↩
Simmons, An Introduction to Category Theory, §4.7 — the tagged disjoint union, the eventual-agreement relation, its transitivity via confluence, and the verification that the quotient is the colimit. ↩