Limits and Colimits — Mathematics Knowledge System

Overview

Limits and colimits are universal constructions that unify many familiar mathematical objects: products, coproducts, pullbacks, pushouts, equalizers, coequalizers, kernels, and direct/inverse limits all arise as special cases. A limit is the 'most economical' way to map into a diagram; a colimit is the 'most economical' way to map out of one. The duality between them — obtained by reversing all arrows — is one of the deepest symmetries in mathematics.

Intuition

A limit of a diagram is like a 'compatible snapshot': a new object that simultaneously sees all objects in the diagram and all relations between them, in the most efficient way possible. A colimit is dual: an object that is simultaneously seen by all objects in the diagram. Concretely, products are limits of discrete diagrams; coproducts are colimits. Pullbacks are limits of 'cospan' diagrams; pushouts are colimits of 'span' diagrams.

Formal Definition

Definition

Let $J$ be a small category (the 'index category') and $D: J \to \mathcal{C}$ a functor (a 'diagram of shape $J$'). A cone over $D$ is an object $L \in \mathcal{C}$ together with morphisms $\lambda_j: L \to D(j)$ for each $j \in J$, such that for each morphism $u: j \to k$ in $J$, we have $D(u) \circ \lambda_j = \lambda_k$. A limit of $D$ is a terminal cone — a cone $(\lim D, \{\pi_j\})$ such that every other cone factors uniquely through it:

\text{Cone}(C, D) = \left\{ (\lambda_j)_{j \in J} : \lambda_j: C \to D(j),\; D(u) \circ \lambda_j = \lambda_k \text{ for all } u: j \to k \right\}

cone-definition

\text{Hom}_{\mathcal{C}}(C, \lim D) \cong \text{Cone}(C, D) \quad \text{naturally in } C

Universal property of the limit

limit-universal-property

\text{Colim}: \quad \text{Hom}_{\mathcal{C}}(\text{colim}\, D, C) \cong \text{Cone}(D, C) \quad \text{naturally in } C

Universal property of the colimit (cocones mapping out)

colimit-universal-property

Notation

Notation	Meaning
$\lim_{j \in J} D(j)$	Limit of diagram D: J → C
$\text{colim}_{j \in J} D(j)$	Colimit of diagram D: J → C
$\pi_j: \lim D \to D(j)$	j-th projection morphism of the limit (limit leg)
$\iota_j: D(j) \to \text{colim}\, D$	j-th inclusion morphism of the colimit (colimit leg)
$A \times_C B$	Pullback (fibered product) of A and B over C
$A \sqcup_C B$	Pushout (amalgamated sum) of A and B under C

Properties

Products and coproducts

The product A \times B is the limit of the discrete diagram \{A, B\}. The coproduct A \sqcup B is the colimit.

Example: In $\text{Set}$: product = Cartesian product, coproduct = disjoint union. In $\text{Ab}$: product = direct product, coproduct = direct sum.

Equalizers and coequalizers

The equalizer of f, g: A \rightrightarrows B is the limit of the diagram A \rightrightarrows B: an object E with e: E \to A such that f \circ e = g \circ e, universal among such. The coequalizer is the colimit.

Example: In $\text{Set}$, the equalizer is $\{a \in A \mid f(a) = g(a)\}$. The coequalizer is $B / {\sim}$ where $f(a) \sim g(a)$.

Pullbacks

The pullback of f: A \to C and g: B \to C is the limit of the cospan A \xrightarrow{f} C \xleftarrow{g} B: an object A \times_C B = \{(a,b) \mid f(a)=g(b)\}.

Example: In $\text{Set}$: $A \times_C B = \{(a,b) \in A \times B \mid f(a) = g(b)\}$. In algebraic geometry, the fiber product is a pullback.

Pushouts

The pushout of f: C \to A and g: C \to B is the colimit of the span A \xleftarrow{f} C \xrightarrow{g} B: an object A \sqcup_C B.

Example: In $\text{Grp}$, the pushout of $G \leftarrow H \rightarrow K$ is the amalgamated free product $G *_H K$. In topology, the pushout of two spaces glued along a subspace gives the union.

Worked Examples

Define $P = A \times_C B = \{(a, b) \in A \times B \mid f(a) = g(b)\}$ with projections $p_1: P \to A$, $p_2: P \to B$ given by $p_1(a,b)=a$, $p_2(a,b)=b$.
$P = \{(a,b) \in A \times B \mid f(a) = g(b)\}$
The square commutes: $f \circ p_1(a,b) = f(a) = g(b) = g \circ p_2(a,b)$.
$\begin{array}{ccc} P & \xrightarrow{p_2} & B \\ \downarrow_{p_1} & & \downarrow_{g} \\ A & \xrightarrow{f} & C \end{array}$
Universal property: Suppose $Q$ has maps $q_1: Q \to A$ and $q_2: Q \to B$ with $f \circ q_1 = g \circ q_2$. Define $u: Q \to P$ by $u(x) = (q_1(x), q_2(x))$. We must check $u(x) \in P$: $f(q_1(x)) = g(q_2(x))$ — yes, by assumption.
$u(x) = (q_1(x), q_2(x)) \in A \times_C B$
Uniqueness: If $v: Q \to P$ also satisfies $p_1 \circ v = q_1$ and $p_2 \circ v = q_2$, then $v(x) = (p_1(v(x)), p_2(v(x))) = (q_1(x), q_2(x)) = u(x)$. So $u$ is unique.
Hence $P = A \times_C B$ with the projections is a pullback in $\text{Set}$.

Answer: The pullback in $\text{Set}$ is $\{(a,b) \in A \times B \mid f(a)=g(b)\}$ with coordinate projections. It satisfies the universal property: any compatible pair of maps into $A$ and $B$ factors uniquely through it.

Practice Problems

Difficulty 7/10

Describe the equalizer of two group homomorphisms $f, g: G \to H$ in $\text{Grp}$. What is it as a subgroup of $G$?

Difficulty 9/10

Prove that limits in a category $\mathcal{C}$ are unique up to unique isomorphism.

Difficulty 8/10

In topology, the pushout of $A \xleftarrow{f} C \xrightarrow{g} B$ is the space $A \sqcup_C B = (A \sqcup B) / {\sim}$ where $f(c) \sim g(c)$. Compute the pushout when $C = \{0,1\}$, $A = [0,1]$, $B = [0,1]$, $f(0)=0, f(1)=1$, $g(0)=0, g(1)=1$.

Common Mistakes

Common Mistake

The limit of a diagram is always a subobject of some product of the diagram's objects.

While this is true in many concrete categories (limits in Set, Grp, Ab, Top can be computed as subsets of products), it fails in general categories. Limits are defined by a universal property, not by a specific construction.

Common Mistake

Products and coproducts are always different.

In abelian categories (like Ab, Mod_R, Vect_k), finite products and coproducts coincide (biproducts or direct sums). In Set, they are different (Cartesian product vs. disjoint union).

Common Mistake

A limit/colimit always exists.

Not every diagram has a limit or colimit in every category. For example, the category of fields does not have all limits. A category is called complete (resp. cocomplete) if all small diagrams have limits (resp. colimits).

Quiz

Summary

A limit of a diagram $D: J \to \mathcal{C}$ is a universal cone over $D$: an object $L$ with projection maps to each $D(j)$ through which any other cone factors uniquely.
Special cases of limits: terminal objects (empty diagram), binary products, equalizers, pullbacks, and inverse limits.
Colimits are dual: initial objects, coproducts, coequalizers, pushouts, and direct limits are all colimits.
Limits are unique up to unique isomorphism; right adjoints preserve limits and left adjoints preserve colimits.
A category with all small (co)limits is called (co)complete; Set, Grp, Ab, Top, and most algebraic categories are bicomplete.

References

BookMac Lane, S. Categories for the Working Mathematician, 2nd ed. Springer, 1998. Ch. V.
BookRiehl, E. Category Theory in Context. Dover, 2016. Ch. 3.
WebsiteWikipedia — Limit (category theory)
WebsitenLab — limit