Category theory; homological algebra Definition

Kernel pair

Definition

Let $\mathcal{C}$ be a category and let $f \colon X \to Y$ be a morphism of $\mathcal{C}$ . A kernel pair of $f$ is an object $K$ with morphisms $p_1, p_2 \colon K \to X$ such that $(K, p_1, p_2)$ is a pullback of $f$ and $f$ . That is:

$f \circ p_1 = f \circ p_2$ .
For every object $W$ and morphisms $q_1, q_2 \colon W \to X$ with $f \circ q_1 = f \circ q_2$ , there is a unique $u \colon W \to K$ with $p_1 \circ u = q_1$ and $p_2 \circ u = q_2$ .

Remarks

Example: in the category of sets, $K = \set{(x, x') \in X \times X : f(x) = f(x')}$ with the two projections is a kernel pair. Clause 1 holds by the definition of $K$ , and for clause 2 the map $u$ is $w \mapsto (q_1(w), q_2(w))$ . For a group homomorphism $f$ with kernel $N$ , this $K$ is the subgroup of pairs $(x, x')$ with $x^{-1} x' \in N$ .

Non-example: in the category of sets, if $f$ is not injective, then $(X, \id_X, \id_X)$ satisfies clause 1 but not clause 2. Take $W$ a point and $q_1, q_2$ picking $x \neq x'$ with $f(x) = f(x')$ ; no $u$ has $u = q_1$ and $u = q_2$ .

A morphism $f$ is a monomorphism if and only if $(X, \id_X, \id_X)$ is a kernel pair of $f$ .

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Category concept
Morphism concept
Pullback concept
Kernel pair definition

Tag	`JS0YH5E9ZR`
URI	`m2://JS0YH5E9ZR`
Kind	definition
Writing	W0 stub
MSC	18A30, 18A20
Defines	Kernel pair
Uses	Category, Morphism, Pullback
Changed	2 November 2026

Kernel pair

Definition

Remarks

Reading path

Further reading

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