DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-13

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Equalizers and coequalizers as limits and colimits of a parallel pair

Definition

For parallel morphisms $f, g : A ⇉ B$ , an equalizer is a limit of that parallel-pair diagram (Limits and colimits as terminal cones and initial cocones, with existence and uniqueness in their universal properties). It is a morphism $e : E \to A$ satisfying $f e = g e$ such that, whenever $h : X \to A$ satisfies $f h = g h$ , there is a unique $u : X \to E$ with $e u = h$ .

A coequalizer is a colimit of the parallel pair. It is a morphism $q : B \to Q$ satisfying $q f = q g$ such that, whenever $h : B \to X$ satisfies $h f = h g$ , there is a unique $u : Q \to X$ with $u q = h$ .

Depends on

Limits and colimits as terminal cones and initial cocones, with existence and uniqueness in their universal properties

Dependency tree · 2 levels

Sources

E. Riehl, Category Theory in Context, Definition 3.1.15