ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 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.

The equalizer of two group homomorphisms is their agreement subgroup

Example

For group homomorphisms $f, g : G ⇉ H$ , their equalizer in $G r p$ is the inclusion $E = {x \in G : f (x) = g (x)} ↪ G$ .

Facts & Assumptions

Given: The parallel group homomorphisms $f, g$ .

[F1]

An equalizer is an equalizing arrow through which every other equalizing arrow factors uniquely (Equalizers and coequalizers as limits and colimits of a parallel pair).

[F2]

Groups and homomorphisms form $G r p$ , and homomorphisms preserve products, identities, and inverses (Groups and group homomorphisms form the large locally small category $G r p$ , Monoid homomorphism and group homomorphism).

Verification

technique · subgroup and factorization

1.1

Since $f (1) = g (1) = 1$ , the identity lies in $E$ . If $x, y \in E$ , then $f (x y) = f (x) f (y) = g (x) g (y) = g (x y)$ , and similarly $f (x^{- 1}) = g (x^{- 1})$ . Thus $E$ is a subgroup and its inclusion is a homomorphism.

2.1

The inclusion equalizes $f, g$ . If $h : K \to G$ satisfies $f h = g h$ , then $h (K) \subseteq E$ , so the unique set-theoretic corestriction $\overset{ˉ}{h} : K \to E$ is a homomorphism and the inclusion composed with $\overset{ˉ}{h}$ is $h$ .

F2step 1.1

3.1

Injectivity of the inclusion makes this factor unique. By [F1], it is the equalizer.

F1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

E. Riehl, Category Theory in Context, Example 3.1.18 (standard reference, not scraped)