LemmaStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The underlying-set functor on unital rings strictly creates split coequalizers

Statement

The underlying-set functor $U : R i n g \to S e t$ strictly creates coequalizers of $U$ -split pairs of unital ring homomorphisms.

Facts & Assumptions

Given: Ring homomorphisms $f, g : A ⇉ B$ and a supplied split coequalizer $q : U B \to Q$ of their underlying functions.

[L1]

A split coequalizer has splitting maps satisfying $q f = q g$ , $q s = 1_{Q}$ , $g t = 1_{B}$ , and $f t = s q$ (Split coequalizer diagrams).

[L2]

A ring has associative and commutative addition, associative multiplication, two-sided additive and multiplicative identities, additive inverses, and multiplication distributing over addition on both sides (Ring: an abelian group under addition and a monoid under multiplication, with multiplication distributing over addition on both sides).

[L3]

A ring homomorphism preserves addition, multiplication, and the multiplicative identity (Ring homomorphism: additive, multiplicative, and required to send $1$ to $1$ ).

[L4]

Every split coequalizer is a coequalizer and is preserved by every functor (Every split coequalizer is a coequalizer and an absolute colimit).

Proof

technique · direct

1.1

By [L4], each finite Cartesian power $q^{n}$ is the coequalizer of $f^{n}, g^{n}$ .

L1L4given

2.1

Each basic ring operation on $B$ , followed by $q$ , coequalizes the appropriate finite powers of $f$ and $g$ . It therefore descends uniquely to $Q$ ; explicitly $0_{Q} = q (0_{B})$ , $1_{Q} = q (1_{B})$ , and negation, addition, and multiplication are the unique operations making $q$ preserve them.

step 1.1construct

3.1

Each ring axiom in [L2] becomes true after precomposition with the relevant surjection $q^{n}$ , because it then becomes the corresponding axiom in $B$ . Hence the descended operations make $Q$ a unital ring, including the zero-ring case.

step 2.1L2algebra

4.1

By construction $q$ preserves addition, multiplication, and one, so it is a ring homomorphism by [L3].

step 2.1step 3.1L3

5.1

If a ring homomorphism $r : B \to C$ coequalizes $f, g$ , the set coequalizer gives a unique $u : Q \to U C$ with $u q = r$ . Precomposing the preservation equations for $u$ with the appropriate $q^{n}$ reduces them to those for $r$ , so $u$ is a ring homomorphism and is the unique algebraic factor.

step 4.1L3

6.1

The descended operations are uniquely forced by the requirement that the supplied $q$ be a ring homomorphism. Thus the lift has exactly the same apex and legs and is unique, which is strict creation.

step 2.1step 5.1∎

Depends on

Used by

Monoids and unital rings are strictly monadic over sets Theorem

Dependency tree · two levels

21 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

D. Mehrle, Category Theory Part III, Example 5.20(b) (standard reference, not scraped)
E. Riehl, Category Theory in Context, 2nd ed., Corollary 5.5.3 (standard reference, not scraped)