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

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FALSE: every $U$ -split pair is split in the domain

Statement

False claim: if a parallel pair in $D$ is $U$ -split for a functor $U : D \to C$ , then that pair is split in $D$ .

Facts & Assumptions

Given: The Eilenberg–Moore forgetful functor for the free-monoid monad.

[L1]

A parallel pair is $U$ -split when its image under $U$ extends to a split coequalizer diagram ( $U$ -split pairs and ordinary or strict creation of their coequalizers).

[L2]

The underlying canonical presentation is split in the base, but its canonical splittings need not be algebra homomorphisms (The canonical algebra presentation is split in the base, but its canonical splittings need not be algebra homomorphisms).

Refutation

technique · direct

1.1

Take the canonical presentation of the two-element monoid $M = {1, e}$ with $e^{2} = e$ as an algebra for the free-monoid monad. By [L1] and [L2], its underlying pair is $U^{T}$ -split.

L1L2

1.2

If the coequalizer evaluation $M^{*} \to M$ had a monoid-homomorphic section $s$ , then $s (e)^{2} = s (e)$ because $e^{2} = e$ . The only idempotent word in a free monoid is the empty word, since a nonempty word has positive length and its square has twice that length. But evaluation of the empty word is $1$ , not $e$ , so no such section exists.

2.1

Hence this pair becomes split after applying $U^{T}$ but is not split in the algebra category. The $U$ -split condition therefore does not imply a splitting in the domain.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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, 2nd ed., Example 5.4.7 (standard reference, not scraped)

FALSE: every UU-split pair is split in the domain