TheoremStatement: 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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Beck's monadicity theorem in data-supplied form

Statement

Let $U : D \to C$ be a right adjoint.

If $U$ is monadic, then it creates coequalizers of $U$ -split pairs.
Conversely, suppose $U$ creates coequalizers of $U$ -split pairs and, for every algebra of the induced monad, a specific created coequalizer of its lifted canonical pair is supplied. Then $U$ is monadic.

Here monadic means that the comparison functor is an equivalence, and creation is ordinary isomorphism-invariant creation, not strict creation. The supplied family in the converse is data; it is not manufactured by global choice.

Facts & Assumptions

Given: A right adjoint $U : D \to C$ , a left adjoint $F$ , the induced monad $T = U F$ , and comparison functor $K : D \to C^{T}$ .

[L1]

The functor $U$ is monadic when its comparison functor $K$ is an equivalence of categories (Monadic and strictly monadic functors).

[L2]

The Eilenberg–Moore forgetful functor strictly creates coequalizers of $U^{T}$ -split pairs (The Eilenberg–Moore forgetful functor strictly creates coequalizers of $U^{T}$ -split pairs).

[L3]

If $U$ creates coequalizers of $U$ -split pairs and a created coequalizer is supplied for every lifted canonical algebra pair, those supplied presentations give a quasi-inverse to the comparison functor (Supplied created canonical presentations give a quasi-inverse to the comparison functor).

[L4]

An equivalence of categories preserves, reflects, and creates existing colimits in the ordinary isomorphism-invariant sense (Equivalences preserve, reflect, and create limits and colimits in the isomorphism-invariant sense).

Proof

technique · direct

1.1

For the forward direction, suppose $U$ is monadic. Then $K$ is an equivalence by [L1] and $U = U^{T} K$ . Transporting the strict-creation result [L2] across $K$ by [L4] shows that $U$ creates coequalizers of $U$ -split pairs in the ordinary sense.

L1L2L4

1.2

For the converse, suppose $U$ creates coequalizers of $U$ -split pairs and the stated family of created canonical coequalizers is supplied. By [L3], those data define a quasi-inverse to $K$ .

L3given

2.1

A functor with a quasi-inverse and the two natural isomorphisms is an equivalence, so $K$ is an equivalence and $U$ is monadic by [L1].

step 1.2L1

3.1

Step 1.1 proves the monadic-to-creation implication, while steps 1.2 and 2.1 prove the data-supplied converse.

step 1.1step 2.1∎

Depends on

Used by

FALSE: ordinary Beck creation characterizes strict monadicity False statement

Dependency tree · two levels

18 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., Theorem 5.5.1 (standard reference, not scraped)
D. Mehrle, Category Theory Part III, Theorem 5.16 (standard reference, not scraped)