TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Under dependent choice, a finitary monad on a complete cocomplete locally small category has complete and cocomplete algebras

Statement

Assume dependent choice. If $T$ is a finitary monad on a complete, cocomplete, locally small category $C$ , then its Eilenberg–Moore category $C^{T}$ is complete and cocomplete.

The completeness conclusion itself uses no choice; dependent choice enters only in the construction of coequalizers used for cocompleteness.

Facts & Assumptions

Given: Dependent choice and a finitary monad $T$ on a complete cocomplete locally small category $C$ .

[L1]

The Eilenberg–Moore forgetful functor strictly creates every limit that exists in the base (The Eilenberg–Moore forgetful functor strictly creates every limit that exists in the base).

[L2]

Under dependent choice, algebras for such a finitary monad have coequalizers (Under dependent choice, algebras for a finitary monad on a complete cocomplete locally small category have coequalizers).

[L3]

Over a cocomplete base, a monadic category is cocomplete exactly when it has coequalizers (Over a cocomplete base, a monadic category is cocomplete exactly when it has coequalizers).

[L4]

The Eilenberg–Moore adjunction induces the given monad on the nose (The free–forgetful Eilenberg–Moore adjunction induces the given monad).

Proof

technique · direct

1.1

Since $C$ has every small limit, [L1] creates every such limit in $C^{T}$ , including the empty limit. Thus $C^{T}$ is complete without using dependent choice.

1.2

Under the stated dependent-choice hypothesis, [L2] gives every coequalizer in $C^{T}$ , including equal parallel maps and the zero-stage case of its construction.

2.1

By [L4], the Eilenberg–Moore adjunction induces $T$ on the nose, so its forgetful functor is monadic. Applying [L3] to the cocomplete base and step 1.2 makes $C^{T}$ cocomplete, including the empty colimit.

step 1.2L3L4

3.1

Combining step 1.1 with step 2.1 proves that $C^{T}$ is complete and cocomplete.

step 1.1step 2.1∎

Depends on

Used by

Under dependent choice, categories of models for algebraic theories are complete and cocomplete Corollary

Dependency tree · two levels

30 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 5.6.12 (standard reference, not scraped)