CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17‡ 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.

Every category monadic over Set is complete

Statement

If $U : D \to S e t$ is monadic, then $D$ has every small limit.

Facts & Assumptions

Given: A monadic functor $U : D \to S e t$ in the sense of Monadic and strictly monadic functors.

[L1]

Every small diagram in $S e t$ has a limit (Set has all small limits, realized as compatible tuples in a set-indexed product).

[L2]

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

[L3]

Equivalences preserve, reflect, and create all existing limits 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 any small diagram in the Eilenberg–Moore category of the induced monad on $S e t$ , [L1] supplies its underlying limit and [L2] lifts that limit; hence this Eilenberg–Moore category is complete.

L1L2

2.1

Monadicity says that the comparison $K : D \to {S e t}^{T}$ is an equivalence. By [L3], each small limit from step 1.1 transports across $K$ , so $D$ is complete.

step 1.1L3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 40 results over 14 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

E. Riehl, Category Theory in Context, 2nd ed., Corollary 5.6.7 (standard reference, not scraped)