CorollaryStatement: 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, categories of models for algebraic theories are complete and cocomplete

Statement

Assume dependent choice. Every category of models for an algebraic theory is complete and cocomplete.

Facts & Assumptions

Given: Dependent choice and a category $A$ of models for an algebraic theory.

[L1]

A category of models for an algebraic theory is equipped with a finitary monadic functor to $S e t$ (Categories of models for algebraic theories).

[L2]

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

[L3]

Every small diagram in $S e t$ has a colimit (Set has all small colimits, realized as a quotient of a set-indexed disjoint union).

[L4]

Equivalences preserve and reflect existing limits and colimits (Equivalences preserve, reflect, and create limits and colimits in the isomorphism-invariant sense).

[L5]

Under dependent choice, a finitary monad on a complete cocomplete locally small category has a complete and cocomplete Eilenberg–Moore category (Under dependent choice, a finitary monad on a complete cocomplete locally small category has complete and cocomplete algebras).

Proof

technique · direct

1.1

By [L2] and [L3], $S e t$ is complete and cocomplete; it is locally small because each function collection between two sets is a set. This includes empty diagrams.

L2L3given

2.1

The finitary monad induced by [L1] satisfies the hypotheses of [L5], so under dependent choice its Eilenberg–Moore category is complete and cocomplete.

step 1.1L1L5

3.1

The comparison supplied by [L1] is an equivalence over $S e t$ . Transporting the limits and colimits of step 2.1 across it by [L4] proves that $A$ is complete and cocomplete.

step 2.1L1L4∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

28 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., Corollary 5.6.14 (standard reference, not scraped)