CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13‡ 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.

Assuming Choice, every small complete category and every small cocomplete category is a preorder

Statement

Assume Choice. Every small complete category is a preorder, and every small cocomplete category is a preorder.

Facts & Assumptions

Given: A small category $C$ that is complete or cocomplete.

[L1]

If a small category has constant products or coproducts indexed by the cardinality of its morphism set, it is a preorder (Assuming Choice, a small category with products or coproducts indexed by the cardinality of its morphism set is a preorder).

[F1]

Complete categories have every small limit, and cocomplete categories have every small colimit (Finite, small, and large limits and colimits; complete and cocomplete categories).

Proof

technique · direct corollary

1.1

Let $κ = ∣ Mor (C) ∣$ . Since $C$ is small, a discrete category on the set $κ$ is a small indexing category.

given

2.1

If $C$ is complete, [F1] supplies the product of every constant $κ$ -family. The product clause of [L1] makes $C$ a preorder.

F1L1step 1.1

3.1

If $C$ is cocomplete, [F1] instead supplies every constant $κ$ -coproduct, and the coproduct clause of [L1] gives the same conclusion.

F1L1step 1.1∎

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: 30 results over 9 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, Corollary after Proposition 3.7.3 (standard reference, not scraped)