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.

If A is small, then [A,C] is complete or cocomplete whenever C is respectively complete or cocomplete

Statement

Assume Choice. If $A$ is small and $C$ is complete, then $[A, C]$ is complete. If $C$ is cocomplete, then $[A, C]$ is cocomplete.

Facts & Assumptions

Given: A small $A$ and the indicated completeness or cocompleteness of $C$ .

[L1]

Functor-category limits and colimits are computed pointwise when the pointwise choices exist (For small source and index categories, chosen target limits and colimits compute the corresponding functor-category limits and colimits pointwise).

[F1]

Completeness and cocompleteness mean existence of every small limit and colimit (Finite, small, and large limits and colimits; complete and cocomplete categories).

[F2]

Choice permits simultaneous selections from a set-indexed family of nonempty sets (The Axiom of Choice).

Proof

technique · direct corollary

1.1

Let $D : J \to [A, C]$ be small. If $C$ is complete, [F1] supplies a limit of $j \mapsto D (j) (a)$ for every $a$ . Since $A$ is small, [F2] selects these pointwise limits simultaneously.

F1F2

2.1

By [L1], the selected objects form a limit of $D$ . Since $D$ was arbitrary, [F1] says the functor category is complete.

L1F1step 1.1

3.1

If $C$ is cocomplete, make the same set-indexed choices of pointwise colimits and apply the colimit clause of [L1]. This proves cocompleteness, with no assertion for a large source $A$ .

L1F1F2∎

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: 29 results over 10 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 3.3.3 (standard reference, not scraped)