Alphabeta Math
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.

A category is complete exactly when it has all small products and equalizers, and cocomplete exactly when it has all small coproducts and coequalizers

Statement

A category is complete if and only if it has products indexed by every set and has equalizers of all parallel pairs. Dually, it is cocomplete if and only if it has all small coproducts and all coequalizers.

Facts & Assumptions

Given: A category C.

[L1]

Existing object-indexed products and equalizers construct the limit of every small diagram (Every small limit can be constructed as an equalizer between products over the objects and arrows of the index category).

[L2]

Existing object-indexed coproducts and coequalizers construct the colimit of every small diagram (Every small colimit can be constructed as a coequalizer between coproducts over the arrows and objects of the index category).

[F1]

Complete means having all small limits and cocomplete means having all small colimits (Finite, small, and large limits and colimits; complete and cocomplete categories).

Proof

technique · biconditional
1.1

If C is complete, specialize [F1] to every small discrete category and to the finite parallel-pair category. These limits are all set-indexed products, including the empty product, and all equalizers.

F1
1.2

Conversely, if those products and equalizers exist, [L1] constructs a limit for every small diagram, so [F1] says that C is complete.

L1F1
1.3

If C is cocomplete, specialization gives all set-indexed coproducts, including the empty one, and all coequalizers. Conversely those colimits construct every small colimit by [L2].

L2F1
2.1

Steps 1.1 and 1.2 prove both directions of the completeness equivalence; step 1.3 proves both directions of its cocomplete dual.

step 1.1step 1.2step 1.3∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 20 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