Alphabeta Math
PropositionStatement: 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 poset category is complete exactly when every small family has an infimum, and cocomplete exactly when every small family has a supremum

Statement

A poset regarded as a category is complete if and only if every set-indexed family has an infimum, including the empty family. It is cocomplete if and only if every set-indexed family has a supremum, including the empty family. Hence it is both complete and cocomplete exactly when it is a complete lattice.

Facts & Assumptions

Given: A poset P regarded as a category.

[F1]

In the associated category, x→y exists exactly when x≤y, and there is at most one such arrow (A preorder is a category with at most one morphism between any two objects, and its functors are exactly monotone maps).

[F2]

Products and coproducts have their family-of-arrows universal properties (Products and coproducts as limits and colimits of discrete diagrams, including their existence-and-uniqueness equations).

[L1]

Products plus equalizers characterize completeness, and coproducts plus coequalizers characterize cocompleteness (A category is complete exactly when it has all small products and equalizers, and cocomplete exactly when it has all small coproducts and coequalizers).

Proof

technique · translate universal properties into inequalities
1.1

A cone from x to a discrete family (pi) is precisely the collection of inequalities x≤pi. By [F1] and [F2], a product is therefore a lower bound above every lower bound, namely inf⁡ipi. For the empty family this is a greatest element.

F1F2
1.2

Reversing inequalities, a coproduct is sup⁡ipi, with the empty coproduct a least element. Coequalizers are identities for the same at-most-one-arrow reason. The dual half of [L1] proves both directions of the cocompleteness equivalence.

F1F2L1
2.1

Parallel arrows in a poset category are equal whenever they exist. The identity of their common domain is consequently an equalizer, since every factor is unique by [F1]. Hence [L1] and step 1.1 prove both directions of the completeness equivalence.

F1L1step 1.1
3.1

Having all set-indexed infima and suprema, including empty ones, is exactly the complete-lattice condition, which proves the last assertion.

step 2.1step 1.2∎

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: 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