TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16‡ 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.

General adjoint functor theorem, data-supplied functor form

Statement

Let $U : A \to C$ , where $A$ is complete and locally small and $U$ is continuous. Suppose that, for every $C \in C$ , a solution set at $C$ is supplied and the resulting initial object of $(C ↓ U)$ is supplied. Then $U$ has a left adjoint.

The conclusion is data-sensitive: objectwise existence from GAFT does not by itself choose one initial comma object over a proper class of objects.

Facts & Assumptions

Given: The displayed hypotheses and a supplied initial object in every comma category $(C ↓ U)$ .

[L1]

Under completeness, local smallness, continuity, and a solution set at a fixed object, that fixed comma category has an initial object (General adjoint functor theorem, objectwise initial-object form).

[L2]

A left adjoint is supplied exactly by choosing an initial object in every comma category; those choices determine the functor and adjunction (A left adjoint exists exactly when chosen initial objects are supplied in every comma category).

Proof

technique · direct

1.1

The supplied initial comma objects satisfy precisely the hypothesis of [L2], so they assemble into a functor $F : C \to A$ and an adjunction $F ⊣ U$ .

L1L2

2.1

The assembly uses the supplied family, not merely the separate existential conclusions of [L1]; therefore no unrecorded proper-class choice is hidden in the functor form.

step 1.1L1L2∎

Depends on

Used by

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, theorem 4.7.3 (standard reference, not scraped)
T. Leinster, Basic Category Theory, theorem 6.3.10 (standard reference, not scraped)