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.

Freyd's representability theorem for continuous Set-valued functors satisfying a solution set condition

Statement

Let $C$ be complete and locally small, and let $F : C \to S e t$ be continuous. Suppose there is a supplied set of pairs $(S_{i}, y_{i})$ with $y_{i} \in F (S_{i})$ such that, for every $C \in C$ and every $x \in F (C)$ , some $i$ and some $f : S_{i} \to C$ satisfy $F (f) (y_{i}) = x .$ Then $F$ is covariantly representable.

Facts & Assumptions

Given: The category, functor, and supplied set of element-pairs in the Statement.

[L1]

The category of elements $\int F$ has objects $(C, x)$ and morphisms $f : (C, x) \to (D, y)$ satisfying $F (f) (x) = y$ (The category of elements of a covariant functor or a presheaf).

[L2]

For a covariant Set-valued functor, a universal element is exactly an initial object of $\int F$ (Universal elements are initial in a covariant category of elements and terminal in a presheaf category of elements).

[L3]

A covariant $S e t$ -valued functor is representable when it is naturally isomorphic to $C (R, -)$ for some object $R$ (Presheaves, covariantly and contravariantly representable functors, and representations).

[L5]

For locally small $C$ , a pair $(R, u)$ with $u \in F (R)$ is universal for $F : C \to S e t$ if and only if, for every object $c$ and every $x \in F (c)$ , there is a unique morphism $f : R \to c$ with $F (f) (u) = x$ (A representation is equivalently a universal element with a unique factorisation property).

[L4]

The objectwise GAFT constructs an initial comma object from completeness, local smallness, continuity, and a supplied solution set (General adjoint functor theorem, objectwise initial-object form).

Proof

technique · constructive

1.1

By [L1], each pair $(S_{i}, y_{i})$ is an object of $\int F$ , and the displayed factorisation condition says exactly that every $(C, x)$ receives a morphism from some $(S_{i}, y_{i})$ . Thus these pairs form a supplied jointly weakly initial set in $\int F$ .

L1construct

2.1

The category $\int F$ is the comma category $(1 ↓ F)$ for a singleton $1$ . Since $F$ is continuous, [L4] applies to the supplied set from step 1.1 and gives an initial object $(R, u)$ , without selecting over a proper class.

step 1.1L4choose

3.1

By [L2], $(R, u)$ is a universal element of $F$ . By [L5] the map $Φ_{c} : C (R, c) \to F (c)$ , $f \mapsto F (f) (u)$ , is then a bijection for every object $c$ ; it is natural in $c$ because for $g : c \to c^{'}$ functoriality gives $F (g) (Φ_{c} (f)) = F (g) (F (f) (u)) = F (g \circ f) (u) = Φ_{c^{'}} (g \circ f)$ . Hence $C (R, -) ≅ F$ as functors, which is representability in the sense of [L3].

step 2.1L2L3L5discharge-construct∎

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: 43 results over 13 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 4.7.15 (standard reference, not scraped)