TheoremStatement: 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.

Set has all small limits, realized as compatible tuples in a set-indexed product

Statement

Every small diagram $D : J \to S e t$ has a limit. It is the set

$L = {(x_{j})_{j \in Ob J} \in \prod_{j} D (j) : D (u) (x_{j}) = x_{k} for every u : j \to k}$

with its coordinate projections.

Facts & Assumptions

Given: A small category $J$ and a diagram $D : J \to S e t$ .

[F1]

A small category has sets of objects and morphisms, and completeness means existence of limits for all small diagrams (Finite, small, and large limits and colimits; complete and cocomplete categories).

[F2]

Sets and functions form $S e t$ (Sets and functions form the large locally small category $S e t$ ).

[F3]

The product of a set-indexed family consists of functions choosing one element from each member (The product $\prod_{i \in I} A_{i} : = {f : I \to ⋃_{i \in I} A_{i} ∣ f (i) \in A_{i} for every i \in I}$ ).

[F4]

A limit requires a unique mediating morphism from every cone (Limits and colimits as terminal cones and initial cocones, with existence and uniqueness in their universal properties).

Proof

technique · construction

1.1

By [F1] and [F3], the displayed product and its subset $L$ are sets. For each $j$ , let $p_{j} : L \to D (j)$ be the coordinate function. The defining equalities give $D (u) p_{j} = p_{k}$ , so $(L, p)$ is a cone.

F1F2F3

1.2

If $Ob J$ is empty, the product is the singleton containing the empty function and all compatibility conditions are vacuous. Thus the construction still gives the terminal set.

1.3

Let $(X, ξ_{j})$ be any cone. Define $h : X \to \prod_{j} D (j)$ by $h (x)_{j} = ξ_{j} (x)$ . The cone equations imply $h (x) \in L$ , so $h$ corestricts to a function $\overset{ˉ}{h} : X \to L$ satisfying $p_{j} \overset{ˉ}{h} = ξ_{j}$ .

F2given

2.1

If $g : X \to L$ has the same composites, then for every $x$ and $j$ , $g (x)_{j} = p_{j} g (x) = ξ_{j} (x) = p_{j} \overset{ˉ}{h} (x)$ . Equality of functions gives $g = \overset{ˉ}{h}$ . This remains true for the empty index, where there is one function to the singleton.

step 1.2step 1.3

3.1

By [F4], steps 1.1, 1.3, and 2.1 prove that $(L, p)$ is a limit, with step 1.2 covering the empty boundary. Since $D$ was an arbitrary small diagram, $S e t$ is complete.

F1F4step 1.1step 1.3step 2.1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 37 results over 11 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 3.2.4 (standard reference, not scraped)