LemmaStatement: 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 cone over an identity diagram is weakly initial, and the identity diagram has a limit exactly when the category has an initial object

Statement

A cone $λ : Δ L \Rightarrow 1_{C}$ supplies a morphism from $L$ to every object of $C$ , so $L$ is weakly initial. The possibly large identity diagram $1_{C} : C \to C$ has a limit if and only if $C$ has an initial object; in that event every limiting apex is initial.

Facts & Assumptions

Given: A category $C$ and its identity diagram.

[F1]

Completeness concerns all small diagrams and makes no assertion about a large identity diagram (Finite, small, and large limits and colimits; complete and cocomplete categories).

[F2]

An initial object $I$ has exactly one morphism $I \to C$ for every object $C$ (Initial object, terminal object, and zero object).

Proof

technique · universal property

1.1

A cone $λ$ has a leg $λ_{C} : L \to C$ for every object $C$ , so its apex is weakly initial.

given

1.2

Suppose $λ$ is limiting. Both $1_{L}$ and $λ_{L}$ are morphisms from the cone $λ$ to itself, because naturality gives $λ_{C} λ_{L} = λ_{C}$ . Limit uniqueness yields $λ_{L} = 1_{L}$ .

given

1.3

Conversely, let $I$ be initial. The unique maps $i_{C} : I \to C$ form a cone: for $f : C \to C^{'}$ , both $f i_{C}$ and $i_{C^{'}}$ are maps $I \to C^{'}$ , hence equal by [F2].

2.1

For any $f : L \to C$ , cone naturality for $f$ says $f λ_{L} = λ_{C}$ . By step 1.2, $f = λ_{C}$ . Thus exactly one morphism $L \to C$ exists, and [F2] makes $L$ initial.

F2step 1.2

2.2

For any cone $ξ : Δ X \Rightarrow 1_{C}$ , take $u : = ξ_{I} : X \to I$ . Naturality along $i_{C} : I \to C$ gives $i_{C} u = ξ_{C}$ . If $v : X \to I$ is another cone morphism, its equation at $I$ is $i_{I} v = ξ_{I}$ ; since $i_{I} = 1_{I}$ , $v = u$ . The cone of step 1.3 is limiting.

F2step 1.3

3.1

Steps 1.2, 2.1, 1.3, and 2.2 prove both directions. If $C$ is large, [F1] explains why this conclusion is not supplied merely by completeness.

F1step 2.1step 2.2∎

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: 15 results over 7 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, Lemma 3.7.1 (standard reference, not scraped)