False statementConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26‡ 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.

FALSE: the Yoneda embedding preserves colimits

Statement refuted

That the Yoneda embedding $y : C \to [C^{o p}, S e t]$ (The Yoneda assignment and the small-source Yoneda functor, traditionally called the Yoneda embedding) preserves colimits in general.

The true statement already on disk is the limit half: For a small category, the Yoneda functor preserves and reflects all existing small limits.

Facts & Assumptions

Given: The terminal category $1$ with unique object $*$ .

[F1]

In the terminal category, the unique object is in particular initial (Initial object, terminal object, and zero object).

[F2]

Colimits in a functor category are computed pointwise, so an initial object in $[1^{o p}, S e t]$ has the empty set at $*$ (For small source and index categories, chosen target limits and colimits compute the corresponding functor-category limits and colimits pointwise, Sets and functions form the large locally small category $S e t$ ).

[F3]

The Yoneda embedding sends $*$ to the representable presheaf $1 (-, *)$ (The Yoneda assignment and the small-source Yoneda functor, traditionally called the Yoneda embedding).

Refutation

technique · direct

1.1

By [F1], the object $*$ is a colimit of the empty diagram in $1$ . If the Yoneda embedding preserved colimits, then $y (*)$ would be an initial object of the presheaf category on $1$ .

F1assume-hyp

1.2

But by [F3], the presheaf $y (*)$ takes the value $1 (*, *)$ , which is the singleton set ${1_{*}}$ . By [F2], an initial presheaf has value $\emptyset$ at $*$ . Therefore $y (*)$ is not initial.

F2F3

2.1

So the Yoneda embedding does not preserve colimits in general. The published limit-preservation theorem is not contradicted, because it is about limits, not colimits.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

24 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

T. Leinster, Basic Category Theory, Warning 6.2.14 (standard reference, not scraped)