False statementConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-26

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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

1.1F1assume-hyp

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 $1$ .

1.2F2F3

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.

2.1step 1.1step 1.2∎

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.

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)