False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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The Yoneda lemma requires its category to be small

Statement

False claim: the Yoneda lemma can be stated and proved only when its category is small.

Facts & Assumptions

Given: The false claim above and the category $S e t$ .

[L1]

For every locally small category $C$ , object $a$ , and functor $F : C \to S e t$ , evaluation at the identity is a bijection $Nat (C (a, -), F) ≅ F (a)$ , with an explicit inverse; smallness is not a hypothesis (Evaluation at the identity gives $Nat (C (a, -), F) ≅ F (a)$ and proves that the natural-transformation collection is a set).

[L2]

The same bijection is natural in both $a$ and $F$ for every locally small $C$ , without forming a functor category on a large source (The Yoneda bijection $Nat (C (a, -), F) ≅ F (a)$ is natural in both $a$ and $F$ ).

[L3]

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

[F1]

A category is small when its objects and morphisms form sets, locally small when each hom-collection is a set, and large when it is not small (Small, locally small, and large categories).

Refutation

technique · direct

1.1

By [L3] and [F1], $S e t$ is a locally small category that is not small.

L3F1

2.1

Apply [L1] and [L2] to $C = S e t$ , any set $a$ , and any functor $F : S e t \to S e t$ . The Yoneda bijection exists and has both naturalities despite the category being large.

step 1.1L1L2

3.1

Thus local smallness, which makes each hom-collection a set, suffices for the pointwise Yoneda lemma; the large locally small category $S e t$ refutes the asserted need for smallness.

step 1.1step 2.1∎

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: 30 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

Emily Riehl, Category Theory in Context, Theorem 2.2.4 and Remark 2.2.7 (standard reference, not scraped)