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

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Non-isomorphic objects can have naturally isomorphic representable presheaves

Statement

False claim: in a locally small category, two non-isomorphic objects can have naturally isomorphic representable presheaves $C (-, a) ≅ C (-, b)$ .

Facts & Assumptions

Given: A locally small category $C$ and the false claim above.

[L1]

Objects $a$ and $b$ are isomorphic if and only if the representable presheaves $C (-, a)$ and $C (-, b)$ are naturally isomorphic (Objects $a$ and $b$ are isomorphic exactly when $C (-, a)$ and $C (-, b)$ are naturally isomorphic).

[L2]

The Yoneda hom-map $C (a, b) \to Nat (C (-, a), C (-, b))$ is a bijection and respects identities and composition (The Yoneda functor is fully faithful, and it is a full embedding when its object map is injective).

Refutation

technique · contradiction

1.1

Suppose there were non-isomorphic objects $a, b$ and a natural isomorphism $α : C (-, a) \overset{≅}{\Rightarrow} C (-, b)$ .

assume-contra

2.1

By [L2], $α$ and its natural inverse lift uniquely to morphisms $f : a \to b$ and $g : b \to a$ . Because the Yoneda map respects identities and composition and is injective, the two equations $α^{- 1} α = 1$ and $α α^{- 1} = 1$ imply $g f = 1_{a}$ and $f g = 1_{b}$ .

step 1.1L2

3.1

Thus $a$ and $b$ are isomorphic, also the reverse implication of [L1], contradicting the choice in step 1.1.

step 1.1step 2.1L1

4.1

No such pair of non-isomorphic objects exists, so the claim is false.

step 3.1discharge-contradiction∎

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: 21 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

Emily Riehl, Category Theory in Context, Proposition 2.3.1 (standard reference, not scraped)
Tom Leinster, Basic Category Theory, Corollary 4.3.10 (standard reference, not scraped)