CounterexampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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The inclusion of one object into a discrete two-object category is fully faithful but not essentially surjective

Statement refuted

Full faithfulness alone does not imply essential surjectivity.

Facts & Assumptions

Given: The one-object discrete category $C=\{0\}$ and the two-object discrete category $D=\{0,1\}$ .

[L1]

Full faithfulness means bijectivity on every hom-collection, while essential surjectivity requires every target object to be isomorphic to an image object (Faithful, full, fully faithful, essentially surjective, and split essentially surjective functors).

Counterexample

technique · direct

1.1

Let $I:C\to D$ send $0$ to $0$ . The sole hom-map is the bijection $\{1_0\}\to\{1_0\}$ , so $I$ is fully faithful by [L1].

1.2

In the discrete category $D$ , the only isomorphisms are identities. Therefore the object $1$ is not isomorphic to $I(0)=0$ , so $I$ is not essentially surjective.

2.1

This finite inclusion is fully faithful by step 1.1 but not essentially surjective by step 1.2.

step 1.1step 1.2∎

Depends on

Faithful, full, fully faithful, essentially surjective, and split essentially surjective functors

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: 8 results over 5 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.