CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-11

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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.

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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 · two levels

4 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.