DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-11

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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

Definition

Let $F : C \to D$ be a functor (Covariant functor, identity functor, composite functor, and contravariant functor). For each $A, B$ , it induces

$F_{A, B} : C (A, B) \to D (F A, F B), f \mapsto F f .$

The functor is faithful when every $F_{A, B}$ is injective, full when every $F_{A, B}$ is surjective, and fully faithful when every $F_{A, B}$ is bijective (Injection, surjection, bijection).

It is essentially surjective when every object $D$ of $D$ is isomorphic to some $F C$ (Isomorphism, groupoid, and connected category). It is split essentially surjective when the data include, for every $D$ , a specified object $C_{D}$ and specified isomorphism $ε_{D} : F C_{D} \to D$ . The word split records these witnesses and is strictly stronger as data than their mere existence.

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

Definition

Depends on

Dependency tree · 2 levels

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