Alphabeta Math
PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 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.

Every fully faithful functor reflects isomorphisms

Statement

If F:C→D is fully faithful and F(f) is an isomorphism, then f is an isomorphism.

Facts & Assumptions

Given: A fully faithful functor F and a morphism f:A→B such that F(f) is invertible.

[L1]

Fullness lifts every morphism between FA and FB, while faithfulness reflects equality between parallel morphisms (Faithful, full, fully faithful, essentially surjective, and split essentially surjective functors).

[L2]

An isomorphism has a two-sided inverse (Isomorphism, groupoid, and connected category).

Proof

technique · direct
1.1

Let h:FB→FA be the inverse of F(f); fullness gives g:B→A with F(g)=h.

givenL1L2
2.1

Then F(g∘f)=h∘F(f)=1FA=F(1A) and F(f∘g)=F(f)∘h=1FB=F(1B).

step 1.1L1L2
3.1

Faithfulness gives g∘f=1A and f∘g=1B, so f is an isomorphism.

step 2.1L1L2∎

Depends on

Used by

Dependency tree · two levels

5 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.

Sources