TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck 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 theorem about categories has a formal dual obtained by reversing morphisms and composition

Statement

A formal theorem derivable solely from the category axioms has a formal dual: reverse every morphism, reverse the order of every composite, and exchange each defined notion with its opposite-category version.

Facts & Assumptions

Given: A derivation in the formal language of categories.

[L1]

Passing from $\mathcal C$ to $\mathcal C^{\mathrm{op}}$ reverses arrows and composition, preserves identities, and is involutive (Opposite category $\mathcal C^{\mathrm{op}}$ ).

Proof

technique · direct

1.1

Translate every typed arrow $f:A\to B$ to $f^{\mathrm{op}}:B\to A$ , every $g\circ f$ to $f^{\mathrm{op}}\circ_{\mathrm{op}}g^{\mathrm{op}}$ , and leave equality and logical connectives unchanged.

givenL1

2.1

Under this translation, each category axiom becomes the corresponding category axiom in the opposite category, so every permitted inference in the original derivation remains a permitted inference after translation.

step 1.1L1

3.1

Translating the complete derivation therefore proves the translated conclusion; applying the translation twice returns the original theorem.

step 2.1L1∎

Depends on

Opposite category $\mathcal C^{\mathrm{op}}$

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 4 results over 3 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, Chapter 1 (standard reference, not scraped)