Alphabeta MathHow 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.
Passing from to reverses arrows and composition, preserves identities, and is involutive (Opposite category ).
Proof
Translate every typed arrow to , every to , and leave equality and logical connectives unchanged.
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.
Translating the complete derivation therefore proves the translated conclusion; applying the translation twice returns the original theorem.
Depends on
Used by
- For a presheaf P, Nat(C(-,a),P)≅ P(a) naturally in a and P Corollary
- Left adjoints preserve every colimit that exists Corollary
- Coreflective full subcategory and coreflector Definition
- Initial object, terminal object, and zero object Definition
- A morphism is an isomorphism exactly when postcomposition, equivalently precomposition, induces bijections on every hom-collection Lemma
- A limiting cone for a diagram is exactly a colimiting cocone for the formally dual diagram in the opposite category Proposition
- Identities and composites of monomorphisms or epimorphisms retain cancellation; split monomorphisms are monic and split epimorphisms are epic Proposition
- Initial and terminal objects are unique up to a unique isomorphism Theorem
Dependency tree · two levels
2 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
- Emily Riehl, Category Theory in Context, Chapter 1 (standard reference, not scraped)