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.
Split monomorphism, split epimorphism, retraction, and section
Definition
Let be a morphism.
If there is with , then is a split monomorphism and a section, while is a retraction of . If there is with , then is a split epimorphism and a retraction, while is a section of .
When both identities hold, and its splitting are inverse isomorphisms in the sense of Isomorphism, groupoid, and connected category.
Depends on
Used by
- Under the Axiom of Choice, every epimorphism in Set is a split epimorphism Corollary
- The injection ∅→{*} is monic in Set but is not split Counterexample
- Identities and composites of monomorphisms or epimorphisms retain cancellation; split monomorphisms are monic and split epimorphisms are epic Proposition
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)