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.

Split monomorphism, split epimorphism, retraction, and section

Definition

Let $f : A \to B$ be a morphism.

If there is $r : B \to A$ with $r \circ f = 1_{A}$ , then $f$ is a split monomorphism and a section, while $r$ is a retraction of $f$ . If there is $s : B \to A$ with $f \circ s = 1_{B}$ , then $f$ is a split epimorphism and a retraction, while $s$ is a section of $f$ .

When both identities hold, $f$ and its splitting are inverse isomorphisms in the sense of Isomorphism, groupoid, and connected category.

Depends on

Isomorphism, groupoid, and connected category

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
Thick subcategory Definition
A distinguished triangle is split up to rotation exactly when one map vanishes Proposition
A distinguished triangle with zero first map is split Proposition
Identities and composites of monomorphisms or epimorphisms retain cancellation; split monomorphisms are monic and split epimorphisms are epic Proposition
The total kernel of a cohomological functor is thick Proposition
Fullness and faithfulness of a right adjoint are detected by its counit 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)