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.

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Identities and composites of monomorphisms or epimorphisms retain cancellation; split monomorphisms are monic and split epimorphisms are epic

Statement

Identity morphisms are monic and epic; composites of monomorphisms are monic, composites of epimorphisms are epic; every split monomorphism is monic, and every split epimorphism is epic.

Facts & Assumptions

Given: Composable morphisms in a category.

[L1]

Monomorphisms and epimorphisms are defined by left and right cancellation (Monomorphism and epimorphism by left and right cancellation).

[L2]

A split monomorphism has a left inverse and a split epimorphism has a right inverse (Split monomorphism, split epimorphism, retraction, and section); the two halves are dual by Every theorem about categories has a formal dual obtained by reversing morphisms and composition.

Proof

technique · direct

1.1

Identities cancel trivially; if $f$ and $g$ are monic and $(g \circ f) u = (g \circ f) v$ , cancellation by $g$ and then by $f$ gives $u = v$ , so composites of monomorphisms are monic.

givenL1

2.1

Applying the dual argument of [L2] proves that identities and composites of epimorphisms are epic.

step 1.1L1L2

3.1

If $r \circ f = 1$ , then $f u = f v$ implies $u = r f u = r f v = v$ , so a split monomorphism is monic; dually, $f \circ s = 1$ makes a split epimorphism epic.

step 2.1L1L2∎

Depends on

Used by

A complete locally small category with a small coseparating set and intersections of all subobject collections has an initial object Theorem

Dependency tree · two levels

6 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)