TheoremStatement: 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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

In $S e t$ , monomorphisms are exactly injections and epimorphisms are exactly surjections

Statement

In $S e t$ , a function is monic exactly when it is injective, and it is epic exactly when it is surjective.

Facts & Assumptions

Given: A function $f : A \to B$ regarded as a morphism of $S e t$ .

[L1]

The morphisms of $S e t$ are functions (Sets and functions form the large locally small category $S e t$ ), monic and epic mean cancellation (Monomorphism and epimorphism by left and right cancellation), and injective, surjective, and bijective have their usual fibrewise meanings (Injection, surjection, bijection).

Proof

technique · direct

1.1

If $f$ is injective, $f \circ g = f \circ h$ implies $g = h$ pointwise, so $f$ is monic; if $f$ is not injective, choose $a_{0} \neq a_{1}$ with $f (a_{0}) = f (a_{1})$ , and the two maps from a singleton selecting $a_{0}, a_{1}$ show that $f$ is not monic.

givenL1

2.1

If $f$ is surjective and $g \circ f = h \circ f$ , then for each $b \in B$ choose an $a$ only for this fixed $b$ with $f (a) = b$ , giving $g (b) = h (b)$ ; hence $g = h$ and $f$ is epic.

step 1.1L1

3.1

If $f$ is not surjective, let $g : B \to {0, 1}$ be constantly $0$ and let $h$ be $0$ on $f [A]$ and $1$ outside $f [A]$ ; then $g \neq h$ but $g \circ f = h \circ f$ , so $f$ is not epic.

step 2.1L1∎

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
The two-element set is a subobject classifier for Set Theorem

Dependency tree · two levels

9 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)

In SetSet, monomorphisms are exactly injections and epimorphisms are exactly surjections