TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-29

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.

Monicity is detected by members

Statement

For a morphism $f : A \to B$ in an abelian category, the following are equivalent:

$f$ is monic.
For every member $x : X \to A$ , the implication $f x \equiv 0 ⟹ x \equiv 0$ holds.

Facts & Assumptions

Given: A morphism $f : A \to B$ .

[L1]

Monomorphisms are left-cancellable (Monomorphism and epimorphism by left and right cancellation).

[L2]

A member equivalent to zero is literally the zero morphism, and every member has a zero comparison member (Each object has a zero member and each member has a negative).

[L3]

Postcomposition preserves member equivalence (A morphism carries members to members and preserves equivalence).

Proof

technique · direct

1.1L1L2assume-hypalgebra

Assume $f$ is monic, and let $x : X \to A$ satisfy $f x \equiv 0$ . Choose an epic $u : W \to X$ witnessing this, so $f x u = 0$ . Since $f$ is monic, $x u = 0$ , and the same epic $u$ witnesses $x \equiv 0$ .

1.2L1L2L3assume-hypalgebra

Assume condition 2. If $u, v : U \to A$ satisfy $f u = f v$ , then for the member $x : = u - v$ one has $f x = f u - f v = 0$ , hence $f x \equiv 0$ by [L2] and [L3]. Condition 2 gives $x \equiv 0$ , so [L2] makes $x = 0$ , namely $u = v$ . Thus $f$ is monic by [L1].

2.1step 1.1step 1.2∎

Therefore the two conditions are equivalent.

Depends on

Used by

A member chase verifying monicity Example
Monicity by member cancellation Theorem

Dependency tree · two levels

8 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

Saunders Mac Lane, Categories for the Working Mathematician, Theorem VIII.4.3(i) (standard reference, not scraped)