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

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Monicity by member cancellation

Statement

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

$f$ is monic.
For all members $x$ and $y$ of $A$ , $f x \equiv f y ⟹ x \equiv y .$

Facts & Assumptions

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

[L1]

Monicity is equivalent to the rule $f z \equiv 0 \Rightarrow z \equiv 0$ (Monicity is detected by members).

[L2]

Members have negatives, and equivalence to zero means literal zero after a common epic comparison (Each object has a zero member and each member has a negative).

Proof

technique · direct

1.1L1L2assume-hypalgebra

Assume $f$ is monic, and suppose $f x \equiv f y$ . Choose a common witness $u : W \to X$ , $v : W \to Y$ with $f x u = f y v$ . Then $f (x u - y v) = 0$ , so [L1] gives $x u - y v \equiv 0$ . By [L2], this means $x u = y v$ , hence $x \equiv y$ .

1.2L1L2assume-hyp

Conversely, assume condition 2. Apply it with $y$ equal to the zero member on the same domain as $x$ . Then $f x \equiv 0$ implies $x \equiv 0$ , so [L1] makes $f$ monic.

2.1step 1.1step 1.2∎

Therefore conditions 1 and 2 are equivalent.

Depends on

Used by

A member chase verifying monicity Example
Half nine lemma Lemma
Four lemma in an abelian category Theorem
Nine lemma in an abelian category Theorem
Nine lemma variants by which rows are assumed exact Theorem
Sharp nine lemma Theorem
Short five lemma in an abelian category Theorem
Weak four lemma with the exactness hypotheses named 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(ii) (standard reference, not scraped)