CounterexampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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Torsion-free abelian groups do not form an abelian category

Statement refuted

The full subcategory of torsion-free abelian groups is an abelian category.

Facts & Assumptions

Given: The full subcategory ${A b}_{t f}$ of torsion-free abelian groups.

[L1]

Torsion-free abelian groups form a full subcategory of $A b$ (Torsion-free abelian groups form a reflective full subcategory of abelian groups, Abelian groups and $Z$ -modules have the same objects and morphisms).

[L2]

Abelian categories are balanced (An abelian category is balanced).

Counterexample

1.1L1

In ${A b}_{t f}$ , multiplication by $2$ on $Z$ is monic and epic. Indeed, if $2 u = 2 v$ for maps into or out of a torsion-free group $T$ , then $2 (u (x) - v (x)) = 0$ for every $x$ , so torsion-freeness forces $u = v$ .

2.1L2step 1.1∎

The map $2 : Z \to Z$ is not an isomorphism in ${A b}_{t f}$ , because its inverse would have to send $1$ to $1 / 2$ , which is not an integer. If ${A b}_{t f}$ were abelian, [L2] would force every bimorphism to be an isomorphism. So the subcategory is not abelian.

Depends on

Used by

FALSE: if coimage and image happen to be isomorphic as objects, then the canonical map is automatically an isomorphism False statement

Dependency tree · two levels

26 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, Example 4.5.13 (standard reference, not scraped)