CounterexampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 2026-08-28

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The opposite of abelian groups does not satisfy AB5

Statement refuted

The opposite category ${A b}^{o p}$ satisfies AB5.

Facts & Assumptions

Given: The abelian category $A b$ .

[L1]

The category $A b$ does not satisfy AB5* (Abelian groups do not satisfy AB5*).

[L2]

The opposite of an abelian category is abelian (The opposite of an abelian category is abelian).

Counterexample

1.1L2algebra

By [L2], the opposite category ${A b}^{o p}$ is again abelian. Passing to the opposite exchanges joins with meets and AB5 with AB5*.

2.1L1step 1.1∎

If ${A b}^{o p}$ satisfied AB5, then $A b$ would satisfy AB5*. This contradicts [L1]. Therefore ${A b}^{o p}$ does not satisfy AB5.

Depends on

Used by

FALSE under the Axiom of Choice: AB4 implies AB5 False statement

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

Charles A. Weibel, An Introduction to Homological Algebra, Appendix A.4 (standard reference, not scraped)