Alphabeta MathHow 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.
FALSE under the Axiom of Choice: AB4 implies AB5
Statement
Assume the Axiom of Choice. Then AB4 implies AB5.
Facts & Assumptions
Given: The Axiom of Choice and the opposite category .
The Axiom of Choice (The Axiom of Choice).
The category does not satisfy AB5 (The opposite of abelian groups does not satisfy AB5).
AB4 is the coproduct-monomorphism axiom (The axioms AB4 and AB4*).
Assuming the Axiom of Choice, satisfies AB4* (Assuming the Axiom of Choice, abelian groups satisfy AB4*).
The opposite of an abelian category is abelian (The opposite of an abelian category is abelian).
Refutation
By [L4], the opposite category is abelian. Passing to the opposite exchanges AB4 with AB4*, so [L3] implies that satisfies AB4 in the sense of [L2].
But [L1] shows that does not satisfy AB5. So, even under [A1], AB4 does not imply AB5.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
11 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)