Alphabeta MathHow statement and proof provenance work
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The axioms AB3 and AB3*
Definition
An abelian category satisfies AB3 when it has all small coproducts, and it satisfies AB3* when it has all small products (Finite, small, and large limits and colimits; complete and cocomplete categories).
Because an abelian category already has cokernels and kernels, the criterion A category is complete exactly when it has all small products and equalizers, and cocomplete exactly when it has all small coproducts and coequalizers identifies AB3 with cocompleteness and AB3* with completeness inside the abelian setting.
Depends on
Used by
- The axioms AB4 and AB4* Definition
- The axioms AB5 and AB5* Definition
- Equivalences preserve small projective generators Lemma
- Filtered colimits of abelian groups are exact Lemma
- The Hom functor of a small projective generator is exact, coproduct-preserving, and faithful Lemma
- A coproduct of projectives is projective and a product of injectives is injective Theorem
- The cancellation and epimorphism descriptions of a generator agree Theorem
Dependency tree · two levels
7 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)
- The Stacks Project, Tag 079B (standard reference, not scraped)