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The pushout of a monomorphism is a monomorphism
Statement
In an abelian category, the pushout of a monomorphism is a monomorphism.
Facts & Assumptions
Given: An abelian category and a pushout square whose left leg is monic.
The opposite of an abelian category is abelian (The opposite of an abelian category is abelian).
In an abelian category, pullbacks of epimorphisms are epimorphisms (The pullback of an epimorphism is an epimorphism).
Proof
Passing to the opposite category turns the given pushout square into a pullback square, and the given monomorphism into an epimorphism. By [L1], the opposite category is still abelian.
Apply [L2] in the opposite category. The opposite of the resulting epimorphism is exactly the pushout leg in the original square, so that leg is monic.
Depends on
Used by
- Bounded below complexes admit injective replacements Lemma
- The one-step generator map is a functorial monomorphism Lemma
- Higher Yoneda Ext agrees with derived Ext Theorem
- The connecting morphism exists and is unique Theorem
- The pullback and pushout theorems Theorem
- The Yoneda product is associative and unital Theorem
Dependency tree · two levels
9 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
- The Stacks Project, Section 12.5, Lemma 12.5.14 (standard reference, not scraped)