Alphabeta MathHow statement and proof provenance work
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The pullback and pushout theorems
Statement
In an abelian category:
- pullbacks of epimorphisms are epimorphisms;
- pushouts of monomorphisms are monomorphisms;
- in a pullback square, the induced map on kernels of the parallel arrows is an isomorphism;
- a commuting square is cartesian exactly when the associated short sequence is exact;
- a cartesian square over an epimorphism is also cocartesian.
Facts & Assumptions
Given: The named pullback and pushout situations in the statement.
Each of the five claims has already been proved under the displayed names (The pullback of an epimorphism is an epimorphism, The pushout of a monomorphism is a monomorphism, In a pullback square, the induced map on the kernels of the two parallel arrows is an isomorphism, A square is cartesian exactly when a short sequence is exact, A cartesian square over an epimorphism is also cocartesian).
Proof
The first claim is [L1]'s pullback-of-epimorphism theorem. The second claim is its pushout-of-monomorphism dual. The third claim is its kernel-comparison theorem.
The fourth claim is [L1]'s exact-square criterion, and the fifth claim is the cartesian-implies-cocartesian theorem over an epimorphism.
Hence the pullback and pushout results actually used by the diagram-lemma proofs are exactly the previously published theorems listed above.
Depends on
- The pullback of an epimorphism is an epimorphism
- The pushout of a monomorphism is a monomorphism
- In a pullback square, the induced map on the kernels of the two parallel arrows is an isomorphism
- A square is cartesian exactly when a short sequence is exact
- A cartesian square over an epimorphism is also cocartesian
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
17 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
- Peter Freyd, Abelian Categories, Sections 2.5 and 2.6 (standard reference, not scraped)
- The Stacks Project, Section 12.5, Lemmas 12.5.11 to 12.5.13 (standard reference, not scraped)