DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-29

How 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.
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Split short exact sequence in an abelian category

Definition

A short exact sequence $0 \to A \overset{i}{\to} B \overset{p}{\to} C \to 0$ in an abelian category is split when there exist morphisms $s : C \to B, π : B \to A$ such that $p s = 1_{C}, π i = 1_{A}, i π + s p = 1_{B} .$

Equivalently, the five-tuple $(B, i, s, π, p)$ is the biproduct diagram of $A$ and $C$ in the sense of Biproduct. The point of the definition is the displayed compatible structure, not merely an abstract isomorphism $B ≅ A \oplus C$ .

Depends on

Used by

A non-split short exact sequence of abelian groups Counterexample
FALSE: a short exact sequence splits whenever its middle object is isomorphic to the biproduct of the outer two False statement
Splitting lemma in an abelian category Theorem

Dependency tree · two levels

4 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, Definition 12.5.9 (standard reference, not scraped)
Saunders Mac Lane, Categories for the Working Mathematician, VIII.4 (standard reference, not scraped)