DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-28

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.
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.

Direct and inverse image of a subobject

Definition

Let $f : A \to A^{'}$ be a morphism in an abelian category.

If $b : B ↣ A$ represents a subobject of $A$ , its direct image along $f$ is the subobject of $A^{'}$ represented by the image inclusion of the composite

$B ↣ A \overset{f}{\to} A^{'} .$

It is denoted $f_{*} B$ .

If $c : C ↣ A^{'}$ represents a subobject of $A^{'}$ , its inverse image along $f$ is the subobject of $A$ represented by the pullback of $c$ along $f$ (Pullbacks and pushouts as limits and colimits of cospans and spans). It is denoted $f^{*} C$ .

The representative-independence of these assignments is discharged by Direct and inverse image of subobjects form a Galois connection ↗.

Depends on

Used by

Dependency tree · two levels

8 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

Saunders Mac Lane, Categories for the Working Mathematician, Section VIII.3 (standard reference, not scraped)