CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)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.

Inverse image preserves meets and direct image preserves joins

Statement

Let $f : A \to A^{'}$ be a morphism in an abelian category. Then for subobjects $C_{1}, C_{2} \leq A^{'}$ and $B_{1}, B_{2} \leq A$ one has

$f^{*} (C_{1} \land C_{2}) = f^{*} C_{1} \land f^{*} C_{2}, f_{*} (B_{1} \lor B_{2}) = f_{*} B_{1} \lor f_{*} B_{2} .$

Facts & Assumptions

Given: A morphism $f : A \to A^{'}$ and the displayed subobjects.

[L1]

The subobject maps $f_{*}$ and $f^{*}$ form a Galois connection (Direct and inverse image of subobjects form a Galois connection).

[L2]

Subobjects form lattices, so meets and joins are characterized by their order universal properties (The subobjects of an object in an abelian category form a lattice).

Proof

technique · direct

1.1L1L2algebra

For any subobject $B \leq A$ , $B \leq f^{*} (C_{1} \land C_{2}) ⟺ f_{*} B \leq C_{1} \land C_{2} ⟺ f_{*} B \leq C_{1} and f_{*} B \leq C_{2} ⟺ B \leq f^{*} C_{1} and B \leq f^{*} C_{2} .$ By [L2], this says that $f^{*} (C_{1} \land C_{2})$ is the meet of $f^{*} C_{1}$ and $f^{*} C_{2}$ .

1.2L1L2algebra

For any subobject $C \leq A^{'}$ , $f_{*} (B_{1} \lor B_{2}) \leq C ⟺ B_{1} \lor B_{2} \leq f^{*} C ⟺ B_{1} \leq f^{*} C and B_{2} \leq f^{*} C ⟺ f_{*} B_{1} \leq C and f_{*} B_{2} \leq C .$ Again [L2] identifies this with the universal property of the join $f_{*} B_{1} \lor f_{*} B_{2}$ .

2.1step 1.1step 1.2∎

Steps 1.1 and 1.2 are exactly the two displayed identities.

Depends on

Used by

Images and preimages of submodules form a concrete Galois connection Example

Dependency tree · two levels

10 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 V.7 (standard reference, not scraped)