TheoremStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-28‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

The subobject lattice of an abelian category is modular

Statement

For every object $A$ in an abelian category, the lattice of subobjects of $A$ is modular.

Facts & Assumptions

Given: An object $A$ in an abelian category.

[L1]

A modular lattice is one satisfying $x \leq z \Rightarrow x \lor (y \land z) = (x \lor y) \land z$ (Modular lattice).

[L2]

The subobjects of $A$ form a lattice (The subobjects of an object in an abelian category form a lattice).

[L3]

Quotienting by the kernel identifies a morphism with its image (First isomorphism theorem in an abelian category).

[L4]

Quotienting nested subobjects satisfies the third isomorphism theorem (Third isomorphism theorem in an abelian category).

[L5]

In an abelian category, a morphism that is both monic and epic is an isomorphism (An abelian category is balanced).

[L6]

The join of two subobjects is the image of the induced map from their biproduct (The join of two subobjects in an abelian category).

[L7]

The meet of two subobjects is represented by their pullback (The meet of two subobjects is their pullback).

Proof

technique · direct

1.1

Consider any lattice in which comparable complements of a fixed element coincide in every interval. Let $X \leq Z$ and put $U : = X \lor (Y \land Z), V : = (X \lor Y) \land Z .$ Then $U \leq V$ by monotonicity.

construct

1.2

Fix an interval $[B_{1}, B_{2}]$ of subobjects of $A$ , and write $Q : = B_{2} / B_{1}$ with quotient map $q : B_{2} \to Q$ . If $D$ is any intermediate subobject, let $q_{D} : B_{2} \to B_{2} / D$ be its quotient map. Because $B_{1} \leq D$ , the composite $q_{D}$ kills $B_{1}$ , so it factors through $q$ as a map ${\overline{q}}_{D} : Q \to B_{2} / D$ . Define $Φ (D)$ to be the kernel subobject of ${\overline{q}}_{D}$ .

L4construct

2.1

Both $U$ and $V$ are complements of $Y$ in the interval $[Y \land Z, X \lor Y]$ : one has $U \lor Y = X \lor Y = V \lor Y$ , and $Y \land Z \leq U \leq Z$ gives $U \land Y = Y \land Z$ , while $V \leq Z$ and $Y \leq X \lor Y$ give $V \land Y = Y \land Z$ . By step 1.1, the interval hypothesis forces $U = V$ . Therefore the modular-law identity of [L1] holds.

L1step 1.1algebra

2.2

Conversely, if $S \leq Q$ with quotient map $t_{S} : Q \to Q / S$ , define $Ψ (S)$ to be the kernel subobject of $t_{S} q$ in $B_{2}$ . Then $B_{1} \leq Ψ (S) \leq B_{2}$ . If $D$ lies in the interval, the equality $q_{D} = {\overline{q}}_{D} q$ gives $Ψ (Φ (D)) = D$ . If $S \leq Q$ , the morphism $t_{S} q$ is epic, so its image is all of $Q / S$ ; by [L3], $B_{2} / Ψ (S) ≅ Q / S .$ Under this identification, ${\overline{q}}_{Ψ (S)}$ is a cokernel of $S$ , so $Φ (Ψ (S)) = S$ . Thus $Φ$ and $Ψ$ are inverse bijections between $[B_{1}, B_{2}]$ and the subobject lattice of $Q$ .

L3L4step 1.2construct

3.1

If $D \leq E$ in the interval, then $q_{E}$ factors through $q_{D}$ , so ${\overline{q}}_{E}$ kills $Φ (D)$ . Hence $Φ (D) \leq Φ (E)$ . The same argument applied to $Ψ$ shows that $Φ$ is an order isomorphism from $[B_{1}, B_{2}]$ onto $Sub (Q)$ .

step 1.2step 2.2

4.1

By step 3.1, it is enough to prove the comparable-complements property in $Sub (Q)$ . Let $E \leq Q$ , and let $U \leq V$ be two complements of $E$ . Write $t : Q \to Q / E$ for the quotient map.

construct

5.1

Because $U \land E = 0$ , the pullback description [L7] makes the kernel of the restricted map $t ∣_{U} : U \to Q / E$ trivial. Because $U \lor E = Q$ , the canonical map $[i_{U}, i_{E}] : U \oplus E \to Q$ has image $Q$ by [L6], so composing with $t$ shows that $t ∣_{U}$ is epic. By [L5], $t ∣_{U}$ is an isomorphism. The same argument shows that $t ∣_{V}$ is an isomorphism. Let $i : U \to V$ represent the comparison $U \leq V$ . Then $t ∣_{U} = (t ∣_{V}) i$ , so $i$ is an isomorphism. Therefore $U$ and $V$ represent the same subobject of $Q$ .

L5L6L7step 4.1construct

6.1

Step 5.1 proves that every interval in the subobject lattice has the comparable-complements property, and step 2.1 shows that this property implies the modular law of [L1]. With [L2], this proves that the subobject lattice of $A$ is modular.

L1L2step 2.1step 3.1step 5.1∎

Depends on

Used by

A subobject lattice of an abelian category need not be distributive Counterexample
The subobject lattice of a cyclic group of order twelve Example
The published subgroup modular law is the instance Remark
Zassenhaus butterfly lemma in an abelian category Theorem

Dependency tree · two levels

22 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

Daniel Murfet, Abelian Categories, Proposition 73 and Corollary 72 (standard reference, not scraped)