TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-08-28

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Second isomorphism theorem in an abelian category

Statement

Let $B$ and $C$ be subobjects of an object $A$ in an abelian category. Then there is a canonical isomorphism

$(B \lor C) / C ≅ B / (B \land C) .$

Facts & Assumptions

Given: Subobjects $b : B ↣ A$ and $c : C ↣ A$ .

[L1]

The join $B \lor C$ is the image of the induced map $[b, c] : B \oplus C \to A$ (The join of two subobjects in an abelian category).

[L2]

The meet $B \land C$ is represented by the pullback of $b$ and $c$ (The meet of two subobjects is their pullback).

[L3]

A morphism modulo its kernel is canonically isomorphic to its image (First isomorphism theorem in an abelian category).

Proof

technique · direct

1.1L2construct

Let $q : A \to A / C$ be the quotient map. Consider the composite $q b : B \to A / C$ . By [L2], a morphism into $B$ is killed by $q b$ exactly when its composite into $A$ factors through $C$ , which is exactly the pullback condition defining $B \land C$ . So $\ker (q b) = B \land C$ .

2.1L1L3step 1.1

By [L3], step 1.1 gives a canonical isomorphism $B / (B \land C) ≅ im (q b) .$ The map $q$ kills $C$ , so its restriction to the join $B \lor C$ factors through the quotient $(B \lor C) / C$ . Conversely, every summand used in the defining map $[b, c]$ lands in $im (q b)$ after composing with $q$ , because the $C$ -summand dies. Hence $im (q b)$ is exactly the image of $B \lor C$ in $A / C$ , namely $(B \lor C) / C$ .

3.1step 1.1step 2.1∎

Combining steps 1.1 and 2.1 yields the canonical isomorphism $(B \lor C) / C ≅ B / (B \land C)$ .

Depends on

Used by

Length is additive along a subobject Theorem
Zassenhaus butterfly lemma in an abelian category Theorem

Dependency tree · two levels

13 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)