TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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The join of two subobjects is their least upper bound

Statement

Let $B$ and $C$ be subobjects of an object $A$ in an abelian category. Then the subobject $B \lor C$ of The join of two subobjects in an abelian category is the least upper bound of $B$ and $C$ in the subobject order of $A$ .

Facts & Assumptions

Given: Monomorphisms $b : B ↣ A$ and $c : C ↣ A$ representing the two subobjects.

[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 image of a morphism is the least subobject through which that morphism factors (The image is the least subobject through which a morphism factors).

[L3]

A subobject inequality is exactly factorization of representatives (Subobjects and quotient objects form oppositely oriented partially ordered collections).

Proof

technique · direct

1.1L1L2L3

Let $j : J ↣ A$ be the image inclusion of $[b, c]$ . Since $[b, c] ι_{B} = b$ and $[b, c] ι_{C} = c$ for the biproduct injections, the factorization of $[b, c]$ through $j$ makes both $b$ and $c$ factor through $j$ . Thus $B \leq J$ and $C \leq J$ , so $J$ is an upper bound of the two subobjects.

1.2L2L3construct

Let $n : N ↣ A$ be any common upper bound. Then $b = n u$ and $c = n v$ for suitable $u : B \to N$ and $v : C \to N$ . By the universal property of $B \oplus C$ , the induced map satisfies $[b, c] = n [u, v]$ . Now [L2] says that the image inclusion $j$ factors through every monomorphism through which $[b, c]$ factors, so $J \leq N$ .

2.1L3step 1.1step 1.2∎

Step 1.1 gives that $J$ is an upper bound, and step 1.2 gives that it lies below every upper bound. By [L3], this is exactly the least-upper-bound claim.

Depends on

Used by

The subobjects of an object in an abelian category form a lattice Theorem

Cited to discharge well-definedness by The join of two subobjects in an abelian category.

Dependency tree · two levels

11 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, Section 4.2 (standard reference, not scraped)