TheoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-08-27

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Biproduct data characterisation without addition

Statement

Let a category with zero morphisms contain a finite family $(A_{i})_{i \in I}$ and an object $B$ such that $(B, ι_{i})$ is a coproduct and $(B, p_{i})$ is a product. Let $c : B \to B$ be the canonical comparison determined by these supplied structures. Then $c = 1_{B}$ if and only if

$p_{j} ι_{i} = {\begin{cases} 1_{A_{i}}, & i = j, \\ 0_{A_{i}, A_{j}}, & i \neq j . \end{cases}$

Facts & Assumptions

Given: A finite family $(A_{i})_{i \in I}$ with maps $ι_{i} : A_{i} \to B$ and $p_{i} : B \to A_{i}$ in a category with zero morphisms.

[L1]

A biproduct is a coproduct and a product whose canonical comparison is an isomorphism (Biproduct).

[L2]

Products and coproducts have their universal properties (Products and coproducts as limits and colimits of discrete diagrams, including their existence-and-uniqueness equations).

Proof

technique · direct

1.1L1L2given

If $c = 1_{B}$ , then the defining equations $p_{j} c ι_{i} = δ_{i j}$ for the canonical comparison immediately give the displayed zero equations for $p_{j} ι_{i}$ .

1.2L1L2

Conversely, assume the displayed equations hold. For every $i, j$ , the definition of $c$ and the assumed equations give $p_{j} c ι_{i} = p_{j} ι_{i}$ . Therefore $c ι_{i} = ι_{i}$ by the product universal property. Since the $ι_{i}$ form a coproduct, $c = 1_{B}$ . The supplied structures therefore form the normalized biproduct diagram of Biproduct.

2.1step 1.1step 1.2given∎

Steps 1.1 and 1.2 prove the equivalence. No addition on hom-sets was used anywhere; only the given zero morphisms and the product-coproduct universal properties entered.

Depends on

Used by

Dependency tree · two levels

5 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, VIII.2 (standard reference, not scraped)
The Stacks Project, Section 12.3: Preadditive and additive categories (standard reference, not scraped)