False statementConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passaudited 2026-08-27

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FALSE: finite products and finite coproducts already force biproducts

Statement

False claim: if a category has finite products and finite coproducts, then those structures are automatically biproducts.

Facts & Assumptions

Given: The category ${S e t}_{*}$ of pointed sets and pointed maps.

[L1]

A biproduct requires the canonical coproduct-to-product morphism to be an isomorphism (Biproduct, Canonical morphism from a finite coproduct to a finite product).

Refutation

technique · direct

1.1

In ${S e t}_{*}$ , let $A = B = {0, 1}$ with basepoint $0$ . The binary coproduct $A \lor B$ has three points $0, a, b$ , while the binary product $A \times B$ has four points $(0, 0), (1, 0), (0, 1), (1, 1)$ .

given

2.1

The canonical map $A \lor B \to A \times B$ sends $0$ to $(0, 0)$ , $a$ to $(1, 0)$ , and $b$ to $(0, 1)$ , so it misses $(1, 1)$ . Therefore it is not surjective and hence not an isomorphism. By [L1], this pair has product and coproduct without being a biproduct.

L1step 1.1∎

Depends on

Used by

Pointed sets are not additive Counterexample

Dependency tree · two levels

4 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

Emily Riehl, Category Theory in Context, Chapter 3 (standard reference, not scraped)