ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-09-01

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.

Commutative monoid objects in sets are ordinary commutative monoids

Example

A commutative monoid object in $(S e t, \times)$ is exactly an ordinary commutative monoid.

Facts & Assumptions

Given: Sets with cartesian product.

[L1]

Monoid objects in a cartesian monoidal category are ordinary monoids on the underlying sets (Monoid objects in a cartesian category are internal monoids; in Set they are ordinary monoids).

[L2]

In a symmetric monoidal category, monoid objects inherit a symmetric tensor product, so commutativity is expressed by invariance under the symmetry (Monoid objects in a symmetric monoidal category form a symmetric monoidal category).

Verification

technique · direct

1.1

By [L1], a monoid object in $(S e t, \times)$ is a set $M$ with a multiplication map $m : M \times M \to M$ and a unit element satisfying the usual associative and unital equations.

L1given

2.1

The symmetry on $S e t$ is the swap map $(a, b) \mapsto (b, a)$ , so the categorical commutativity condition says $m (a, b) = m (b, a)$ for all $a, b \in M$ . By [L2], that is exactly the ordinary commutativity law.

L2step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, Example 8.2.1 (standard reference, not scraped)