ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 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.

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The swap map on sets is the cartesian braiding

Example

In $S e t$ with cartesian product, the braiding on $X \times Y$ is the swap map

$τ_{X, Y} (x, y) = (y, x) .$

Facts & Assumptions

Given: The category of sets with cartesian product.

[L1]

The cartesian swap braiding is a symmetry in every category with finite products (The cartesian swap braiding is a symmetry).

Verification

technique · direct

1.1

The category $S e t$ has finite products, namely cartesian products of sets.

givenalgebra

2.1

Therefore [L1] applies with $C = S e t$ , and the resulting braiding is exactly the swap map $(x, y) \mapsto (y, x)$ .

L1step 1.1∎

Depends on

The cartesian swap braiding is a symmetry

Used by

Nothing in the library uses this result yet.

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

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