Alphabeta MathHow 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.
Why 'every diagram commutes' is false as stated
Remark
Mac Lane's coherence theorem is not the slogan that every diagram in a monoidal category commutes. The theorem speaks only about diagrams whose arrows are canonical in the sense of Canonical morphisms between parenthesised tensor words and whose vertices are formal parenthesisations of one ordered tensor word.
The warning matters because formally different vertices can evaluate to the same object in a particular monoidal category. Once that happens, one may insert a noncanonical endomorphism of that object, and there is no reason for the resulting square or polygon to commute. So the theorem is about formal structural maps, not arbitrary parallel arrows.
Depends on
Used by
- Two formally distinct words can become the same object Counterexample
- Every diagram in a monoidal category commutes False statement
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
- S. Mac Lane, Categories for the Working Mathematician, Chapter VII.2 (standard reference, not scraped)