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.
Conjugation preserves invertibility but the general mates correspondence need not
When both adjunctions in Natural transformations have mates under a pair of adjunctions are adjoint equivalences, all unit and counit components in the mate formulas are isomorphisms. A mate is then obtained from the original transformation by whiskering and composing with isomorphisms, so invertibility is preserved. For a general adjunction the unit or counit need not be invertible; an identity transformation can have such a component as its mate. The mates bijection therefore preserves equations and typed pasting, not invertibility in general.
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Dependency tree · two levels
6 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, 2nd ed., Exercise 4.3.iv (standard reference, not scraped)