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.
Conservative functor
Definition
A functor is conservative if it reflects isomorphisms: whenever is a morphism of and is an isomorphism in (Isomorphism, groupoid, and connected category), the morphism is an isomorphism in .
Depends on
Used by
- An exact functor preserves quasi-isomorphisms and reflects them when it is conservative Corollary
- Torsion-free abelian groups give a conservative right adjoint that is not monadic Counterexample
- FALSE: every conservative right adjoint is monadic False statement
- FALSE: the underlying-set functor from topological spaces is monadic False statement
- Data-supplied crude monadicity theorem for reflexive coequalizers Theorem
- Every monadic functor is conservative Theorem
- The contravariant power-set functor is monadic Theorem
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
- E. Riehl, Category Theory in Context, 2nd ed., discussion following Lemma 5.6.1 (standard reference, not scraped)