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.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Separating and coseparating sets of objects
Definition
Let be a category. A set of objects of is a separating set if, whenever are distinct morphisms, there are and with .
A set of objects is a coseparating set if, whenever are distinct, there are and with . A single object whose singleton family has the relevant property is called a separating or coseparating object.
Depends on
Used by
- A complete locally small category with no small coseparating set Counterexample
- Generator and cogenerator of a category Definition
- Abelian sheaves form a Grothendieck category Lemma
- Enough injective sheaves of modules Lemma
- A complete locally small category with a small coseparating set and intersections of all subobject collections has an initial object Theorem
- In a locally small category, separating and coseparating sets are equivalently jointly faithful families of representables Theorem
- Special adjoint functor theorem, objectwise form with explicit intersection smallness or preservation data Theorem
- Under dependent choice, the unit interval is a coseparating object in compact Hausdorff spaces 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, section 4.7 (standard reference, not scraped)
- S. Mac Lane, Categories for the Working Mathematician, section V.8 (standard reference, not scraped)