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.
Embedding and full embedding of categories
Definition
An embedding of categories is a functor that is faithful and injective on objects. A full embedding is an embedding that is also full. These terms use the notions of Faithful, full, fully faithful, essentially surjective, and split essentially surjective functors.
Thus a fully faithful functor need not be an embedding: it may send distinct but isomorphic objects to the same object. Conversely, an embedding need not be full.
Depends on
Used by
- Abelian subcategory and exact embedding Definition
- The Yoneda assignment and the small-source Yoneda functor, traditionally called the Yoneda embedding Definition
- Freyd-Mitchell gives a fully faithful exact functor from every small abelian category to a module category Remark
- The Yoneda functor is fully faithful, and it is a full embedding when its object map is injective 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
- Emily Riehl, Category Theory in Context, Chapter 1 (standard reference, not scraped)