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.
Faithful, full, fully faithful, essentially surjective, and split essentially surjective functors
Definition
Let be a functor (Covariant functor, identity functor, composite functor, and contravariant functor). For each , it induces
The functor is faithful when every is injective, full when every is surjective, and fully faithful when every is bijective (Injection, surjection, bijection).
It is essentially surjective when every object of is isomorphic to some (Isomorphism, groupoid, and connected category). It is split essentially surjective when the data include, for every , a specified object and specified isomorphism . The word split records these witnesses and is strictly stronger as data than their mere existence.
Depends on
Used by
- The Kleisli and Eilenberg–Moore categories of an idempotent monad are equivalent Corollary
- A fully faithful left Kan extension that is not pointwise Counterexample
- A limit in a full subcategory need not be the ambient limit Counterexample
- The inclusion of one object into a discrete two-object category is fully faithful but not essentially surjective Counterexample
- Dense subcategory Definition
- Embedding and full embedding of categories Definition
- A left Kan extension along a full subcategory inclusion of preorders Example
- Chosen bases exhibit Mat_F as equivalent to finite-dimensional vector spaces Example
- The idempotent completion of the matrix category gives the finitely generated projective modules Example
- FALSE: a left Kan extension along a fully faithful functor always restricts back to the original functor False statement
- FALSE: every Kan extension is pointwise False statement
- FALSE: The Kleisli and Eilenberg–Moore categories are equivalent for every monad False statement
- Under the Axiom of Choice, essential surjectivity onto a small category admits a splitting Lemma
- Every fully faithful functor reflects isomorphisms Proposition
- Fully faithful functors reflect limits and colimits Proposition
- A functor is an equivalence exactly when it is fully faithful and split essentially surjective, without Choice Theorem
- A pointwise Kan extension along a fully faithful functor genuinely extends the original functor Theorem
- Fullness and faithfulness of a right adjoint are detected by its counit Theorem
- In a locally small category, separating and coseparating sets are equivalently jointly faithful families of representables Theorem
- The comparison from the Kleisli category is fully faithful with image the free algebras Theorem
- The counit of a reflection is an isomorphism Theorem
- The matrix category is fully faithful in modules and, with chosen bases, equivalent to finite free modules Theorem
- The Yoneda functor is fully faithful, and it is a full embedding when its object map is injective Theorem
Dependency tree · two levels
7 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)
- Ahrens, Kapulkin and Shulman, Univalent categories and the Rezk completion, section 6 (standard reference, not scraped)