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.
Subcategory and full subcategory
Definition
A subcategory of a category (Category, object, morphism, domain, codomain, identity, composition, and hom-collection) has a subclass of the objects of and, for each , a subclass . It contains the identities of its objects and is closed under the composition inherited from .
The subcategory is full when for every pair of its objects. Thus a full subcategory is determined entirely by its objects.
Depends on
Used by
- Reflective full subcategory and reflector Definition
- Skeletal category and skeleton Definition
- Chosen bases exhibit Mat_F as equivalent to finite-dimensional vector spaces Example
- For n≥ 1, determinant is a natural transformation det:GLₙ(-)⟹(-)^× from commutative rings to groups Example
- FALSE: every functor on CᵒᵖtimesC has an end False statement
- FALSE: every functor preserves the ends that exist in its domain False statement
- An adjunction restricts to an equivalence on the subcategories fixed by its unit and counit Proposition
- A complete locally small category with a jointly weakly initial set has an initial object, without class-indexed choice Theorem
- The counit of a reflection is an isomorphism Theorem
- Under the ultrafilter lemma and dependent choice, Stone-Cech compactification is left adjoint to the compact-Hausdorff inclusion Theorem
Dependency tree · two levels
3 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)