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.
Small, locally small, and large categories
Definition
Let be a category (Category, object, morphism, domain, codomain, identity, composition, and hom-collection) under the class convention of Class-sized category theory in ZFC: definable-class schemas, small and locally small categories, and why is not formed.
- is small when both and are sets.
- is locally small when every is a set.
- is large when it is not small.
A small category is locally small because each hom-collection is a subclass of the set of all morphisms. A large category may or may not be locally small.
Depends on
Used by
- Every category is locally small False statement
- Under the Axiom of Choice, essential surjectivity onto a small category admits a splitting Lemma
- Groups and group homomorphisms form the large locally small category Grp Proposition
- If mathcal C is small and mathcal D is locally small then [mathcal C,mathcal D] is locally small; if both are small it is small Proposition
- Left modules over a fixed ring and module homomorphisms form the large locally small category R-Mod Proposition
- Posets and monotone maps form the large locally small category Poset Proposition
- Sets and functions form the large locally small category Set Proposition
- Topological spaces and continuous maps form the large locally small category Top Proposition
- Under the Axiom of Choice, a connected small groupoid is equivalent to the automorphism group of any one of its objects Proposition
- Under the Axiom of Choice, every small category has a skeleton Proposition
- Unital rings and unit-preserving ring homomorphisms form the large locally small category Ring Proposition
- Vector spaces over a fixed field and linear maps form the large locally small category Vect_F Proposition
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 17 results over 7 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- Emily Riehl, Category Theory in Context, Chapter 1 (standard reference, not scraped)