Alphabeta Math
PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11
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  • 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.

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Groups and group homomorphisms form the large locally small category Grp\mathbf{Grp}

Statement

Groups and group homomorphisms form a large locally small category Grp\mathbf{Grp}.

Facts & Assumptions

Given: Groups G,H,KG,H,K and homomorphisms f:G→Hf:G\to H, g:H→Kg:H\to K.

[L1]

Groups are sets with the group operations (Group and abelian group), and identity maps and composites of group homomorphisms are group homomorphisms (Monoid homomorphism and group homomorphism).

[L2]

The functions A→BA\to B between fixed sets form the set BAB^A (The set BAB^{A} of all functions A→BA \to B); the category and size conditions are Category, object, morphism, domain, codomain, identity, composition, and hom-collection and Small, locally small, and large categories, while the ordinals form a proper class (Burali-Forti: there is no set of all ordinals).

Proof

technique · direct
1.1

By [L1], identity maps and composites remain group homomorphisms; their associativity and unit equations are the corresponding equations for functions.

givenL1
2.1

Thus the group objects and homomorphisms satisfy the category axioms, and each hom-collection is a set of functions between two fixed underlying sets.

step 1.1L2
3.1

For every ordinal α\alpha, the singleton {α}\{\alpha\} carries a transported trivial group structure, and distinct ordinals give distinct group objects; [L2] therefore rules out a set of all group objects, so Grp\mathbf{Grp} is large and locally small.

step 2.1L2∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 34 results over 15 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