PropositionStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

How 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.

Under the Axiom of Choice, every small category has a skeleton

Statement

Assume the Axiom of Choice. Every small category has a skeleton.

Facts & Assumptions

Given: A small category $\mathcal C$ .

[L1]

A skeleton is a full skeletal subcategory containing one representative of every isomorphism class (Skeletal category and skeleton).

[L2]

The object collection of a small category is a set (Small, locally small, and large categories), and Choice selects from a set-indexed family of nonempty sets (The Axiom of Choice).

Proof

technique · direct

1.1

Isomorphism defines an equivalence relation on the set $\operatorname{Ob}\mathcal C$ , so its quotient is a set whose members are nonempty isomorphism classes.

givenL1L2

2.1

By [L2], choose one object from each class and let $\mathcal S$ be the full subcategory on the selected objects.

step 1.1L1L2choose

3.1

Every object of $\mathcal C$ is isomorphic to its selected representative, while two selected isomorphic objects lie in the same class and are equal; hence $\mathcal S$ is a skeleton.

step 2.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 15 results over 5 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)
Ahrens, Kapulkin and Shulman, Univalent categories and the Rezk completion, section 6 (standard reference, not scraped)