PropositionStatement: AI-adaptedProof: AI-generatedprecheck 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 $C$ .

[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 $Ob C$ , 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 $S$ be the full subcategory on the selected objects.

step 1.1L1L2choose

3.1

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

step 2.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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)