PropositionStatement: Literature-sourcedProof: AI-adaptedSession-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.

The isomorphisms in a category form its maximal subgroupoid

Statement

For every category $\mathcal C$ , the subcategory with all objects and exactly the isomorphisms as morphisms is a groupoid, and it contains every subgroupoid of $\mathcal C$ .

Facts & Assumptions

Given: A category $\mathcal C$ .

[L1]

Isomorphisms and groupoids are defined in Isomorphism, groupoid, and connected category, and category composition is associative and unital (Category, object, morphism, domain, codomain, identity, composition, and hom-collection).

Proof

technique · direct

1.1

Identities are isomorphisms, and if $f$ and $g$ are composable isomorphisms then $g\circ f$ has inverse $f^{-1}\circ g^{-1}$ , so the isomorphisms form a subcategory.

givenL1

2.1

Every morphism of this subcategory is invertible by construction, hence it is a groupoid.

step 1.1L1

3.1

If $\mathcal G$ is any subgroupoid contained in $\mathcal C$ , every morphism of $\mathcal G$ has its two-sided inverse in $\mathcal C$ and is therefore an isomorphism of $\mathcal C$ ; thus $\mathcal G$ lies in this subgroupoid, proving maximality.

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: 12 results over 6 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)