PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck 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 $C$ , the subcategory with all objects and exactly the isomorphisms as morphisms is a groupoid, and it contains every subgroupoid of $C$ .

Facts & Assumptions

Given: A category $C$ .

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

step 2.1L1∎

Depends on

Used by

The inclusion of groupoids into categories is left adjoint to the maximal-subgroupoid functor Example

Dependency tree · two levels

4 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)