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.
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A monoid is a one-object category, and a group is a one-object category in which every morphism is invertible

Statement

Every monoid is a one-object category. Under this identification, the monoid is a group exactly when every morphism is invertible.

Facts & Assumptions

Given: A monoid (M,⋅,e)(M,\cdot,e).

[L1]

A monoid has associative multiplication and a two-sided identity (Semigroup and monoid), exactly the composition and identity laws required by Category, object, morphism, domain, codomain, identity, composition, and hom-collection.

[L2]

A group is a monoid in which every element has a two-sided inverse (Group and abelian group).

Proof

technique · direct
1.1

Take one object ∗*, put Hom⁡(∗,∗)=M\operatorname{Hom}(*,*)=M, define composition by y∘x=y⋅xy\circ x=y\cdot x, and take 1∗=e1_*=e; [L1] verifies the category axioms.

givenL1
2.1

A morphism x:∗→∗x:*\to * is an isomorphism precisely when some yy satisfies yx=e=xyyx=e=xy.

step 1.1L1
3.1

By [L2], every morphism in this one-object category is invertible exactly when MM is a group.

step 2.1L2∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 18 results over 11 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