DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-31

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 category of binary words

Definition

The binary words are generated recursively from:

the empty word $e_{0}$ ;
the one-letter word $(-)$ ;
if $v$ and $w$ are binary words, a new word $(v □ w)$ .

Their length is defined recursively by $∣ e_{0} ∣ = 0, ∣ (-) ∣ = 1, ∣ v □ w ∣ = ∣ v ∣ + ∣ w ∣ .$

The category of binary words $W$ has these words as objects. For objects $v, w$ , $W (v, w) = {\begin{cases} {*_{v, w}}, & ∣ v ∣ = ∣ w ∣, \\ \emptyset, & ∣ v ∣ \neq ∣ w ∣ . \end{cases}$

Composition and identities are forced by this rule: when the hom-collection is nonempty there is exactly one possible composite and exactly one possible identity arrow. Thus $W$ is a category in the sense of Category, object, morphism, domain, codomain, identity, composition, and hom-collection.

Depends on

Category, object, morphism, domain, codomain, identity, composition, and hom-collection

Used by

The free-word formulation implies the canonical-map formulation Corollary
The word category on words of length three Example
The category of binary words is monoidal Theorem
The word category is the free monoidal category on one generator Theorem

Dependency tree · two levels

3 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

S. Mac Lane, Categories for the Working Mathematician, Chapter VII.2 (standard reference, not scraped)