LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-16

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.

Finite words satisfy the free-monoid universal property

Statement

For a set $X$ , let

$X^{*} : = ⋃_{n \in N} X^{n}$

be the set of finite words in letters from $X$ , where $X^{n}$ is the set of functions $n \to X$ . With concatenation and the empty word, $X^{*}$ is a monoid. The one-letter map $i_{X} : X \to X^{*}$ has the universal property that every function $u : X \to M$ into a monoid extends uniquely to a monoid homomorphism $\hat{u} : X^{*} \to M$ .

Facts & Assumptions

Given: A set $X$ , a monoid $(M, \cdot, e)$ , and a function $u : X \to M$ .

[F1]

A monoid is a set with an associative binary operation and a two-sided identity (Semigroup and monoid).

[F2]

A finite product in a monoid is uniquely defined by the recursion $P (0) = e$ and $P (n + 1) = P (n) g_{n}$ (The product $g_{0} g_{1} \dots g_{n - 1}$ of a finite list in a monoid, by recursion, with the empty product ( $n = 0$ ) equal to the identity).

[F3]

The natural numbers form the smallest inductive set (The natural numbers $N$ (von Neumann)).

[F4]

Natural addition satisfies $n + 0 = n$ and $n + (m + 1) = (n + m) + 1$ (Addition of natural numbers).

[F5]

Natural addition is associative: $(n + m) + r = n + (m + r)$ (Addition is associative).

[F6]

For sets $A, B$ , the functions $A \to B$ form a set (The set $B^{A}$ of all functions $A \to B$ ).

[F7]

The indexed union of a family $(A_{i})_{i \in I}$ is $⋃_{i \in I} A_{i} = ⋃ {A_{i} : i \in I}$ ( $⋃_{i \in I} A_{i} : = ⋃ {A_{i} : i \in I}$ , and $⋂_{i \in I} A_{i} : = ⋂ {A_{i} : i \in I}$ for $I \neq \emptyset$ ).

[F8]

If a property holds at $0$ and passes from $n$ to $n + 1$ , it holds for every natural number (The principle of mathematical induction).

Proof

technique · direct

1.1F3F6F7

Each $X^{n}$ is a set by [F6], and [F3] and [F7] make their indexed union $X^{*}$ a set. The unique function $0 \to X$ is the empty word.

1.2F4construct

For $p : n \to X$ and $q : m \to X$ , define $p q : n + m \to X$ by using $p$ on the first $n$ positions and $q$ on the following $m$ positions.

1.3F2construct

For a word $p : n \to X$ , define $\hat{u} (p)$ as the finite product $u (p (0)) \dots u (p (n - 1))$ , with value $e$ when $n = 0$ .

2.1step 1.1step 1.2F1F4F5

Function extensionality and [F5] show $(p q) r = p (q r)$ ; the equations in [F4] show that the empty word is a two-sided identity. Thus $X^{*}$ is a monoid by [F1].

2.2step 1.2step 1.3F1F2F8

Induction on the length of the second word, using [F2] and associativity in $M$ , proves $\hat{u} (p q) = \hat{u} (p) \hat{u} (q)$ . Hence $\hat{u}$ is a monoid homomorphism and extends $u$ on one-letter words.

3.1step 2.1step 1.3F1F8

If $T : X^{*} \to M$ is any homomorphism extending $u$ , induction on word length gives $T (p) = u (p (0)) \dots u (p (n - 1)) = \hat{u} (p)$ ; the base case uses the empty word and the identity of $M$ .

4.1step 2.2step 3.1∎

Thus the extension exists and is unique for every $u$ , including $X = \emptyset$ , where $X^{*}$ contains only the empty word.

Depends on

Used by

The free-monoid functor is left adjoint to the underlying-set functor Theorem

Dependency tree · two levels

27 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, 2nd ed., Example 4.1.10 (standard reference, not scraped)
Tom Leinster, Basic Category Theory, Example 2.1.3 (standard reference, not scraped)