TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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The free unital ring functor is left adjoint to the underlying-set functor

Statement

The free-word-ring construction $X \mapsto Z^{(X^{*})}$ extends to a functor $FreeRing : S e t \to R i n g$ left adjoint to the underlying-set functor. Its unit sends $x \in X$ to the basis element of the one-letter word $[x]$ .

Facts & Assumptions

Given: A set $X$ , its free word ring, a unital ring $R$ , and a function $u : X \to R$ .

[L1]

For every set $X$ , integer-valued finite formal sums of words in $X$ form a unital ring (Integer-valued finite formal sums of words form unital convolution rings).

[L2]

Every map from a basis set to a module extends uniquely to a linear map from the corresponding free module (Universal property of the free module on a set).

[L3]

Supplied objectwise universal arrows assemble uniquely into a left adjoint (Chosen objectwise universal arrows assemble uniquely into a left adjoint).

Proof

technique · direct

1.1

Send a word $[x_{1}, \dots, x_{n}]$ to $u (x_{1}) \dots u (x_{n})$ in $R$ , and send the empty word to $1_{R}$ . The generator $x$ is represented by the basis word $[x]$ .

L1construct

2.1

Apply [L2] over $Z$ to extend this word-evaluation function uniquely to a $Z$ -linear map $\overset{ˉ}{u} : Z^{(X^{*})} \to R$ .

step 1.1L2

3.1

Expanding convolution as a finite sum shows $\overset{ˉ}{u} (α β) = \overset{ˉ}{u} (α) \overset{ˉ}{u} (β)$ , and the empty-word basis vector maps to $1_{R}$ . Thus $\overset{ˉ}{u}$ is a unit-preserving ring homomorphism, including when $R$ is the zero ring.

step 2.1L1algebra

4.1

Any ring homomorphism extending $u$ must send every word basis vector to the corresponding product and is additive, so it equals $\overset{ˉ}{u}$ by uniqueness in [L2].

step 3.1L2

5.1

Hence the generator inclusion is a universal arrow from every set $X$ to the underlying-set functor on rings. By [L3] these universal arrows assemble into the free-ring functor and the asserted adjunction; for $X = \emptyset$ the free ring is $Z$ .

step 4.1L3∎

Depends on

Used by

Monoids and unital rings are strictly monadic over sets Theorem

Dependency tree · two levels

21 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

E. Riehl, Category Theory in Context, 2nd ed., Corollary 5.5.3 (standard reference, not scraped)
E. Riehl, Category Theory in Context, 2nd ed., Example 4.1.10(vi) (standard reference, not scraped)
D. Mehrle, Category Theory Part III, Example 5.18 (standard reference, not scraped)