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

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 comparison functor for the free-group adjunction

Example

For the free-group adjunction, the comparison sends a group $G$ to the algebra whose carrier is $U G$ and whose structure map evaluates a reduced word in elements of $G$ to its product. It sends each group homomorphism to its underlying function.

For $C_{2} = Z / 2 Z$ , the word $[\overset{ˉ}{1}, \overset{ˉ}{1}, \overset{ˉ}{1}]$ evaluates to $\overset{ˉ}{1}$ , while every adjacent inverse pair and every occurrence of the identity may be removed before evaluation.

Facts & Assumptions

Given: The free-group adjunction and a group $G$ .

[L1]

The Eilenberg–Moore category of the free-group monad is isomorphic over $S e t$ to the category of groups (The free-group monad has groups as its Eilenberg–Moore algebras).

[L2]

The comparison sends $d$ to $(U d, U ε_{d})$ and sends a morphism to its image under $U$ (The comparison functor to the Eilenberg–Moore category exists and is unique).

[L3]

The quotient group $(Z, +) / 2 Z$ is $(Z / 2, +)$ with addition of congruence classes (For every $n \in N$ , the congruence-class group $(Z / n, +)$ is the quotient group $(Z, +) / n Z$ ).

[L4]

Choosing a free group $(F (X), i_{X})$ on every set $X$ makes $F$ left adjoint to the underlying-set functor $U$ , the adjunction bijection sending $φ : F (X) \to G$ to $U (φ) i_{X}$ (The free-group functor is left adjoint to the underlying-set functor).

[L5]

An algebra homomorphism is a map commuting with the two algebra structure maps (Algebra and algebra homomorphism for a monad).

Verification

technique · direct

1.1

By [L2], the structure map is the underlying counit $U ε_{G}$ of the free-group adjunction. Under [L4] the counit corresponds to the identity of $U G$ , so $ε_{G}$ is the unique group homomorphism $F (U G) \to G$ carrying each basis element $g$ to $g$ ; a homomorphism out of a free group is determined by its values on the basis, so it evaluates a reduced word in $U G$ to its product in $G$ .

L2L4

1.2

By [L5] the algebra-homomorphism equation says that a function commutes with evaluation of every word. Evaluating the words $[x, y]$ , the empty word and $[x^{- 1}]$ makes such a function preserve product, identity and inverse, and conversely a group homomorphism preserves the value of every word; so the comparison morphisms are exactly group homomorphisms, in agreement with [L1] and with $K (h) = U (h)$ in [L2].

L1L2L5

2.1

Evaluating a one-letter word returns its letter, and evaluating after substitution of words agrees with evaluating the flattened word by associativity of group multiplication. These are the algebra unit and multiplication laws.

step 1.1algebra

3.1

In the group [L3], one has $\overset{ˉ}{1} + \overset{ˉ}{1} = \overset{ˉ}{0}$ and $\overset{ˉ}{0} + \overset{ˉ}{1} = \overset{ˉ}{1}$ , so the displayed three-letter word evaluates to $\overset{ˉ}{1}$ .

step 1.1L3construct∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

26 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., Example 5.1.4(iv) and Section 5.3 (standard reference, not scraped)