TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 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.

Monoids and unital rings are strictly monadic over sets

Statement

The underlying-set functors from monoids and from unital rings to $S e t$ are strictly monadic, and hence monadic.

Facts & Assumptions

Given: The free-monoid and free-ring adjunctions.

[L1]

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

[L2]

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

[L3]

The underlying-set functor on unital rings strictly creates split coequalizers (The underlying-set functor on unital rings strictly creates split coequalizers).

[L4]

A right adjoint is strictly monadic if and only if it strictly creates coequalizers of its split pairs (Strict Beck monadicity theorem).

[L5]

Choosing a free monoid $(X^{*}, i_{X})$ on every set $X$ makes the finite-word functor left adjoint to the underlying-set functor, the adjunction bijection sending $φ : X^{*} \to M$ to $U (φ) i_{X}$ (The free-monoid functor is left adjoint to the underlying-set functor).

[L6]

The comparison functor is $K (d) = (U d, U ε_{d})$ and $K (h) = U (h)$ (The comparison functor to the Eilenberg–Moore category exists and is unique).

[L7]

An algebra $(A, a)$ for a monad $(T, η, μ)$ satisfies $a η_{A} = 1_{A}$ and $a T (a) = a μ_{A}$ , and an algebra homomorphism is a map commuting with the two structure maps (Algebra and algebra homomorphism for a monad).

Proof

technique · direct

1.1

By [L6], the comparison for the free-monoid adjunction is $K (M) = (U M, U ε_{M})$ with $K (h) = U (h)$ . Under [L5] the counit $ε_{M}$ corresponds to the identity of $U M$ , so it is the unique monoid homomorphism $(U M)^{*} \to M$ carrying each one-letter word $[m]$ to $m$ ; a homomorphism out of a free monoid is determined on the letters, so $ε_{M}$ evaluates a word in the elements of $M$ to its product.

L5L6construct

1.2

For rings, [L2] supplies the left adjoint and [L3] supplies strict creation of the required split coequalizers.

L2L3

2.1

Applying strict Beck [L4] to step 1.2 makes the ring underlying-set functor strictly monadic. The construction includes the empty generating set and the zero ring.

step 1.2L4

2.2

A function $f : U M \to U N$ commutes with word evaluation exactly when it is a monoid homomorphism, by evaluating the two-letter words and the empty word in one direction and every word in the other; with [L7] and $K (h) = U (h)$ this makes $K$ bijective on morphisms. It is injective on objects because step 1.1 recovers the product of $M$ from $U ε_{M}$ on two-letter words.

step 1.1L7algebra

2.3

$K$ is surjective on objects: for an algebra $(A, a)$ of the free-monoid monad put $x \cdot y : = a ([x, y])$ and $1 : = a ([])$ . Splitting a word into its first letter and its tail and applying the multiplication law $a T (a) = a μ_{A}$ of [L7] gives $a (w) = a ([x_{1}]) \cdot a (tail)$ , while the unit law gives $a ([x]) = x$ ; induction on length identifies $a$ with evaluation of words in these operations, and substituting the monoid-word identities into the same law gives associativity and the unit laws. Hence $A$ is a monoid $M_{A}$ with $a = U ε_{M_{A}}$ , that is $(A, a) = K (M_{A})$ .

step 1.1L7construct

3.1

By steps 2.2 and 2.3 the monoid comparison is bijective on objects and morphisms, hence an isomorphism of categories over $S e t$ — the isomorphism [L1] asserts may be taken to be it — so the monoid underlying-set functor is strictly monadic. With step 2.1 this proves the assertion for both concrete categories, and strict monadicity implies monadicity.

step 2.1step 2.2step 2.3L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

29 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)
D. Mehrle, Category Theory Part III, Example 5.18 (standard reference, not scraped)