TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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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.
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The Eilenberg–Moore forgetful functor creates every colimit in the base that the monad and its square preserve

Statement

Let $D : J \to C^{T}$ be a diagram of algebras whose underlying diagram has a colimit $(Q, i_{j})$ in $C$ . If both $T$ and $T^{2}$ preserve this colimit, then there is a unique algebra structure on $Q$ making every $i_{j}$ an algebra homomorphism, and the resulting cocone is a colimit in $C^{T}$ . Thus $U^{T}$ creates every such colimit.

Facts & Assumptions

Given: The diagram, its structure maps $a_{j}$ , the base colimit $(Q, i_{j})$ , and preservation of that colimit by $T$ and $T^{2}$ .

[L1]

Creation of colimits is the cocone dual of creation of limits (Preservation, reflection, and creation of limits and colimits; continuous and cocontinuous functors), with colimits as in Limits and colimits as terminal cones and initial cocones, with existence and uniqueness in their universal properties.

Proof

technique · direct

1.1givenL1

Since $T$ preserves the colimit, the maps $i_{j} \circ a_{j} : T U^{T} D j \to Q$ induce a unique $a : T Q \to Q$ satisfying $a \circ T (i_{j}) = i_{j} \circ a_{j}$ for every $j$ .

2.1step 1.1given

Precomposition with every $i_{j}$ proves $a \circ η_{Q} = 1_{Q}$ . Precomposition with every $T^{2} (i_{j})$ proves $a \circ T (a) = a \circ μ_{Q}$ ; these families are jointly epic because $T$ and $T^{2}$ preserve the colimit, so $(Q, a)$ is an algebra and every $i_{j}$ is an algebra homomorphism.

3.1step 1.1step 2.1∎

Any algebra cocone has a unique underlying mediating arrow $u : Q \to X$ ; precomposition with every $T (i_{j})$ shows $u \circ a = x \circ T (u)$ , so $u$ is automatically an algebra homomorphism. This proves the lifted universal property and the conditional creation claim.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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., Theorem 5.6.5(ii) (standard reference, not scraped)
B. Richter, From Categories to Homotopy Theory, Lemma 6.5.2 (standard reference, not scraped)