Alphabeta MathHow 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.
Under dependent choice, a finitary monad on a complete cocomplete locally small category has complete and cocomplete algebras
Statement
Assume dependent choice. If is a finitary monad on a complete, cocomplete, locally small category , then its Eilenberg–Moore category is complete and cocomplete.
The completeness conclusion itself uses no choice; dependent choice enters only in the construction of coequalizers used for cocompleteness.
Facts & Assumptions
Given: Dependent choice and a finitary monad on a complete cocomplete locally small category .
The Eilenberg–Moore forgetful functor strictly creates every limit that exists in the base (The Eilenberg–Moore forgetful functor strictly creates every limit that exists in the base).
Under dependent choice, algebras for such a finitary monad have coequalizers (Under dependent choice, algebras for a finitary monad on a complete cocomplete locally small category have coequalizers).
Over a cocomplete base, a monadic category is cocomplete exactly when it has coequalizers (Over a cocomplete base, a monadic category is cocomplete exactly when it has coequalizers).
The Eilenberg–Moore adjunction induces the given monad on the nose (The free–forgetful Eilenberg–Moore adjunction induces the given monad).
Proof
Since has every small limit, [L1] creates every such limit in , including the empty limit. Thus is complete without using dependent choice.
Under the stated dependent-choice hypothesis, [L2] gives every coequalizer in , including equal parallel maps and the zero-stage case of its construction.
By [L4], the Eilenberg–Moore adjunction induces on the nose, so its forgetful functor is monadic. Applying [L3] to the cocomplete base and step 1.2 makes cocomplete, including the empty colimit.
Combining step 1.1 with step 2.1 proves that is complete and cocomplete.
Depends on
- Finitary functors and finitary monads
- Finite, small, and large limits and colimits; complete and cocomplete categories
- The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain
- The Eilenberg–Moore forgetful functor strictly creates every limit that exists in the base
- Under dependent choice, algebras for a finitary monad on a complete cocomplete locally small category have coequalizers
- Over a cocomplete base, a monadic category is cocomplete exactly when it has coequalizers
- The free–forgetful Eilenberg–Moore adjunction induces the given monad
Used by
Dependency tree · two levels
30 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 5.6.12 (standard reference, not scraped)