Linton's theorem, in category theory, concerns the correspondence between monads and Lawvere theories, the two principal category theoretic formulations of universal algebra. The correspondence was essentially introduced in Linton's own 1969 work, though later authors, notably Eduardo Dubuc, gave the first fully explicit statements and proofs of parts of the result, extending it to the more general setting of enriched categories.
Facts
StatementThe category of finitary monads on Set is equivalent to the category of Lawvere theories and, subject to a generalization in the definition of Lawvere theory, every monad arises thus, uniquely up to coherent isomorphism. 1 Classification
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Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Sources
1. Linton's theorem (equational theory), Wikipedia
Section on the theorems relating monads and Lawvere theories due to Linton
The category of finitary monads on Set is equivalent to the category of Lawvere theories and, subject to a generalization in the definition of Lawvere theory, every monad arises thus, uniquely up to coherent isomorphism.
Opening paragraph, before the theorem statements
Although these results are not explicitly stated, they were essentially introduced by Linton (1969).
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