We investigate the phenomenon that "every monad is a linear state monad". We do this by studying a fully-complete state-passing translation from an impure call-by-value language to a new linear type theory: the enriched call-by-value calculus. The results are not specific to store, but can be applied to any computational effect expressible using algebraic operations, even to effects that are not usually thought of as stateful. There is a bijective correspondence between generic effects in the source language and state access operations in the enriched call-by-value calculus. From the perspective of categorical models, the enriched call-by-value calculus suggests a refinement of the traditional Kleisli models of effectful call-by-value languages. The new models can be understood as enriched adjunctions.

• 03F52 - Proof-theoretic aspects of linear logic and other substructural logics
• 18C15 - Monads (= standard construction, triple or triad), algebras for monads, homology and derived functors for monads
• 18C50 - Categorical semantics of formal languages
• 68N18 - Functional programming and lambda calculus