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Algebraic Operations and Generic Effects

Abstract

Given a complete and cocomplete symmetric monoidal closed category V and a symmetric monoidal V-category C with cotensors and a strong V-monad T on C, we investigate axioms under which an Ob C-indexed family of operations of the form α x :(Tx)v→(Tx)w provides semantics for algebraic operations on the computational λ-calculus. We recall a definition for which we have elsewhere given adequacy results, and we show that an enrichment of it is equivalent to a range of other possible natural definitions of algebraic operation. In particular, we define the notion of generic effect and show that to give a generic effect is equivalent to giving an algebraic operation. We further show how the usual monadic semantics of the computational λ-calculus extends uniformly to incorporate generic effects. We outline examples and non-examples and we show that our definition also enriches one for call-by-name languages with effects.

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Plotkin, G., Power, J. Algebraic Operations and Generic Effects. Applied Categorical Structures 11, 69–94 (2003). https://doi.org/10.1023/A:1023064908962

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  • algebraic operation
  • computational effect
  • Lawvere theory
  • monad