Semantics for Algebraic Operations

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 ObCindexed family of operations of the form αx : (Tx) v −→ (Tx)w provides semantics for algebraic operations, which may be used to extend the usual monadic semantics of the computational λ-calculus uniformly. 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. We outline examples and non-examples and we show that our definition also enriches one for call-by-name languages with effects.