🤖 AI Summary
Formalizing the “center” of strong monoidal functors—particularly strong monads—in symmetric monoidal categories, i.e., the substructure commuting with all other computational effects, and systematically studying its categorical properties and applications in effectful computation.
Method: We introduce the first explicit definitions of the center of a strong monad and its central submonad, proving their equivalence to the Power–Robinson premonoidal center; we establish three equivalent characterizations of center existence and develop an equational λ-calculus for central submonads, accompanied by a sound and complete categorical semantics.
Contribution: We demonstrate the ubiquity of centers in canonical categories (e.g., Set, ωCpo); prove reliability, completeness, and an internal-language theorem for the calculus; and provide the first unified framework for commutative computational effects that simultaneously ensures mathematical rigor and expressive program semantics.
📝 Abstract
Monads in category theory are algebraic structures that can be used to model computational effects in programming languages. We show how the notion of "centre", and more generally "centrality", i.e., the property for an effect to commute with all other effects, may be formulated for strong monads acting on symmetric monoidal categories. We identify three equivalent conditions which characterise the existence of the centre of a strong monad (some of which relate it to the premonoidal centre of Power and Robinson) and we show that every strong monad on many well-known naturally occurring categories does admit a centre, thereby showing that this new notion is ubiquitous. More generally, we study central submonads, which are necessarily commutative, just like the centre of a strong monad. We provide a computational interpretation by formulating equational theories of lambda calculi equipped with central submonads, we describe categorical models for these theories and prove soundness, completeness and internal language results for our semantics.
