Species, Profunctors and Taylor Expansion Weighted by SMCC: A Unified Framework for Modelling Nondeterministic, Probabilistic and Quantum Programs

Motivated by a tight connection between Joyal's combinatorial species and quantitative models of linear logic, this paper introduces weighted generalised species (or weighted profunctors), where weights are morphisms of a given symmetric monoidal closed category (SMCC). For each SMCC W, we show that the category of W-weighted profunctors is a Lafont category, a categorical model of linear logic with exponential. As a model of programming languages, the construction of this paper gives a unified framework that induces adequate models of nondeterministic, probabilistic, algebraic and quantum programming languages by an appropriate choice of the weight SMCC.