Laplace Distributors and Laplace Transformations for Differential Categories
Résumé
In a differential category and in Differential Linear Logic, the exponential conjunction ! admits structural maps, characterizing quantitative
operations and symmetric co-structural maps, characterizing differentiation. In this paper, we introduce the notion of a Laplace distributor,
which is an extra structural map which distributes the linear negation operation ( )^∗ over ! and transforms the co-structural rules into the structural rules. Laplace distributors are directly inspired by the well-known Laplace transform, which is all-important in numerical analysis. In the
star-autonomous setting, a Laplace distributor induces a natural transformation from ! to the exponential disjunction ?, which we then call a
Laplace transformation. According to its semantics, we show that Laplace distributors correspond precisely to the notion of a generalized exponential function ex on the monoidal unit. We also show that many well-known and important examples have a Laplace distributor/transformation, including (weighted) relations, finiteness spaces, K¨othe spaces, and convenient vector spaces.
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