Summary. The introduction of linear logic and its associated proof theory has revolutionized many semantical investigations, for example, the search for fullyabstract models of PCF and the analysis of optimal reduction strategies for lambda calculi. In the present paper we show how proof nets, a graph-theoretic syntax for linear logic proofs, can be interpreted as operators in a simple calculus. This calculus was inspired by Feynman diagrams in quantum field theory and is accordingly called the φ-calculus. The ingredients are formal integrals, formal power series, a derivative-like construct and analogues of the Dirac delta function. Many of the manipulations of proof nets can be understood as manipulations of formulas reminiscent of a beginning calculus course. In particular, the “box ” construct behaves like an exponential and the nesting of boxes phenomenon is the analogue of an exponentiated derivative formula. We show that the equations for the multiplicative-exponential fragment of linear logic hold. 1

A category with biproducts is enriched over (commutative) additive monoids. A category with tensor products is enriched over scalar multiplication actions. A symmetric monoidal category with biproducts is enriched over semimodules. We show that these extensions of enrichment (e.g. from hom-sets to homsemimodules) are functorial, and use them to make precise the intuition that “compact objects are finitedimensional” in standard cases. Keywords: Semimodules, enriched categories, biproducts, scalar multiplication, compact objects.

Abstract not found

We show that the category of convenient vector spaces in the sense of Frölicher

A categorical framework for the quantum harmonic oscillator

A categorical framework for the quantum harmonic oscillator