Cartesian effect categories are Freyd-categories

TL;DR

This paper introduces Cartesian effect categories, a new categorical framework that models evaluation order and side-effects in programming languages, and shows their equivalence to Freyd-categories with a specific product structure.

Contribution

It defines Cartesian effect categories and the sequential product, establishing their equivalence to Freyd-categories with a premonoidal structure, and compares them with existing models like monads and Arrows.

Findings

Cartesian effect categories are equivalent to Freyd-categories with a sequential product.

The sequential product has a universal property facilitating constructions and proofs.

Both effect categories and sequential products are newly introduced concepts.

Abstract

Most often, in a categorical semantics for a programming language, the substitution of terms is expressed by composition and finite products. However this does not deal with the order of evaluation of arguments, which may have major consequences when there are side-effects. In this paper Cartesian effect categories are introduced for solving this issue, and they are compared with strong monads, Freyd-categories and Haskell's Arrows. It is proved that a Cartesian effect category is a Freyd-category where the premonoidal structure is provided by a kind of binary product, called the sequential product. The universal property of the sequential product provides Cartesian effect categories with a powerful tool for constructions and proofs. To our knowledge, both effect categories and sequential products are new notions.