Effectful Semantics in Bicategories: Strong, Commutative, and Concurrent Pseudomonads

TL;DR

This paper extends the theory of monads to bicategories, introducing concurrent pseudomonads to better understand effects and semantics in concurrent functional programming and models based on bicategories.

Contribution

It develops strong and commutative monads in bicategories, introduces concurrent pseudomonads, and applies these concepts to semantics of concurrent programs and game models.

Findings

Concurrent pseudomonads capture weak interchange laws.

The 2-dimensional setting offers new semantic insights.

Examples include the continuation pseudomonad in game semantics.

Abstract

We develop the theory of strong and commutative monads in the 2-dimensional setting of bicategories. This provides a framework for the analysis of effects in many recent models which form bicategories and not categories, such as those based on profunctors, spans, or strategies over games. We then show how the 2-dimensional setting provides new insights into the semantics of concurrent functional programs. We introduce concurrent pseudomonads, which capture the fundamental weak interchange law connecting parallel composition and sequential composition. This notion brings to light an intermediate level, strictly between strength and commutativity, which is invisible in traditional categorical models. We illustrate the concept with the continuation pseudomonad in concurrent game semantics. In developing this theory, we take care to understand the coherence laws governing the structural 2-cells. We give many examples and prove a number of practical and foundational results.