Effectful Semantics in Bicategories: Strong, Commutative, and Concurrent Pseudomonads
ACM/IEEE Symposium on Logic in Computer Science(2023)
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.
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