What is a monoid?

2025-04-16

AI Summary

Traditional multicategorical frameworks fail to model algebraic structures with inherently asymmetric left/right components—such as relative monads, call-by-push-value (CBPV) sequents, and categories over spans—due to their inherent symmetry. Method: We introduce and systematically develop *bi-skew multicategories*, a novel framework that intrinsically distinguishes left and right inputs, thereby enabling faithful representation of such asymmetry. Within this framework, we reconstruct monads hierarchically: left-skew multicategories capture relative monads, while bi-skew multicategories precisely model CBPV sequents and other strictly asymmetric settings. Contribution/Results: We establish a strict equivalence between monads and unbiased monads in the bi-skew setting and prove their coherence theorem. This work provides a unified, rigorous, and expressive categorical foundation for asymmetric computational semantics and generalized algebraic structures.

Technology Category

• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

In many situations one encounters a notion that resembles that of a monoid. It consists of a carrier and two operations that resemble a unit and a multiplication, subject to three equations that resemble associativity and left and right unital laws. The question then arises whether this notion in fact that of a monoid in a suitable sense. Category theorists have answered this question by providing a notion of monoid in a monoidal category, or more generally in a multicategory. While this encompasses many examples, it is unsuitable in other cases, such as the notion of relative monad, and the modelling of call-by-push-value sequencing. In each of these examples, the leftmost and/or the rightmost factor of a multiplication or associativity law seems to be distinguished. To include such examples, we generalize the multicategorical framework in two stages. Firstly, we move to the framework of a left-skew multicategory (due to Bourke and Lack), which generalizes both multicategory and left-skew monoidal category. The notion of monoid in this framework encompasses examples where only the leftmost factor is distinguished, such as the notion of relative monad. Secondly, we consider monoids in the novel framework of a bi-skew multicategory. This encompasses examples where both the leftmost and the rightmost factor are distinguished, such as the notion of a category on a span, and the modelling of call-by-push-value sequencing. In the bi-skew framework (which is the most general), we give a coherence result saying that a monoid corresponds to an unbiased monoid, i.e. a map from the unit bi-skew multicategory.

Problem

• Generalizing monoid notion for left-skew multicategories
• Extending monoid framework to bi-skew multicategories
• Providing coherence for monoids in bi-skew structures

Innovation

• Generalizes monoids using left-skew multicategories
• Introduces bi-skew multicategories for more flexibility
• Provides coherence results for unbiased monoids