What is a monoid?

Abstract

In many situations one encounters an entity that resembles 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 entity is, in fact, 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 these encompass many examples, there remain cases which do not fit into these frameworks, 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 terminal bi-skew multicategory.