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.

Differentiation in logic has several sources of inspiration. The most recent is differentiable programming, models of which demand functoriality and good typing properties. More historical is reverse denotational semantics, taking inspiration from models of Linear Logic to differentiate proofs and λ-terms. In this paper, we take advantage of the rich structure of categorical models of Linear Logic to give a new functorial presentation of differentiation. We define differentiation as a functor from a coslice of the category of smooth maps to the category of linear maps. Extending linear-non-linear adjunction models of Linear Logic, this produces models of Differential Linear Logic. We use these functorial presentations to shed new light on integration in differential categories.