Cartesian Linearly Distributive Categories

Linearly distributive categories (LDC) were introduced by Cockett and Seely to provide alternative categorical semantics for multiplicative linear logic [2], generalizing $\ast$-autonomous categories, by taking multiplicative conjunction and disjunction as primitive notions. So briefly, a LDC is a category with two monoidal products, tensor $\otimes$ and par $\oplus$, whose interaction is mediated by linear distributivities. A cartesian linearly distributive category (CLDC) is a LDC whose tensor is the categorical product $\otimes =\times$ and par is the coproduct $\oplus =+$.

Two monoidal products is not unusual, perhaps the most well-know definition in this branch being the distributive category [1]. It was initially believed that LDCs were a weakening of distributive categories, more precisely that the notion of a CLDC and of a distributive category would coincide. This was later found out not to be the case [2]. Consequently, the study on CLDCs was not pursued further at the time.

With recent developments for and applications of LDCs, there has been renewed interest in CLDCs. In particular, the development of a linearly distributive Fox theorem [3] (presented at CT2025) lead to further investigation into the internal workings of CLDCs and a search for new examples.

In this talk, we will discuss various important structural properties of CLDCs which we’ve uncovered [4]. While the definition of a CLDC seems straightforward, it has become clear that linear distributivity between cartesian and cocartesian structures is distinctive, resulting in rather nuanced behavior within CLDCs. One such key observation is that in a CLDC, its terminal object is always preinitial and, dually, its initial object is always subterminal.

Moreover, we will discuss two key classes of examples: bounded distributive lattices and semi-additive categories. A CLDC must often fall into one of these two categories via collapse theorems. For example, a CLDC must be semi-additive if it either has invertible linear distributivities or if it is isomix. Additionally, by applying these collapse theorems, we revisit a previously assumed class of CLDCs, the Kleisli categories of exception monads of distributive categories, and show that they are not, in fact, CLDCs. That said, while these collapse theorem may seem to constrain the landscape of possible CLDCs, we will provide a Grothendieck construction to generate new examples of CLDCs, producing some which are neither bounded distributive lattices nor semi-additive categories.