Cartesian Linearly Distributive Categories: Revisited
Rose Kudzman-Blais and Jean-Simon Pacaud Lemay
Applied Categorical Structures 34, 47 (2026)
doi: 10.1007/s10485-026-09880-0 - pdf
arXiv: arXiv:2509.04435 [math.CT]
Linearly distributive categories (LDC) were introduced by Cockett and Seely to provide an alternative categorical semantics for multiplicative linear logic. In contrast to Barr's ∗-autonomous categories, LDCs take multiplicative conjunction and disjunction as primitive notions. Thus, a LDC is a category with two monoidal products that interact via linear distributors. A cartesian linearly distributive category (CLDC) is a LDC whose two monoidal products coincide with categorical products and coproducts. Initially, it was believed that CLDCs and distributive categories would coincide, but this was later found not to be the case. 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. This paper revisits CLDCs, demonstrating strong structural properties they all satisfy and investigating two key classes of examples: posetal distributive categories and semi-additive categories. Additionally, we re-examine a previously assumed class of CLDCs, the Kleisli categories of exception monads of distributive categories, and show that they are not, in fact, CLDCs.
Constructing linear bicategories
Richard Blute, Rose Kudzman-Blais and Susan Niefield
Mathematical Structures in Computer Science, 36 (2026), e20
doi: 10.1017/S0960129526100553 - pdf
arXiv: arXiv:2209.05693 [math.CT]
Linearly distributive categories were introduced to model the tensor/par fragment of linear logic, without resorting to the use of negation. Linear bicategories are the bicategorical version of linearly distributive categories. Essentially, a linear bicategory has two forms of composition, each determining the structure of a bicategory, and the two compositions are related by a linear distribution. After the initial paper on the subject, there was little further work as there seemed to be a lack of examples. The main goal of this paper is to demonstrate that there are in fact a great many examples, which are obtained by considering quantales and quantaloids and by extending familiar constructions from the (ordinary) bicategorical setting. It is standard in the field of monoidal topology that the category of quantale-valued relations is a bicategory. Here, we begin by showing that a quantale is Girard if and only if the corresponding bicategory is a Girard quantaloid, which is an example of a linear bicategory. The tropical and arctic semiring structures fit together into a Girard quantale, so this construction is likely to have multiple applications. More generally, we define LD-quantales, which are suplattices with two quantale structures related by a linear distribution, and their bicategorical analog, linear quantaloids. We show that Q-Rel is a linear quantaloid if and only if Q is a LD-quantale. We then consider several standard constructions from enriched bicategory theory and show that these lift to the linear quantaloid setting and produce new examples of linear bicategories. In particular, we consider linear Q-categories, matrices in Q, and linear monads in Q, where Q is a linear quantaloid. We develop non-locally posetal examples as well: Quant, the bicategory of quantales, modules, and module homomorphisms; and Qtld, the bicategory of quantaloids, modules, and module homomorphisms. These turn out to be cyclic ∗-autonomous bicategories, which are in essence a closed version of linear bicategories.