Related papers: Categorical Realizability for Non-symmetric Closed…
A type theory is presented that combines (intuitionistic) linear types with type dependency, thus properly generalising both intuitionistic dependent type theory and full linear logic. A syntax and complete categorical semantics are…
Applied category theory often studies symmetric monoidal categories (SMCs) whose morphisms represent open systems. These structures naturally accommodate complex wiring patterns, leveraging (co)monoidal structures for splitting and merging…
Applied category theory often studies symmetric monoidal categories (SMCs) whose morphisms represent open systems. These structures naturally accommodate complex wiring patterns, leveraging (co)monoidal structures for splitting and merging…
We give a general notion of combinatory completeness with respect to a faithful cartesian club and use it systematically to obtain characterisations of a number of different kinds of applicative system. Each faithful cartesian club…
Implicative algebras have been recently introduced by Miquel in order to provide a unifying notion of model, encompassing the most relevant and used ones, such as realizability (both classical and intuitionistic), and forcing. In this work,…
This paper introduces categories of assemblies which are closely connected to realizability interpretations and which are based on an important subcategory of the effective topos. There is a list of properties which characterize these…
It is common to encounter symmetric monoidal categories $\mathcal{C}$ for which every object is equipped with an algebraic structure, in a way that is compatible with the monoidal product and unit in $\mathcal{C}$. We define this formally…
We combine two recent ideas: cartesian differential categories, and restriction categories. The result is a new structure which axiomatizes the category of smooth maps defined on open subsets of $\R^n$ in a way that is completely algebraic.…
A symmetric monoidal category naturally arises as the mathematical structure that organizes physical systems, processes, and composition thereof, both sequentially and in parallel. This structure admits a purely graphical calculus. This…
This paper presents preliminary work on a general system for integrating dependent types into substructural type systems such as linear logic and linear type theory. Prior work on this front has generally managed to deliver type systems…
We examine various categorical structures that can and cannot be constructed. We show that total computable functions can be mimicked by constructible functors. More generally, whatever can be done by a Turing machine can be constructed by…
Building on work of Derksen-Fei and Plamondon, we formulate a conjectural correspondence between additive and monoidal categorifications of cluster algebras, which reveals a new connection between the additive reachability conjecture and…
We give an exposition of the semantics of the simply-typed lambda-calculus, and its linear and ordered variants, using multi-ary structures. We define universal properties for multicategories, and use these to derive familiar rules for…
We show that a compact rigid balanced braided monoidal category with enough compact projective objects gives rise to a system of mapping class group representations compatible with the gluing along marked intervals. A motivation to consider…
In this paper, we apply the machinery developed in arXiv:2401.06641(2) to study the behavior of computable categoricity relativized to non-c.e. degrees. In particular, we show that we can build a computable structure which is not computably…
In this document, we collect a list of categorical structures on the category $\mathbf{Poly}$ of polynomial functors. There is no implied claim that this list is in any way complete. It includes: infinitely many monoidal structures, all but…
We establish a formal correspondence between resource calculi an appropriate linear multicategories. We consider the cases of (symmetric) representable, symmetric closed and autonomous multicategories. For all these structures, we prove…
In this paper we present cartesian structure for symmetric Gray-monoidal double categories. To do this we first introduce locally cubical Gray categories, which are three-dimensional categorical structures analogous to classical, locally…
Following the analogy between algebras (monoids) and monoidal categories the construction of nucleus for non-associative algebras is simulated on the categorical level. Nuclei of categories of modules are considered as an example.
We instal homological algebra, including derived functors, on certain non-additive categories like categories of pointed CW-complexes, modules of monoids or sheaves thereof. We apply this theory to Monoid schemes and sheaves on them,…