Related papers: Ultraproducts in abstract categorical logic

We study the expressive power of fragments of inclusion and independence logic defined either by restricting the number of universal quantifiers or the arity of inclusion and independence atoms in formulas. Assuming the so-called lax…

We extend classical work by Janusz Czelakowski on the closure properties of the class of matrix models of entailment relations - nowadays more commonly called multiple-conclusion logics - to the setting of non-deterministic matrices…

We show that numerous distinctive concepts of constructive mathematics arise automatically from an "antithesis" translation of affine logic into intuitionistic logic via a Chu/Dialectica construction. This includes apartness relations,…

This article is an introduction to the basic generalized category theory used in recent work on an extension of the theory of categories and categorical logic, including parts of topos theory. We discuss functors, equivalences, natural…

Deep learning, despite its remarkable achievements, is still a young field. Like the early stages of many scientific disciplines, it is marked by the discovery of new phenomena, ad-hoc design decisions, and the lack of a uniform and…

Effectus theory is a new branch of categorical logic that aims to capture the essentials of quantum logic, with probabilistic and Boolean logic as special cases. Predicates in effectus theory are not subobjects having a Heyting algebra…

Abstracting from a low level to a more explanatory high level of description, and ideally while preserving causal structure, is fundamental to scientific practice, to causal inference problems, and to robust, efficient and interpretable AI.…

We examine the use of classes to formulate several categorical notions. This leads to two proposals: an explicit structure for working with subobjects, and a hierarchy of $k$-classes. We apply the latter to both ordinary and higher…

This paper explores the interplay between category theory, topology, and the algebraic theory of finite groups. Our analysis unfolds in three stages. First, we establish the foundational universe of our objects: the complete and cocomplete…

We define the concept of a regular object with respect to another object in an arbitrary category. We present basic properties of regular objects and we study this concept in the special cases of abelian categories and locally finitely…

We bring an abstract model theory perspective to interpolation. We ask, what is the role of interpolation in the study of extensions of first order logic, such as infinitary logics, generalized quantifiers and higher order logics? The…

The analysis and control of stochastic dynamical systems rely on probabilistic models such as (continuous-space) Markov decision processes, but large or continuous state spaces make exact analysis intractable and call for principled…

Although the categorical arithmetic is not effectively axiomatizable, the belief that the incompleteness Theorems can be apply to it is fairly common. Furthermore, the so-called "essential" (or "inherent") semantic incompleteness of the…

For a locally compact group $G$, we show that it is possible to present the class of continuous unitary representations of $G$ as an elementary class of metric structures, in the sense of continuous logic. More precisely, we show how…

The study of complex systems through the lens of category theory consistently proves to be a powerful approach. We propose that cognition deserves the same category-theoretic treatment. We show that by considering a highly-compact cognitive…

We describe a notion of categorical model for unitless fragments of (multiplicative) linear logic. The basic definition uses promonoidal categories, and we also give an equivalent elementary axiomatisation.

This paper presents an abstract, mathematical formulation of classical propositional logic. It proceeds layer by layer: (1) abstract, syntax-free propositions; (2) abstract, syntax-free contraction-weakening proofs; (3) distribution; (4)…

Building over some ideas of Ren\'e Guitart, we provide a categorical framework towards some deviation notions in abstract logic.

We introduce prime products as a generalization of ultraproducts for positive logic. Prime products are shown to satisfy a version of {\L}o\'s's Theorem restricted to positive formulas, as well as the following variant of Keisler…

We prove a single category-theoretic result encapsulating the notions of ultrafilters, ultrapower, ultraproduct, tensor product of ultrafilters, the Rudin--Kiesler partial ordering on ultrafilters, and Blass's category of ultrafilters UF.…