Related papers: Algebraic models of dependent type theory
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…
The purpose of this text is to prove all technical aspects of our model for dependent type theory with parametric quantifiers [Nuyts, Vezzosi and Devriese, 2017]. It is well-known that any presheaf category constitutes a model of dependent…
Coloured probability tree models are statistical models coding conditional independence between events depicted in a tree graph. They are more general than the very important class of context-specific Bayesian networks. In this paper, we…
The main objective of this paper is to show that the notion of type which was developed within the frames of logic and model theory has deep ties with geometric properties of algebras. These ties go back and forth from universal algebraic…
The most commonly known triangulated categories arise from chain complexes in an abelian category by passing to chain homotopy classes or inverting quasi-isomorphisms. Such examples are called `algebraic' because they originate from abelian…
Conditional independence has been widely used in AI, causal inference, machine learning, and statistics. We introduce categoroids, an algebraic structure for characterizing universal properties of conditional independence. Categoroids are…
Denotational models of type theory, such as set-theoretic, domain-theoretic, or category-theoretic models use (actual) infinite sets of objects in one way or another. The potential infinite, seen as an extensible finite, requires a dynamic…
Dynamical systems---by which we mean machines that take time-varying input, change their state, and produce output---can be wired together to form more complex systems. Previous work has shown how to allow collections of machines to…
Group-based models arise in algebraic statistics while studying evolution processes. They are represented by embedded toric algebraic varieties. Both from the theoretical and applied point of view one is interested in determining the ideals…
We generalize Franz' independence in tensor categories with inclusions from two morphisms (which represent generalized random variables) to arbitrary ordered families of morphisms. We will see that this only works consistently if the unit…
We consider Hilbert series associated to modules over various categories of trees. Using the technology of Sam and Snowden, we show that these Hilbert series must be algebraic. We then apply these technical theorems to prove facts about…
The attempt is to give a formal concpet of system, and with this provide a definition of category, that will also satisfy the definition of a system. An axiomatic base is given, for constructing the group of integers. In the process, we…
In this paper we explain certain systematic differences between algebraic and topological triangulated categories. A triangulated category is algebraic if it admits a differential graded model, and topological if it admits a model in the…
Usual math sets have special types: countable, compact, open, occasionally Borel, rarely projective, etc. Each such set is described by a single Set Theory formula with parameters unrelated to other formulas. Exotic expressions involving…
Universal algebra uniformly captures various algebraic structures, by expressing them as equational theories or abstract clones. The ubiquity of algebraic structures in mathematics and related fields has given rise to several variants of…
We study links between first-order formulas and arbitrary properties for families of theories, classes of structures and their isomorphism types. Possibilities for ranks and degrees for formulas and theories with respect to given properties…
It is becoming increasingly important to explain complex, black-box machine learning models. Although there is an expanding literature on this topic, Shapley values stand out as a sound method to explain predictions from any type of machine…
This is the first of a pair of papers where we construct and investigate a closed monoidal structure on the category of generalized algebraic theories (in the sense of Cartmell). In the present text, as a starting point, we define the…
A notion of morphism that is suitable for the sheaf-theoretic approach to contextuality is developed, resulting in a resource theory for contextuality. The key features involve using an underlying relation rather than a function between…
We introduce a class of theories called metastable, including the theory of algebraically closed valued fields (ACVF) as a motivating example. The key local notion is that of definable types dominated by their stable part. A theory is…