Related papers: Object generators, categories, and everyday set th…
Category theory provides a powerful tool to organize mathematics. A sample of this descriptive power is given by the categorical analysis of the practice of "classes as shorthands" in ZF set theory. In this case category theory provides a…
The foundations of mathematics have long been considered settled by the Zermelo-Fraenkel-Choice axioms. But set theory abounds in models with different truths and even classical questions such as the measurability of projective sets can…
It is useful to have a criterion for when the predictions of an operational theory should be considered classically explainable. Here we take the criterion to be that the theory admits of a generalized-noncontextual ontological model.…
Category theory is the language of homological algebra, allowing us to state broadly applicable theorems and results without needing to specify the details for every instance of analogous objects. However, authors often stray from the realm…
Two successive generalizations of the usual tensor products are given. One can be constructed for arbitrary binary operations, and not only for semigroups, groups or vector spaces. The second one, still more general, is constructed for…
The construction of first-order logic and set theory gives rise to apparent circularities of mutual dependence, making it unclear which can act as a self-contained starting point in the foundation of mathematics. In this paper, we carry out…
Generalized contextuality is a hallmark of nonclassical theories like quantum mechanics. Yet, three fundamental computational problems concerning its decidability and complexity remain open. First, determining the complexity of deciding if…
This is a short introductory course to Set Theory, based on axioms of von Neumann--Bernays--G\"odel (briefly NBG). The text can be used as a base for a lecture course in Foundations of Mathematics, and contains a reasonable minimum which a…
Consider a subfield of the field of rational functions in several indeterminates. We present an algorithm that, given a set of generators of such a subfield, finds a simple generating set. We provide an implementation of the algorithm and…
We study and relate categories of modules, comodules and contramodules over a representation of a small category taking values in (co)algebras, in a manner similar to modules over a ringed space. As a result, we obtain a categorical…
The presence of contextuality in quantum theory was first highlighted by Bell, Kochen and Specker, who discovered that for quantum systems of three or more dimensions, measurements cannot be viewed as revealing pre-existing properties of…
Answering a question of Junker and Ziegler, we construct a countable first order structure which is not omega-categorical, but does not have any proper non-trivial reducts, in either of two senses (model-theoretic, and group-theoretic). We…
Providing a human-understandable explanation of classifiers' decisions has become imperative to generate trust in their use for day-to-day tasks. Although many works have addressed this problem by generating visual explanation maps, they…
This paper argues that mathematical objects are constructions and that constructions introduce a flexibility in the ways that mathematical objects are represented (as sets of binary sequences for example) and presented (in a particular…
In this paper, we formally address universal object detection, which aims to detect every scene and predict every category. The dependence on human annotations, the limited visual information, and the novel categories in the open world…
Classification theory of elementary classes deals with first order (elementary) classes of structures (i.e. fixing a set T of first order sentences, we investigate the class of models of T with the elementary submodel notion). It tries to…
Although foundation models (FMs) claim to be powerful, their generalization ability significantly decreases when faced with distribution shifts, weak supervision, or malicious attacks in the open world. On the other hand, most domain…
We outline a definition of accessible and presentable objects in a 2-category $\mathcal K$ endowed with a "KZ context", that is to say a pair of lax-idempotent monads interacting in a prescribed way; this perspective suggests a unified…
Generic sentences express generalisations about the world without explicit quantification. Although generics are central to everyday communication, building a precise semantic framework has proven difficult, in part because speakers use…
Recent research on Generalized Zero-Shot Learning (GZSL) has focused primarily on generation-based methods. However, current literature has overlooked the fundamental principles of these methods and has made limited progress in a complex…